## Playing Recursion V2 >> demo examples for one liners programs

### Preface

gold 2020-10-08: Here is some source code to supplement the TCL Wiki page Playing with Recursion by RS. This supplemental code is intended for study of McCarthy theorems on computer arithmetic into one-line programs.

### Introduction

Playing with Recursion translates the McCarthy theorems into one-line Tcl scripts. To further qualify, some of the one-line programs require additional helper functions. The "easy eye" console was used as a testbed for one-line programs in Tcl on Windows 10. Using the easy eye testbed, the code in Playing with Recursion was extended with a few more functions. The "Easy Eye" console testbed has large black type on green background for bad eyes. The "easy eye" console testbed includes an extra Self_help button under the normal Help button of console display.

### Discussion

Professor McCarthy advocates "conditional expressions", where a conditional expression is a sequence alternating condition and result clauses. Each condition may represent one or more domains on the number line. In Tcl, the condition is a domain defined by if or the ? in a?b:c operator of expr. In Playing with Recursion, the conditional was mostly ?. The result, or estimated variable function f(v), is sort of hovering and ambiguous cloud of values over the condition domain. For the triangle function, the condition domains are:

• \$variable < 0
• \$variable = 0
• 0 < \$variable < 1
• \$variable > 1

However, since the Playing with Recursion is concerned primarily with positive integer arithmetic, the number of condition domains may be abbreviated, merged, or constrained to the positive domain only. As another example for the rectangle step function, the condition domains might be:

• \$variable < -1
• -1 < \$variable < 0
• \$variable = 0
• 0 < \$variable < 1
• \$variable > 1

Yet another example is the triangle wave function centered at the origin. After the algorithm using "conditional expressions" was developed and "rigorously proved" though known functions, parts of the recursion algorithm could possibly be sped up with equivalent methods of iteration, mapping tables, or executing compiled rather than interpreted code.

This page on developing McCarthy based algorithms is not a replacement for the current Tcl core and Tcllib, which is much improved since Tcl version 4, and other <faster> language constructs. math ops, Tcllib routines, and other compiled routines can reduce the cost of big-data tasks by about 1/3. The time savings are not always obvious on small quantities of data. Performance of one-line programs may suffer degradation due to lengthy recursion calls, and may be limited by constraints on recursion. Dependence on math operator notation, helper procedures, math check examples, degradation due to lengthy recursion calls, and special library functions should be noted in the comment lines.

Some functions were added in the Easy Eye testbed. Factorial recursion quickly exceeds domain. fac is limited by recursion constraints, and excessive time delay beyond factorial 4. Functions min and max are basic integer functions, but may need some error checks for entry zeros and equality, \$m == \$n. Logic tested for <= as puts {< 5 0} returns 0, puts {<= 0 5} returns 1, and puts {<= 10 10} returns 1. Logic tested for func < as puts {< 5 0} returns 0 and puts {< 0 5} returns 1. Swap brackets for {}. zero_killer replaces 0 with 0.0000001 to avoid division by zero. It proved useful for observed data loaded from Tcl programs into spreadsheet calculation charts. An RS proc with the x?y:z operator seemed a good model in regular Tcl as proc sgn5 x {set temp expr {\$x>0? 1 : \$x<0? -1 : 0}}. In the Playing_Recursion code, the proc sgn5 seemed to work well as func sgn5 n { expr {\$n>0? 1 : \$n<0? -1 : 0}}. But when > \$n 0? and < \$n 0? was substituted, could not get the func sgn5 to work. The most understandable RS derived code was proc sgn7 nn { if {\$nn>0} { return 1 } elseif {\$nn<0} { return -1 } else { return 0 }}. The timimg was time { sgn7 7 } 5000 , which returned 0.4554 microseconds per iteration. I have to say that McCarthy and RS have a different take on setting up the algorithms as <conditional regions> of variable space, which seems to apply well to regular Tcl coding.

### Naughty Nots in Knots

In the original set of Playing with Recursion, there was the positive equal, proc eq {m n} {string equal \$m \$n} ;# RS. What was intriguing is that the proc eq works on strings as well as integers. This proc eq is very fast at 0.65 microseconds delay and does not rely on expr or math. It seemed desirable to have an equally fast not equal or ne. Most of the algorithm textbooks and McCarthy papers assume 6 comparison operators are available. For example, Fortran 77 has logic operators EQ,NE,LT,LE,GT, and GE as well as 3 relational operators AND,OR, and NOT. Qbasic has Operator Relation Expressions as = Equality, <> not equal , > Greater than, < Less than, >= Greater Than or Equal, and <= Less than. For Qbasic, the logical operators are NOT, AND, XOR, NOR, or EQV. Several variants of ne were loaded and tested in the Playing Recursion testbed for performance and speed.

```;# testing negative logic for =! and ne
;# procs =! and ne work on strings too
proc =! {m n} { if {[string equal \$m \$n]} then {return 0} else {return 1} }
;# Usage liner::=! 5 5 r> 0, liner::=! 5 6 r> 1
;# Usage time {liner::=! 5 5 } 5 returns 8.6 microseconds per iteration
proc ne {m n} { abs [string compare \$m \$n]}
;# Usage liner::ne 5 5 r> 0, liner::ne 5 6 r> 1
time {liner::ne 5 5 } 5 returns 12.6 microseconds per iteration
proc fie {m n} { abs [string compare \$m \$n]}
time {fie b b } 5 ;# returns 1.6 microseconds per iteration
proc fibber {m n} { if {[string equal \$m \$n]} {return 0} {return 1} }
;# fibber works on strings but slow
time {fibber b b } 5  ;# returns 8.0 microseconds per iteration
;# gold standard proc for eq in global space
;# liner::= is same proc  in namespace liner::
proc eq {m n} {string equal \$m \$n}  ;# RS
time {eq 1 1 } 1000  returns 0.608 microseconds per iteration in global space
time { liner::= 1 1} 1000 returns 0.436 microseconds per iteration in namespace ::liner```

### Playing Recursion V2 in a Namespace

One can load the the bulk of Playing Recursion V2 into a namespace called ::liner. The advantage of the namespace ::liner is that Playing Recursion is no longer overwriting the math commands for +,-,*, and others in the global space of the TCL script. In the global side of the testbed, one can write regular TCL procs that either call the global math commands in the TCL core and Tcllib library or else call the func commands in the ::liner namespace. Somewhat like the expr command, the func commands in the ::liner::Playing_Recursion namespace have their own set of rules, logic, and a limited set of commands that may conflict with the different set of commands in the global namespace. So a typical console session in the Playing Recursion testbed might have the entire command set of global commands, the limited ::liner::Playing_Recursion set of namespace commands, and the temporary Playing_Recursion funcs and procs under test. The command to invoke the namespace would be ::liner::convert and the command for the demo would be ::liner::demo. The original func command of RS in installed inside the namespace, so ::liner::func would be used to create a new func inside the namespace ::liner::. The available commands, procs, and funcs in the namespace are listed with info commands ::liner::* While good for studying the McCarthy algorithm math theorems, the Playing_Recursion funcs are recursion limited with lengthy return times compared to the precompiled core in the TCL core. So it is suspected that the testbed session on the console would have to develop a McCarthy based algorithm in ::liner::Playing_Recursion and then transfer the algorithm into the regular TCL commands in the global namespace.

### Conclusions

Professor McCarthy and RS have a different take on setting up the algorithms as <conditional regions> of variable space, which seems to apply well to regular Tcl coding. McCarthy developed his own notation for the mathematical theory of computation in several papers. And later, McCarthy applied much of his mathematical theory of computation to the development of the Lisp computer language.

There are pros and cons to one-line programs. Working with recursion, primes, and timing the procedures will quickly show the warts on the one-line programs. To gain speed and shorter computation times, one will generally have to access the Tcl core distribution and Tcllib. Functions math::numtheory::isprime , math::numtheory::firstNprimes, and math::numtheory::primesLowerThan are available in the Tcllib math library. See Tcllib & Tklib core contents & math::numtheory Category Numerical Analysis.

### Important Note

gold 2020-09-15: Important Note. This page is not a replacement for the current Tcl core and Tcllib with much improvement since Tcl version 4 and other <faster> language constructs. See better routines and current methods for angle reduction, sin, cos, pi, etc in the Tcl core distribution and Tcllib. As of Jul 2018, Tcllib has developed code for trig angles in degrees, trig inverse, hyper functions in degrees, and angle reduction in radians and degrees. This supplemental trig.tcl, trigtest.tcl, and trig.man code is posted on the Tcllib website. This math::trig.tcl seems really exciting work, which will keep Tcl in pace with some of the other brand name languages (math oriented, I mean). Some of the Tcl library code is posted as pending on the Tcllib website, and sometimes not really published in the main Tcl distribution yet, so its worthwhile to investigate and run searches on the pending Tcllib code also.

## Pseudocode Section

```;#  **********Extensions from gold **********
source playing_recursion_V2.tcl
;# Suggest maintain air gaps, dead spaces, brackets
;#--  >= greater than or equal to logic function
func >= {m n} {[= \$n 0]? 1: [= \$m 0]? 0: [>= [pred \$m] [pred \$n]]}
;#--  >  greater than logic function
func > {m n} {[= \$m \$n]? 0: [>= \$m \$n]}
proc flipper n { [expr {\$n * -1 } ] }
;# func abs n { [> \$n 0]? \$n: [< \$n 0]? [ flipper \$n]:0}
set sgn5 [expr {\$x>0? 1 : \$x<0? -1 : 0}]    ;# RS
proc sgn5 x {set temp [expr {\$x>0? 1 : \$x<0? -1 : 0}]} ;# RS
proc sgn x {expr {(\$x>0) - (\$x<0)}}  ;# RS & rmax
proc sgn7 nn { if {\$nn>0} { return  1 } elseif {\$nn<0} { return -1 } else { return  0 }}
time { sgn7 7   } 5000 ;# 0.4554 microseconds per iteration.
;#-- triangular number. Tricky code
;#   and Suggest maintain air gaps, spaces, brackets
func tri  { n} {[< \$n 2]? 1: [/  [* \$n [+ \$n 1] ] 2 ]}
test {tri 3} 6 {tri 2} 3 {tri 5} 15
;#-- factorial recursion, but quickly exceeds domain
;#   fac is recursion limited, & excessive time beyond factorial 7
func fac n {\$n<2? 1: [* \$n  [fac  [pred \$n ]]]}
;#-- min, max, basic integer funcions
;#  but may need some error checks
;#  for entry zeros and equality, \$m = \$n
func min {m n} {\$m<\$n? \$m: \$n}
func max {m n} {\$m>\$n? \$m: \$n}
;# zero_killer function swaps points of zero
;# for 0.0000001 in observed data to avoid
;# computer division by zero
func kzero { n } {[= \$n 0]? 0.0000001: [= \$n 0.]? 0.0000001: \$n }
;#  ******* end extensions ************```

## Console Session from Namespace ::liner

```    ;# under Pseudocode Section
namespace eval ::liner {
namespace export math
}
proc ::liner::convert {  } {  ;# before  Playing Recursion deck
*****  enter >::liner::convert< for invoking namespace **********
*****  enter >::liner::demo< for code test **********
*********  but considerable lengthy recursions ****
(System) 1 % ::liner::convert
(System) 2 % ::liner::demo
example >>  gcd 12 24 -> 12, & expected was 12
example >>  gcd 12 18 -> 6, & expected was 6   ;# ommitted lengthy reply here
info commands ::liner::*```

(System) 2 % ::liner::sgn ::liner::fac ::liner::sum_tri ::liner::pred ::liner::prime ::liner::abs ::liner::gcd ::liner::' ::liner::test ::liner::<= ::liner::test1 ::liner::convert ::liner::flipper ::liner::* ::liner::kzero ::liner::>= ::liner::+ ::liner::min ::liner::| ::liner::rem ::liner::< ::liner::func ::liner::demo ::liner::- ::liner::pred2 ::liner::= ::liner::max ::liner::> ::liner::eq ::liner::succ ::liner::tri ::liner::prime2 ::liner::/

```(System) liner::func square  { n} {[< \$n 2]? 1:  [* \$n  \$n  ] }
;# liner::square has lengthy recursion and too many nested evaluations beyond 30
(System) 4 % liner::square 5 ;# returns 25
(System) 4 % liner::square 30 ;# returns 900
;# liner::cube has lengthy recursion and too many nested evaluations beyond 10
(System) 8 %  liner::func cube  { n} {[< \$n 2]? 1: [* [* \$n  \$n  ]  \$n ]}
(System) 9 % liner::cube 2  ;# returns 8
(System) 10 % liner::cube 4 ;#returns 64
(System) 11 % liner::cube 5 ;# returns 125
;# func rectangle_area is test of redundant logic with 5 elements
liner::func rectangle_area {l w} {[= \$l 0]? 0: [= \$w 0]? 0: [* \$l \$w]}
(System) 13 % liner::rectangle_area 5 10  ;# returns 50
;# constant for golden section, have to exactly name sqrt inside mathfunc::
liner::func gm  { n} {[< \$n 2]? 1:    [expr  (1 + [::tcl::mathfunc::sqrt 5] )*.5]}
liner::test {liner::gm 4 } 1.618033988749895
example >>  liner::gm 4  -> 1.618033988749895, & expected was 1.618033988749895
;# Playing Recursion was written originally for integers.
;# But now passing pointing point calculations thru ::liner::func
:# The global namespace is recognizing the func liner::gm.
set rook  [liner::gm 4] ;# returns 1.618033988749895
:# guess could put in a switch for ::liner::func to pass on floating points?
;# or manually swap function calls
liner::func rectangle_area2 {l w} {[= \$l 0]? 0: [= \$w 0]? 0: [::tcl::mathop::* \$l \$w]}
(System)  set house [ liner::rectangle_area2 5.2 10.1 ] ;# returns 52.52
;# reciprocal func r!  has several exceptions , [= \$mm 0]? avoid division by zero
;# reciprocal r! has exception , [= \$mm 1]? defaults to 1
;# exception tests performed before main calculation of 1/\$mm
;#  maybe case liner::r! -2 -> -1 on negative integers has problem?
liner::func r! {mm} {  [= \$mm 0]? 0:  [= \$mm 1]? 1: [::tcl::mathop::/ 1  \$mm] }
;# Usage liner::r! 1  -> 1 ;# liner::r! 0 -> 0 ;# liner::r! 0.000001 -> 1000000.0
;# liner::r! 0.000001 ->  1000000.0 ;# liner::r! -2. -> -0.5 ```

### if, if, and iffy Table

See section Noise Words of if by AMG. Seemed to call for a table, if one is studying recursion on one liner programs.

table printed inTCL format
elementsshort hand for if long hand for if comment, if any
2if a b ;# if {a} then {b}
3if a b c ;# if {a} then {b} else {c};# 2*n+1 elements odd
4if a b c d ;# if {a} then {b} elseif {c} then {d}
5if a b c d e ;# if {a} then {b} elseif {c} then {d} else {e};# 2*n+1 elements odd
***** alternate expressions for expr ****
8 if a b c ...c1,c2,c3... d e;# if {a} then {b} elseif {c} ...elseif {c1} elseif {c2} elseif {c3} .... then {d} else {e}multiple successive elseif?, but not seen example
3 a?b:c expr equivalent;# if {a} then {b} else {c} ;# 2*n+1 elements odd expr command has an internal if conditional ? in a?b:c
5 a?b:c?d:e expr equivalent ;# 2*n+1 elements odd expr command has an internal if conditional ? in a?b:c
7 a?b:c?d:e?f:g expr equivalent ;# 2*n+1 elements odd expr command has an internal if conditional ? in a?b:c
9 a?b:c?d:e?f:g?h:i expr equivalent ;# 2*n+1 elements odd expr command has an internal if conditional ? in a?b:c
11 a?b:c?d:e?f:g?h:i?j:k expr equivalent ;# 2*n+1 elements odd expr command has an internal if conditional ? in a?b:c

### McCarthy Symbolic Logic Table

Meaning symbols used in the McCarthy books and papers, along with equivalents mentioned on this TCL wiki page.

table printed inTCL format
order of precedence for symbolic logic name conventional definition possible McCarthy, alternate liner::proc on page , or TCL ‘equivalent comment, if any
1 ( ) connectives within parentheses innermost parentheses first
2 negation =!, ne, ! hard to see on computer screen [L1 ] ,! = U+0021
3 ^ conjunction U+2038 U+2227 ‸ CARET [L2 ][L3 ]
3 disjunction U+2228 bottom hat not on some keyboards [L4 ][L5 ]
4 -> implication [L6 ][L7 ]
5 <-> equivalence = U+21D4, U+2261, U+2194
******* **** working list from symbolic logic **** *****
NA. Universal quantification (For all X )
NA. Existential quantification (There exists X )
NA. Uniqueness quantification (There is a unique X )
NA. Non-existence quantification (There is no X ))
NA. Numerical quantification (There are exactly n * X )
NA. Numerical quantification (There are at least n * X )
NA. Numerical quantification (There are at most n * X )

### Table for List of Recursion Procs and Funcs

Includes procs and funcs mentioned on this TCL wiki page [L8 ] & Playing with recursion from RS.

table printed inTCL format
subject name conventional definition possible McCarthy func, alternate liner::proc on page , or TCL ‘equivalent comment, if any
math +,-,*,/ math precedence
math func proc loads func expressions ::liner::func proc func {name argl body} {proc \$name \$argl list expr \$body} ;# RS
math test proc test args ::liner::test multi-case tester: -RS
math + plus ::liner::+ add 2 elements, integer sum
math - minus ::liner::- subtract 2 integers, N1 => N2, difference defined only for ( m >= n)
math * multiply ::liner::* multiply 2 integers
math / divide ::liner::/ divide 2 integers , integer division
math % Integer division remainder ::liner::rem Integer division remainder (% in expr)
math 1 yes logic value returned from some funcs
math 0 no logic value returned from some funcs
math -1 negative logic value returned from some funcs
math = equal to ::liner::eq , :liner::= compare 2 integers, proc eq works for strings too
math > greater than ::liner::> compare 2 integers
math < lesser than ::liner::< compare 2 integers
math => lesser than or equal to ::liner::>= compare 2 integers
math =< lesser than or equal ::liner::<= compare 2 integers
math =! not equal ::liner::=! , ::liner::=! compare 2 integers
math n+1 increment ::liner::succ, ::liner::' similar function to TCL incr command
math n-1 decrement, the non-negative integer before n ::liner::pred similar function to TCL incr command
math 2X double ::liner::int_2x
math 3X triple ::liner::int_3x
math 1/X reciprocate reciprocate ::liner::r!
math n! factorial ::liner::fac
math fib(n) Fibonacci Number n ::liner::fib
math * -1. flips sign ::liner::flipper
math \$X * \$X square ::liner::square
math \$X * \$X * \$X cube ::liner::cube
math Phi golden ratio constant ::liner::gm -> 1.618033988749895
math phi golden ratio conjugate ::liner::golden_ratio_conjugate -> 0.618033988749895
math gcd greatest common denominator ::liner::gcd argument of two elements
math tri triangle number ::liner::tri
math sgn sign returns sign logic, 1 for positive number, 0 for 0, -1 for negative, used in validation logic Modeling COND with expr
math sum_int sum of integers to n ::liner::sum_int {n} returns sum of integers to n
math sum_squares sum of squares ::liner::sum_squares {n}returns sum of squares to n
math sum_cubes sum of cubes ::liner::sum_cubes {n} ;# formula for Sum of Cubes is (n**2) ((n + 1)**2) / 4
math sum_4th_power sum of 4th powers ::liner::sum_4th_power {n} formula sum_4th_power = n* (n+1)* (2*n+1 )* ( 3*n**2 + 3*n -1)/30
math sum_5th_power sum of 5th powers liner::sum_5th_power ;# formula sum_5th_power = n*n* (n+1)* (n+1 )* ( 2*n**2 + 2*n -1)/12
math kzero zero killer ::liner::kzero multiplies n by 0.000001
math max maximum ::liner::max maximum of 2 numbers
math min minimum ::liner::min minimum of 2 numbers
math prime test n for prime ::liner::prime 1 for prime zero for non-prime, defined for integers, similar to TCL wiki isprime Primes
math prime2 helper func to prime ::liner::prime2 integer helper func to prime
math abs absolute value ::liner::abs returns absolute value from one number, TCL expr abs
math pi circle constant liner::pi -> 3.141592653589793 proc pi {} {expr acos(-1)} ;# AMG
math fix circle constant experiment ;# returns pi from atan formula with 2 terms ::liner::fix proc fix {} {expr {4. * (atan(1./fib2 3) + atan(1./fib2 4))}}
math

Also Ref. Mathematical Structures for Computer Science, Judith L. Gersting, Indiana University, 2014

### Figure 1. Playing Recursion Screenshot 1 ### Figure 2. Playing Recursion Screenshot Two ### Figure 3. Playing Recursion Screenshot Three ### Figure 4. Playing Recursion V2 triangle function ### Figure 5. Playing Recursion V2 step function ### Figure 6. Playing Recursion V2 triangle wave function

An example is the triangle wave function centered at the origin. These are linear equations for the wave near the origin, but the slope constants of the wave can be different in sign and magnitude. Slopes with different -/+ signs and equal magnitude are shown. The condition domains for the triangle wave function might be:

• \$variable < -1
• -1 < \$variable with negative slope -\$s1 < 0
• \$variable = 0
• 0 < \$variable with positive slope +\$s2 < 1
• \$variable > 1 ### Figure 7. Playing Recursion V2 positive step function

As another example for the positive rectangle step function, the condition domains might be:

• \$variable = 0 d < 0
• \$variable = 0 d = 0
• 0 < positive \$variable < 1
• \$variable = 0 d > 1 ### Figure 8. Iteration on Canvas ### Figure 9. Factorial Function N! ### Testcases Section

#### Testcase 1, step_function Starts at -1

table printed in Tcl format
session proc & mean value comment, if any
-2.0 step_function 0
-1.8 step_function 0
-1.6 step_function 0
-1.4 step_function 0
-1.2 step_function 0
-1.0 step_function 1 1st breakpoint
-0.8 step_function 1
-0.6 step_function 1
-0.4 step_function 1
-0.2 step_function 1
0.0 step_function 1 zero point
0.2 step_function 1
0.4 step_function 1
0.6 step_function 1
0.8 step_function 1
1.0 step_function 1 2nd breakpoint
1.2 step_function 0
1.4 step_function 0
1.6 step_function 0
1.8 step_function 0
2.0 step_function 0
2.2 step_function 0
2.4 step_function 0
2.6 step_function 0
2.8 step_function 0
3.0 step_function 0

#### Testcase 2, step_function Starts at Zero

table printed in Tcl format
session proc & mean value comment, if any
-2.0 step_function 0
-1.8 step_function 0
-1.6 step_function 0
-1.4 step_function 0
-1.2 step_function 0
-1.0 step_function 0
-0.8 step_function 0
-0.6 step_function 0
-0.4 step_function 0
-0.2 step_function 0
0.0 step_function 1 1st breakpoint
0.2 step_function 1
0.4 step_function 1
0.6 step_function 1
0.8 step_function 1
1.0 step_function 1 2nd breakpoint
1.2 step_function 0
1.4 step_function 0
1.6 step_function 0
1.8 step_function 0
2.0 step_function 0

#### Testcase 3, Impulse step_function Starts at Zero

table printed in Tcl format
session proc & mean value comment, if any
-2.0 step_function 0
-1.8 step_function 0
-1.6 step_function 0
-1.4 step_function 0
-1.2 step_function 0
-1.0 step_function 0
-0.8 step_function 0
-0.6 step_function 0
-0.4 step_function 0
-0.2 step_function 0
0.0 step_function 1 1st breakpoint at zero
0.2 step_function 1 2nd breakpoint at zero plus small amount
0.4 step_function 0
0.6 step_function 0
0.8 step_function 0
1.0 step_function 0
1.2 step_function 0
1.4 step_function 0
1.6 step_function 0
1.8 step_function 0
2.0 step_function 0

### Testcase 4, One Liner Programs Approach to the Fibonaci series

AMG posted me a really good recursion proc on the Fibonaci series. I have recast this proc into the Playing Recursion namespace ::liner. Now the construction of a recursion problem can be sequenced into generic steps. -> is the return symbol. fib(n) is the Fibonaci function and f(n) is the generic function.

• n! = n*(n-1)*(n-2)*(n-3) ... 3*2*1
• (n-1)! = (n-1)*(n-2)*(n-3) ...3*2*1
• int fac (int 0) -> 1
• fib (1) = f(1) if n=> 1, -> 1
• fib (0) = f(0) else {-> 1 }
• fib (n) = f(1) -> 1
• fib (4) = f(4) -> 4 * fib (3)
• fib (3) = f(3) -> 3 * fib (2)
• fib (2) = f(3) 2 * fib (2-1) -> 2! * fib (1) = 2 * 1 = 2
```;# AMG proc w/a?b.c recast to func inside namespace liner::
proc fib2 {n} {expr {\$n < 2 ? \$n : [fib2 [expr {\$n - 1}]] + [fib2 [expr {\$n - 2}]]}}  ;# AMG
;# func has little different notation but understandable
func fib  { n } {[< \$n 2 ]? \$n: [+ [fib [- \$n 1 ]] [fib [- \$n 2 ]] ]}
;# Usage fib  2 ;# returns 1 fib  5 ;# returns 5
;# Usage fib  10 ;# returns 55```

### Testcase 5, One Liners Programs Approach to the Golden Section

Continuing with the really good recursion proc AMG posted me. This proc can be recast or morphed into the Playing Recursion namespace ::liner. In the higher Fibonaci numbers, the ratio or dividend of successive Fibonaci numbers fib(n)/ fin(n-1) begins to approximate' the Golden Section or Golden Mean. An exact solution func gm was loaded into the namespace::liner (1+sqrt(5))*.5 = 1.618033988749895. One may develop a liner::func for approximating the Golden Section using recursion.

However, since lengthy recursion calls are anticipated, lets check the speed of the AMG proc which used the expr method. Using mathop in part of the proc fib3 in global space does save about 1 microsecond.

Initial trial calculations: Hand calculations for fib3 (38) = 39088169 over (fib3 37) = 24157817, expr (1.*39088169) / 24157817 returns 1.618033988749894. The textbook value for phi was 1.618033988749895. The golden ratio conjugate approximates (fib3 37) = 24157817 over fib3 (38) = 39088169, expr (1.*24157817) / 39088169 -> 0.6180339887498951. The textbook value for 1/phi was 0.618033988749895

The golden ratio conjugate is 1/phi or phi-1 is 0.618033988749894. One can develop the golden ratio conjugate by installing the reciprocal fib(n-1)/ fin(n) into liner::func golden. The formula pi =~ 4 * sqrt phi can be used to approximate pi through the golden section. Steve Lautizar posted pi =~ 6/5 * Phi^2, which looks easier to implement in Playing Recursion. expr { (6./5)*1.618033988749894*1.618033988749894 } returns 3.1416407864998708.

```;# AMG proc w/a?b.c recast to func inside namespace liner::
proc fib2 {n} {expr {\$n < 2 ? \$n : [fib2 [expr {\$n - 1}]] + [fib2 [expr {\$n - 2}]]}}  ;# AMG
time { fib2 10 } 1000 ;#  fib2 10  ->   65.842 microseconds per iteration
proc fib3 {n} {expr {\$n < 2 ? \$n : [fib3 [- \$n 1]] + [fib3 [- \$n 2]]}}  ;# AMG
time { fib3 10 } 1000  ;#  fib3 10  ->   64.722 microseconds per iteration
;# using mathop in part of the proc does save about 1 microsecond
;# func has little different notation from the liner:: rules but understandable
liner::func golden { n }  { [::tcl::mathop::/ [fib \$n]  1. [fib [- \$n 1]]  ]}
;# liner::golden 10 ;#  -> 1.6176470588235294, liner::golden 18 ;# -> 1.6180338134001253
;# but the liner::golden was recursion limited to 18,
;# analysis went back to a proc in global space to further check  the solution
proc golden2 { n }  { [/ [fib3 \$n]  1. [fib3 [- \$n 1]]  ]}
;#  golden2 20  ;# ->  1.6180339631667064
golden2 38  ;# -> 1.618033988749894 was highest fib(\$n)/fib(\$n-1) without cobwebs on programmer
;# installing  the reciprocal '''fib(n-1)/ fin(n)''' into   liner::func golden and proc golden2
liner::func golden_ratio_conjugate { n }  { [::tcl::mathop::/ [fib [- \$n 1]] 1. [fib \$n]   ]}
;# Usage liner::golden_ratio_conjugate 18 -> 0.6180340557275542
proc golden2_ratio_conjugate  { n }  { [/ [fib3 [- \$n 1]]  1. [fib3  \$n ]  ]}
;# Usage golden2_ratio_conjugate 18 ;# -> 0.6180340557275542 reasonable time delay
;# Usage golden2_ratio_conjugate 38  ;# -> 0.6180339887498951 cobwebs falling```

### Figure 10. Fibonaci small numbers ### Figure 11. Fibonaci large numbers ### Testcase 6, One Liners Programs Approach to the Approximate Pie

The analysis starts out with with the AMG proc for pi using acos -1, which returns 3.141592653589793 correct. We would like to develop a recursive formula for pi in the liner:: testbed. The formula pi oc 4 * sqrt (phi_conjugate) or pi oc 4 * sqrt (1/phi) can be used to approximate pi through the golden section phi. The formula expr 4. * sqrt (1/1.618033988749894) returns 3.144605511029694. Something wrong, contra-check was expr 4.* sqrt (0.618) -> 3.144. Steve Lautizar posted pi =~ 6/5 * Phi^2, which looks easier to implement in Playing Recursion. The trial calculation { (6./5)*\$phi*\$phi } returns 3.14164, pi to 2 decimal places. Subbing the exact formula for phi gives another avenue, phi = expr (1+sqrt(5))*.5, pi =~ expr (6./5)*(((1+sqrt(5))*.5)**2)

The positive root and negative conjugate root to the minimal polynomial x**2 − x − 1 is phi 1.61803 and -0.618033. There was a graphical method for finding square roots using circle and inner right triangle on the number line. Rene Descartes gave a proof from symmetric triangles. The circle radius was n; the circle radius was 2*n; the center of circle was positioned at N on the number line;the adjacent side of the triangle was 2*N-1; the opposite side of the triangle was sqrt (2n-1). For our problem and somewhat expanding the unit circle with radius of 1 from X**2 + y**2 = 1. The new circle equation is X**2 + y**2 = 1.616. The opposite side of triangle is sqrt phi. We can see the vector lies in or near the pi/4 sector or arc length, which may give the pi/4 or 45 degree factor in the problem.

```;# procs below should should be pastable into TCL Console
proc pi {} {expr acos(-1)} ;# AMG
;# Usage pi ;# returns 3.141592653589793 correct
;# expr 4.* sqrt (0.618) ->  3.1445190411253674 phi conjugate formula
proc approx_pie2 { }  {expr 4.* sqrt (.618033988749894) }
;# Usage approx_pie2 #;   3.144605511029691 approximate pi to 2 decimal places
;# Note approx_pie2 number slightly above correct answer?
proc approx_pie { n }  { [* [/ 6. 5] [** [/ [fib3 \$n]  1. [fib3 [- \$n 1]]] 2]]}
;# Usage approx_pie 20 ;# -> 3.141640 approximate pi to 3 decimal places
;# Note approx_pie number slightly above correct answer?
proc pix  { } {return  [expr { (4. * ( [atan  .5] + [atan [/ 1 3.] ]))}]}
;# exact double atangent formula from Euler
;# Using expr methods on *,+,& atan  archtangent
proc pieatan2  { } {return  [expr { (4. * ( atan(.5) +  atan(1/ 3.) ))}]}
;# using mathop method on *&+ and mathfunc method on atan
proc pieatan  { } { return [* 4.   [+ [atan  .5]  [atan [/ 1 3.] ] ]  ]}
;# Usage proc pieatan  { } ;# returns 3.141592653589793  exact
time {pieatan} 5000  ;# 1.1686  microseconds per iteration,
;# fairly fast using both mathop and mathfunc calls
;# pi formula from Jorge Xerxes
[* 2 [+ [atan [/ 1. [sqrt .618033988749894]]]   [atan  [sqrt .618033988749894]] ]]
;# returns  3.141592653589793
;# piex passes through namespace liner::
liner::func piex {} {[* 2. [::tcl::mathfunc::+ [::tcl::mathfunc::atan [/ 1. [::tcl::mathfunc::sqrt .618033988749894]]]   [::tcl::mathfunc::atan  [::tcl::mathfunc::sqrt .618033988749894]] ]]}```

### testing recursion on infinite series pi = 4* atan(fib(2*\$n+1)], 1 < f(\$n) terms < \$n

```;# following under test
;# proc fib3 has to be loaded first
proc fib3 {n} {expr {\$n < 2 ? \$n : [fib3 [- \$n 1]] + [fib3 [- \$n 2]]}}  ;# AMG
proc pie5 {} {expr {4* (atan(1./[fib3 3]) + atan(1./[fib3 4]))}}
;# Usage pie5 ;#    3.141592653589793  ;# 2 terms for atan(1./[fib3 \$n] series
;# checking  terms for infinite series atan(fib(2*\$n+1)]
proc fix2 {n} {expr {\$n < 2 ? \$n : [fix2 [expr 4* (atan(1./[fib3 [expr 2*\$n+1]]))]]}}
;# Usage fix2 3   -> 0.30708756507911217   terms  of  infinite series
proc fix3 {n} {expr {\$n < 2 ? \$n :[- [fix [expr 4* (atan(1./[fib3 [expr 2*\$n+1]]))]]  [fix [expr 4* (atan(1./[fib3 [expr 2*\$n+1+2]]))]]  ] }}
foreach i {1 2 3 4 5 6 7 8 9} {lappend rook [fix \$i] };\$rook' set rook {}
proc vik {n} { return [expr { 8./ ((4.*\$n+1. )*( 4.*\$n+3. ))}]}
;# Usage  vik 7 -> 0.008898776418242492 ;# individual terms for pi series
foreach i {1 2 3 4 5 6 7 8 9} {lappend rook " [vik \$i] + " };\$rook ;  set rook {}
;# following is experimental kludge combination,
;# but some of the atan series for pi have
;# 2,3,4, or more terms, which can be
;# zeroed out or fill in the blanks.
proc fix {n} {expr {\$n < 2 ? \$n : [fix  [expr {.000001* atan ( 1. / (2.*\$n-1) )+ .0000001* atan ( 1. / (2.*\$n-1) )}]] +  [- [* 16. [atan [/ 1. 5]]] [* 4. [atan [/ 1. 239]]  ] ]}}
;# Usage  fix 3 ;# -> 3.1415928509853535
;# Chinese fractions for pi from Zu Chongzhi, fifth century, 429-500 CE
;# Jiu zhang suanshu,  Nine chapters of the mathematical art
liner::func pie_poly2 {} {[::tcl::mathop::/ 355. 113. ] } ;# liner:: namespace use
proc pie_poly {} {[/ 355. 113.]}  ;# global namespace use
;# Usage  pie_poly ;# -> 3.1415929203539825 accurate to 6 places
;# TCLLIB integration of expr expression in x
;# of OEIS formula for pi, ref notes on sequence A003881
proc pie_tcllib {} {set return [::math::calculus::integralExpr 0 100 10000 {(4.* \$x)/(\$x**4 + 1)}]}
;# Usage pie_tcllib  -> 3.1413926535904273
proc piez {} {[expr {4.*atan (1./sqrt(2))+2.*atan(1./sqrt(8))}]}
;# Usage piez ;# returns 3.141592653589793```

• Phi**3=2*phi+1
• Phi**4=3*phi+2
• Phi**5=5*phi+3
• Phi**6=8*phi+5

### Figure 12. minimal polynomial x**2 − x − 1

• minimal polynomial x**2 − x − 1 ### Figure 13. Unit Circle

• unit circle, radius =1, circle equation x**2 + y**2 = 1**2 ### Figure 14. Circle with radius of phi

• for circle with sqrt phi, radius =1.616 , circle equation x**2 + y**2 = phi
• subbing sqrt into equation x**2 + y**2 = (sqrt phi)**2 ### Figure 15. MACHIN PI 2 TERM FORMULA ed2 ### Testcase 7, Polynomials, Sum of Integers, Sum of Squares, Sum of Cubes, and Sum of Nth Powers

The formula for Sum of Integers is n*(n+1)/2. The formula for Sum of Squares is n*(n+1)*(2*n+1)/6. Initial code reporting too many nested evaluations with 4 terms in row. Added helper procs to contend with this.

```;# base functions loaded into namespace liner::
proc eq {m n} {string equal \$m \$n}  ;# RS
proc succ x {incr x}  ;# RS
;#  liner:: has shorter symbolic names
;#  for the two basic functions:
proc '  x    {incr x}    ;# RS
proc = {m n} {string equal \$m \$n}  ;# RS
;# formula for Sum of Integers is n*(n+1)/2
;# initial sum of integers code
;# reporting endless loops with w/4 terms
liner::func sum_int  { n} {[< \$n 2]? 1:    [expr [/ [* \$n [succ \$n ]] 2 ]]}
;# Usage liner::sum2 3 ;#  -> 6 correct for sum 1  + 2 + 3
;# helper functions 2X and 3X,
;# following liner:: rules
liner::func int_2x  { n} {[< \$n 1]? 1:    [expr  [* \$n  2 ]]}
liner::func int_3x  { n} {[< \$n 1]? 1:    [expr  [* \$n  3 ]]}
;# formula for Sum of Squares is n*(n+1)*(2*n+1)/6
liner::func sum_squares  { n} {[< \$n 1]? 1:    [expr [/ [* [* \$n [succ \$n ]]  [succ [int_2x \$n] ]   ] 6 ]]}
;# Usage liner::sum_squares 3  ;# -> 14 correct for sum 1 + 4 + 9```

Scratch Code on Sum of Cubes

```;# formula for Sum of Cubes is (n**2) ((n + 1)**2) / 4
func sum_cubes { n} {[< \$n 1]? 0:[=  \$n 1]?   1: [expr [/ [* [* \$n \$n]  [* [succ \$n] [succ \$n] ]   ] 4 ]]}
;# defined for positive integers, two ? conditionals in expr
;# Usage sum_cubes 0 -> 0, sum_cubes 1 -> 1, sum_cubes 2 -> 9```

Scratch Code on Sum of 4th Powers

func sum_4th_power2 reporting errors on too many nested evaluations (infinite loop?) inside the liner:: namespace. The TCL proc using the expr method is working well. The two helper funcs are working well inside liner:: namespace, but when the helper funcs are combined, the func sum_4th is reporting too many nested evaluations. Maybe have to go to conventional multiple statement proc to work around.

```;# formula sum_4th_power = n* (n+1)* (2*n+1 )* (      3*n**2 + 3*n -1)/30
;# using integer arithmetic for the long digit answers
proc  sum_4th_power  {n}   { expr { \$n* (\$n+1)* (2*\$n+1 )* (3*\$n**2 + 3*\$n -1)/30}}
;# Usage sum_4th_power 0  -> 0  ;# Usage sum_4th_power 1  -> 1
;# Usage sum_4th_power 1000  -> 200500333333300
;# Usage  sum_4th_power 200 -> 64802666660
;# Usage sum_4th_power 1000000000
;#     -> 200000000500000000333333333333333333300000000
func helper   {n } {  [ +   [ int_3x [square \$n] ]    [pred  [ int_3x \$n ]]]}
func helper2 {n}     { [* \$n [*  [ succ \$n ] [succ [int_2x \$n ]]]] }
func sum_4th_power2 { n} {[< \$n 1]? 0:[=  \$n 1]?   1: [expr [/ [* [helper \$n]  [helper2 \$n] ] 30 ]]}
proc tester_4x {n} {
if { [< \$n 1]  } { set res 0; break}
if { [= \$n 1]  } { set res 1; break}
set term1 [helper \$n]
set term2 [helper2 \$n]
set res [expr { (\$term1 * \$term2 ) / 30 } ]
}
;# Usage liner::tester_4x 5 -> 979
;# reporting nested evals after 5```

Scratch Code on Sum of 5th Powers

```;# formula sum_5th_power = n*n* (n+1)* (n+1 )* (  2*n**2 + 2*n -1)/12
proc sum_5th {n} {return [expr {(\$n*\$n* (\$n+1)*(\$n+1 )*(2*(\$n**2)+2*\$n-1))/12}] }
;# internal ? in expr testing for 2 conditions
proc sum_5th_power  {n} {expr { \$n < 1? 0: \$n > 1 ? 1 : (\$n*\$n* (\$n+1)*(\$n+1 )*(2*(\$n**2)+2*\$n-1))/12}}
;# Usage sum_5th_power 0  -> 0  ;# Usage sum_5th_power 1  -> 1
;# Usage sum_5th_power 2  -> 33```

### Testcase 8, Triangular Number Multiplication

The Glaisher formula returns the triangular multiplication product TNX as an integer from two positive natural numbers. Triangular multiplication is formulated as aa * bb = TN <aa-1> + TN <bb> - TN <aa-bb-1>. The Glaisher formula does not define the Triangular Number for negative numbers. As understood here, the term TN <aa-bb-1> might return negative numbers for bb > aa, so suggest swap or reorder to avoid TN<-n>. The original liner::- subtraction calls for \$m > \$n. However, it should be possible to redefine subtraction in liner::- for both cases \$m > \$n and \$n > \$m, but hate to monkey too much with original RS code. Maybe a dual subtraction based on liner::func max could be called as a helper function. For now, using m > n in liner::func for triangular multiplication.

Test. for TNX 6 3, the first term is 15, the second term is 6, and the third term is 3. Glaisher formula gives 15+6-3 equals 18, correct.

Scratch Code on triangular multiplication product

```;#  TNX =  aa * bb = TN <aa-1> +  TN <bb> - TN <aa-bb-1>
;# may need some helper functions inside liner:: if n>m
;# for now, m > n
liner::func max {m n} {\$m>\$n? \$m: \$n}
;#-- the difference (defined only for m >= n)
func - {m n} {[= \$n 0]? \$m: [- [pred \$m] [pred \$n]]}
;# test {- 12 7} 5 {- 1 1} 0 {- 0 0} 0
liner::func first_term  {m n} { [+ [tri [pred \$m ]] [tri \$n]] }
liner::first_term 5 3  ;# -> 16
liner::func third_term  {m n} {  [tri [- [- \$m  \$n ] 1 ]  ]  }
liner::third_term 5 3  ;#   ->  1
liner::func trix {m n} { [- [+ [tri [pred \$m ]] [tri \$n]]  [tri [- [- \$m  \$n ] 1 ]  ]    ]}
;# Usage liner::trix 5 3 -> 15   ;# m > n because subtraction call for this
;# Usage liner::trix 30 5 -> 150   ;# m > n
;# but too many  nested evals beyond m = 30```

### References on Golden Ratio and pi Math

• Golden Ratio pdf [L21 ]
• Golden section ratio: Phi [L22 ]
• High quality Wolfram, Golden_Ratio [L23 ][L24 ]
• Mathematics Galore!, pub 2012 by James Tanton
• The golden ratio, Fibonacci numbers and BBP-type formulas,Kunle Adegoke†
• Department of Physics, Obafemi Awolowo University, Ile-Ife, Nigeria
• Random Fibonacci sequences and the number 1.13198824
• Author: Divakar Viswanath, Journal: Math. Comp. 69 (2000), 1131-1155
• Hayashi, Ko, Fibonacci Numbers and the Arctangent Function, manuscript (1995).
• Abel-Plana Formula [L25 ]
• The encyclopedia of integer sequences, hardcopy, , 1995 ,
• by N. J.A. Sloane & Simon Plouffe
• non-alternating infinite series pi/4 = expr 2./ (4.*\$n+1. )*( 4.*\$n+3. )[L26 ]
• The On-Line Encyclopedia of Integer Sequences® (OEIS®) Wiki [L27 ]
• OEIS notes on infinite series pi/4 = expr 2./ (4.*\$n+1. )*( 4.*\$n+3. )
• sequence A003881 [L28 ]
• From Peter Bala, Nov 05 2019: (Start)
• Quote . Pi/4 = k!*Sum_{n = -inf..inf} 1/((4*n+1)*(4*n+3)*...*(4*n+2*k+1))
• Equals Integral_{x = 0..inf} sin(x)^4/x^2 dx = Sum_{n >= 1} sin(n)^4/n^2,
• by the Abel-Plana formula.
• Equals Integral_{x = 0..inf} sin(x)^3/x dx = Sum_{n >= 1} sin(n)^3/n,
• by the Abel-Plana formula. Unquote (End)
• individual terms for pi series
• From Amiram Eldar, Aug 19 2020: (Start)
• Quote. Equals arcsin(1/sqrt(2)).
• Equals Product_{k>=1} (1 - 1/(2*k+1)^2).
• Equals Integral_{x=0..oo} x/(x^4 + 1) dx.
• Equals Integral_{x=0..oo} 1/(x^2 + 4) dx. unquote (End)
• A Viète-like formula for pi based on infinite sum of the arctangent functions with nested radicals
• posted on 03.01.2017, 21:34 by Sanjar Abrarov, Brendan Quine
• OEIS notes on Fibonaci sequence A000045 [L29 ]
• OEIS notes on atan related "sequence" expansion, A333691 [L30 ]
• OEIS notes on pi 3.14... related "sequence" expansions plural, A000796 [L31 ] & [L32 ]
• Pi: A Source Book, Lennart Berggren, Jonathan Borwein, and Peter Borwein,
• Third Edition, Simon Fraser University, July 5, 1999
• Mathematical Structuresfor Computer Science, 2014
• Discrete MatheMatics anD its applications, Judith L. Gersting,
• Indiana University-Purdue University at Indianapolis

## Appendix Code

### Console Program for Playing Recursion V2

demo examples for one liner programs

```;# pretty-print autoindent from ased editor
;# trial console operation
;# Playing Recursion V2
;# demo examples for one liner programs
;# written on Windows 10 on Tcl
;# working under Tcl version 8.6
;# gold on Tcl Club , 2020 05 10
;# console has large black type on green
;# used as testbed for integer arithmetic
;# Playing with Recursion V2 ???
;# Playing with Recursion is Tcl wiki page by RS
;# < positive integers > function procs written by RS
;# gold added cosmetics in console deck
package require Tk
package require math::numtheory
package require math::constants
package require math::trig
package require math
namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory math::trig math::constants }
namespace path {::tcl::mathop ::tcl::mathfunc}
set tclprecision 17
;# RS main deck
proc succ x {incr x}  ;# RS
;# Note. procs succ and proc eq loaded in both decks
;# for global and namespace ::liner::
;# equality of integers in canonical form can,
;# without using expr, be had as
proc eq {m n} {string equal \$m \$n}  ;# RS
;# proc eq  works with strings and numbers
;# alternating sequences of {condition} {result} like
;#  A B C D ...
;# which can in Tcl (and many other languages)
;# be implemented as variables & conditionals
;# if \$A then \$B elseif \$C then \$D ...
;# or more compactly with expr's ?: ternary operator:
;# \$A? \$B: \$C? \$D: ...
;# using expr in these experiments only for the ?: operator
;# before  Playing Recursion deck
;# begin namespace liner
namespace eval ::liner {
namespace export math
}
proc ::liner::convert {  } {
;# before  Playing Recursion deck
proc eq {m n} {string equal \$m \$n}  ;# RS
proc succ x {incr x}  ;# RS
proc func {name argl body} {proc \$name \$argl [list expr \$body]}
;# here's a tiny multi-case tester:  -RS
;# added puts result to original proc test1
proc test1 args {
foreach {case expected} \$args {
catch {uplevel 1 \$case} res
if {\$res != \$expected} {error "\$case->\$res, expected \$expected"}
}
}

proc test args {
foreach {case expected}  \$args  {
catch {uplevel 1 \$case} res
if {\$res != \$expected} {error "\$case->\$res, expected \$expected"}
puts "example >>  \$case -> \$res, & expected was \$expected"
}
}

;# proc test reworked for console print out (puts line)
;# Following McCarthy's notation,  use   -RS
;# shorter symbolic names for the two basic functions:
proc '  x    {incr x}    ;# RS
;# test {' 0} 1 {' 42} 43

proc = {m n} {string equal \$m \$n}  ;# RS
;# test {= 0 0} 1 {= 0 42} 0 {= 42 42} 1

;# The partial converse of the successor function is the predecessor
;# "the non-negative integer before"),
;# where the auxiliary function pred2 uses recursion
;#  - it calls itself (as most of the following will).
;# The next definitions are a mechanical
;# transcription to Tcl of McCarthy's:
func pred   n    {[pred2 \$n 0]}  ;# RS
func pred2 {n m} {[= [' \$m] \$n]? \$m: [pred2 \$n [' \$m]]}  ;# RS
;# test {pred 42} 41 {pred 1} 0

;#-- This allows us to define the sum:
func + {m n} {[= \$n 0]? \$m: [+ [' \$m] [pred \$n]]} ;# RS
;# test {+ 3 4} 7

;#-- the product
func * {m n} {[= \$n 0]? 0: [+ \$m [* \$m [pred \$n]]]}
;# test {* 3 4} 12

;#-- the difference (defined only for m >= n)
func - {m n} {[= \$n 0]? \$m: [- [pred \$m] [pred \$n]]}
;# test {- 12 7} 5 {- 1 1} 0 {- 0 0} 0

;#-- the inequality predicate <=
func <= {m n} {[= \$m 0]? 1: [= \$n 0]? 0: [<= [pred \$m] [pred \$n]]}  ;# RS
;# test {<= 1 2} 1 {<= 2 2} 1 {<= 3 2} 0

#-- leading to the strict inequality <
func < {m n} {[= \$m \$n]? 0: [<= \$m \$n]}
;# test {< 1 2} 1 {< 2 2} 0 {< 3 2} 0

;#-- Integer division goes like this:
func / {m n} {[< \$m \$n]? 0: [' [/ [- \$m \$n] \$n]]}
;# test {/ 1 2} 0 {/ 2 2} 1 {/ 3 2} 1 {/ 42 2} 21

;#-- Integer division remainder (% in expr)
func rem {m n} {[< \$m \$n]? \$m: [rem [- \$m \$n] \$n]}  ;# RS
;# test {rem 1 2} 1 {rem 2 2} 0 {rem 3 2} 1

;#-- Divisibility of a number n by a number m
func | {m n} {[= \$n 0]? 1: [< \$n \$m]? 0: [| \$m [- \$n \$m]]}  ;# RS
;# test {| 2 42} 1 {| 2 43} 0 {| 2 3} 0

;# Primeness of a number uses another auxiliary function.
func prime   n    {[= \$n 0]? 0: [= \$n 1]? 0: [prime2 \$n 2]}
func prime2 {m n} {[= \$m \$n]? 1: [| \$n \$m]? 0: [prime2 \$m [' \$n]]}
;# test {prime 2} 1 {prime 3} 1 {prime 4} 0 {prime 5} 1 {prime 6} 0 {prime 7} 1

;#-- Greatest common denominator, according to Euclid's algorithm:
func gcd {m n} {[<= \$n \$m]?        [gcd \$n \$m]:
[= [rem \$n \$m] 0]? \$m:
[gcd [rem \$n \$m] \$m]}

;#  ****Extensions from gold ****10/30/2020******
;# Suggest maintain air gaps, dead spaces, brackets
;#--  >= greater than or equal to logic function
func >= {m n} {[= \$n 0]? 1: [= \$m 0]? 0: [>= [pred \$m] [pred \$n]]}
;#--  >  greater than logic function
func > {m n} {[= \$m \$n]? 0: [>= \$m \$n]}
;#--  flipper reverses +/- signs
;#    flipper is used in abs function
func flipper n { [expr {\$n * -1 } ] }
;#--  abs absolute value, uses flipper function
;#    abs 5 returns 5, abs -5 returns 5
;#    abs by rules above  is sticking,
;# so needs work invoking RS code
;# func abs1 n { [> \$n 0]? \$n: [ flipper \$n ] }
;# func abs2 n { [> \$n 0]? \$n: [< \$n 0]? [ flipper \$n]:0}
;# func abs5 n { \$n>0? \$n: [ flipper \$n ] }
;# following line cheats in expr > use
func abs n { \$n>0? \$n: [ flipper \$n ] }
;#    sgn is logic on +/- sign  ;# RS & rmax
;#    1 is true for positive sign
;#    -1  is for negative sign
;#   sgn 0 returns zero
proc sgn x {expr {(\$x>0) - (\$x<0)}} ;# RS & rmax
;#-- triangular number. Tricky code
;#   and Suggest maintain air gaps, spaces, brackets
func tri  { n} {[< \$n 2]? 1: [/  [* \$n [+ \$n 1] ] 2 ]}
;# test {tri 3} 6 {tri 2} 3 {tri 5} 15
;#-- factorial recursion, but quickly exceeds domain
;#   fac is recursion limited, & excessive time beyond factorial 7
func fac n {\$n<2? 1: [* \$n  [fac  [pred \$n ]]]}
;#-- min, max, basic integer funcions
;#  but may need some error checks
;#  for entry zeros and equality, \$m = \$n
func min {m n} {\$m<\$n? \$m: \$n}
func max {m n} {\$m>\$n? \$m: \$n}
;# zero_killer function swaps points of zero
;# for 0.0000001 in observed data to avoid
;# computer division by zero
func kzero { n } {[= \$n 0]? 0.0000001: [= \$n 0.]? 0.0000001: \$n }
;# One Liner Approach to the Fibonaci series
;# recursion test
;# AMG proc w/a?b.c recast to func inside namespace liner::
proc fib2 {n} {expr {\$n < 2 ? \$n : [fib2 [expr {\$n - 1}]] + [fib2 [expr {\$n - 2}]]}}  ;# AMG
;# func has little different notation but understandable
func fib  { n } {[< \$n 2 ]? \$n: [+ [fib [- \$n 1 ]] [fib [- \$n 2 ]] ]}
;# Usage fib  2 ;# returns 1 fib  5 ;# returns 5
;# Usage fib  10 ;# returns 55
func square  { n } {[< \$n 2]? 1:  [* \$n  \$n  ] }
;# Usage square 5 ;# returns 25
;# Usage square 30 ;# returns 900
;# liner::square has lengthy recursion and
;# too many nested evaluations beyond 30
func cube  { n} {[< \$n 2]? 1: [* [* \$n  \$n  ]  \$n ]}
;# Usage cube 2  ;# returns 8,cube 4 ;#returns 64
;# Usage cube 5 ;# returns 125
func rectangle_area {l w} {[= \$l 0]? 0: [= \$w 0]? 0: [* \$l \$w]}
;# Usage rectangle_area 5 10  ;# returns 50
;# testing negative logic for =! and ne
;# procs =! and ne work on strings too
proc =! {m n} { if {[string equal \$m \$n]} then {return 0} else {return 1} }
;# Usage liner::=! 5 5 r> 0, liner::=! 5 6 r> 1
proc ne {m n} { abs [string compare \$m \$n]}
;# Usage liner::ne 5 5 r> 0, liner::ne 5 6 r> 1
;# Playing Recursion was written originally for integers.
;# But now passing pointing point calculations thru ::liner::func
;# constant for golden section, have to exactly name sqrt inside mathfunc::
func gm  { n} {[< \$n 2]? 1:    [expr  (1 + [::tcl::mathfunc::sqrt 5] )*.5]}
;# Usage gm 5 ;# returns  1.618033988749895
;# test on floating points for rectangle
;# added logic for zero inputs may be unnecessary
func house {l w} {[= \$l 0]? 0: [= \$w 0]? 0: [::tcl::mathop::* \$l \$w]}
;# Usage house 5.2 10.1 ;# returns 52.52
;# reciprocal func r!  has several exceptions , [= \$mm 0]? avoid division by zero
;# reciprocal r! has exception , [= \$mm 1]? defaults to 1
;# exception tests performed before main calculation of 1/\$mm
;#  maybe case liner::r! -2 -> -1 on negative integers has problem?
func r! {mm} {  [= \$mm 0]? 0:  [= \$mm 1]? 1: [::tcl::mathop::/ 1.  \$mm] }
;# Usage liner::r! 1  -> 1 ;# liner::r! 0 -> 0 ;# liner::r! 0.000001 -> 1000000.0
;# liner::r! 0.000001 ->  1000000.0 ;# liner::r! -2. -> -0.5
;# installing  the reciprocal  fib(n-1)/ fin(n)
;# into   liner::func golden and proc golden2
;# either use func fib for McCarthy  recursion testing
;# or use AMG proc fib2 to avoid
;# multiple loop limit beyond n=18 on fib
func golden_ratio_conjugate { n }  { [::tcl::mathop::/ [fib2 [- \$n 1]] 1. [fib2 \$n]   ]}
;# Usage golden2_ratio_conjugate 18 ;# -> 0.6180340557275542 reasonable time delay
;# Usage golden2_ratio_conjugate 38  ;# -> 0.6180339887498951 cobwebs falling
;# formula for Sum of Integers is n*(n+1)/2
;# initial sum of integers code
;# reporting endless loops with w/4 terms
func sum_int  { n} {[< \$n 2]? 1:    [expr [/ [* \$n [succ \$n ]] 2 ]]}
;# Usage liner::sum2 3 ;#  -> 6 correct for sum 1  + 2 + 3
;# helper functions for 2X and 3X,
;# following liner:: rules
func int_2x  { n} {[< \$n 1]? 1:    [expr  [* \$n  2 ]]}
func int_3x  { n} {[< \$n 1]? 1:    [expr  [* \$n  3 ]]}
;# formula for Sum of Squares is n*(n+1)*(2*n+1)/6
func sum_squares  { n} {[< \$n 1]? 1:    [expr [/ [* [* \$n [succ \$n ]]  [succ [int_2x \$n] ]   ] 6 ]]}
;# formula for Sum of Cubes is (n**2) ((n + 1)**2) / 4
func sum_cubes { n} {[< \$n 1]? 0:[=  \$n 1]?   1: [expr [/ [* [* \$n \$n]  [* [succ \$n] [succ \$n] ]   ] 4 ]]}
;# defined for positive integers, two ? conditionals in expr
;# Usage sum_cubes 0 -> 0, sum_cubes 1 -> 1, sum_cubes 2 -> 9
;# formula sum_4th_power = n* (n+1)* (2*n+1 )* (      3*n**2 + 3*n -1)/30
proc  sum_4th_power  {n}   { expr { \$n* (\$n+1)* (2*\$n+1 )* (3*\$n**2 + 3*\$n -1)/30}}
;# Usage sum_4th_power 0  -> 0  ;# Usage sum_4th_power 1  -> 1
;# Usage sum_4th_power 1000  -> 200500333333300
;# Usage  sum_4th_power 200 -> 64802666660
;# Usage sum_4th_power 1000000000 ->
;#  200000000500000000333333333333333333300000000
func helper   {n } {  [ +   [ int_3x [square \$n] ]    [pred  [ int_3x \$n ]]]}
func helper2 {n}     { [* \$n [*  [ succ \$n ] [succ [int_2x \$n ]]]] }
proc tester_4x {n} {
if { [< \$n 1]  } { set res 0; break}
if { [= \$n 1]  } { set res 1; break}
set term1 [helper \$n]
set term2 [helper2 \$n]
set res [expr { (\$term1 * \$term2 ) / 30 } ]
}
;# Usage liner::tester_4x 5 -> 979
;# reporting nested evals after 5
;# formula sum_5th_power = n*n* (n+1)* (n+1 )* (  2*n**2 + 2*n -1)/12
;# internal ? in expr testing for 2 conditions
proc sum_5th_power  {n} {expr { \$n < 1? 0: \$n > 1 ? 1 : (\$n*\$n*(\$n+1)*(\$n+1 )*(2*(\$n**2)+2*\$n-1))/12}}
;# Usage sum_5th_power 0  -> 0  ;# Usage sum_5th_power 1  -> 1
;# Usage sum_5th_power 2  -> 33
proc pi {} {expr acos(-1)} ;# AMG
;# Usage pi -> 3.141592653589793
;# pi used for comparison with homebrew procs
;# Usage liner::sum_squares 3  ;# -> 14 correct for sum 1 + 4 + 9
;# The Glaisher formula returns the triangular  multiplication product
;#  TNX =  aa * bb = TN <aa-1> +  TN <bb> - TN <aa-bb-1>
;# may need some helper functions inside liner:: if n>m
;# for now, m > n
liner::func trix {m n} { [- [+ [tri [pred \$m ]] [tri \$n]]  [tri [- [- \$m  \$n ] 1 ]  ]    ]}
;# Usage liner::trix 5 3 -> 15   ;# m > n because subtraction call for this
;# Usage liner::trix 30 5 -> 150   ;# m > n
;# but too many  nested evals beyond m = 30
;# following are experimental kludge combinations,
;# but some of the atan series for pi have
;# 2,3,4, or more terms, which can be
;# zeroed out or fill in the blanks.
;# error reporting too many nested evaluations
;# (infinite loop?) with 4 terms
;# returns pi from atan formula
;# with  2 terms of Fibonacci function
proc fix {} {expr {4. * (atan(1./[fib2 3]) + atan(1./[fib2 4]))}}
;# Usage  fix  ;# -> 3.1415928509853535
;#  ******* end extensions ************
proc demo {} {
test {gcd 12 24} 12 {gcd 12 18} 6 {gcd 17 19} 1
test {' 0} 1 {' 42} 43
test {= 0 0} 1 {= 0 42} 0 {= 42 42} 1
test {pred 42} 41 {pred 1} 0
test {+ 3 4} 7
test {* 3 4} 12
test {- 12 7} 5 {- 1 1} 0 {- 0 0} 0
test {<= 1 2} 1 {<= 2 2} 1 {<= 3 2} 0
test {< 1 2} 1 {< 2 2} 0 {< 3 2} 0
test {/ 1 2} 0 {/ 2 2} 1 {/ 3 2} 1 {/ 42 2} 21
test {rem 1 2} 1 {rem 2 2} 0 {rem 3 2} 1
test {| 2 42} 1 {| 2 43} 0 {| 2 3} 0
test {prime 2} 1 {prime 3} 1 {prime 4} 0 {prime 5} 1 {prime 6} 0 {prime 7} 1
;# extensions from gold on Tcl Club
puts " *** demo extensions from gold on Tcl Club *****  "
test {flipper -1 } 1 {flipper 2 } -2 {flipper 10 } -10 {flipper -10} 10
test {abs -1 } 1 {abs 2 } 2 {abs 10 } 10 {abs -10} 10
test {sgn -1 } -1 {sgn 2 } 1 {sgn 10 } 1 {sgn -10} -1
test {>= 1 2} 0 {>= 2 2} 1 {>= 10 5} 1 {>= 10 10} 1
test {> 1 2} 0 {> 9 2} 1 {> 10 5} 1 {> 5 10} 0
test {kzero 5} 5 {kzero 500} 500 {kzero 0} .0000001 {kzero 0.} .0000001
test {tri 3} 6 {tri 2} 3 {tri 5} 15
test {min 4 5} 4 {min 400 500} 400 {min 3 5} 3 {min 36 35} 35
test {max 4 5} 5 {max 400 500} 500 {max 3 5} 5 {max 36 35} 36
test {fac 1} 1 {fac 3} 6 {fac 4} 24 {fac 5} 120 {fac 6} 720  }
;#   yada yada inside namespace ::liner:: }         ;# after   Playing Recursion deck
proc !! {} {info commands ::liner::* }
proc \$\$ {} {::liner::convert }
proc && {} {::liner::demo }
puts " *****  enter  liner::convert  to invoke namespace ::liner *** or \$\$    "
puts " *****  enter  liner::func_name to invoke single func **********  "
puts " *****  list cmds with info commands ::liner::* or !! "
puts " *****  enter  liner::demo  for code test *** or &&   "
puts " *******  but considerable lengthy recursions **** "
#; demo
console show
console eval {.console config -bg palegreen}
console eval {.console config -font {fixed 20 bold}}
console eval {wm geometry . 40x20}
console eval {wm title . "  Playing Recursion V2, screen grab and paste from console 2 to texteditor"}
console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
console eval { proc self_helpx {} {
set msg "in Tcl, large black type on green
from Tcl Wiki,
self help listing
Conventional text editor formulas grabbed
from internet screens can be pasted
into green console
;# testbed & demo_recursion_functional_programming
;# demo command is demo or liner::demo
;# suggest maintain dead spaces and air gaps
;# near expr, brackets, etc in following statements
;# note on liner:: >> brackets  for
;# liner::math +-*/ and global mathop may conflict
;# written on Windows 10 on Tcl
;# working under Tcl version 8.6
;# gold on Tcl Club , 2020 10 05
;# easy eye console has large black type on green
;# used as testbed for integer arithmetic
;# Playing Recursion V2
;# Playing with Recursion by RS is Tcl wiki page
;# < positive integers > function procs written by RS
;# equality of integers in canonical form can,
;# without using expr, be written
;# proc eq  is Tcl string function
;# McCarthy advocates <conditional expressions>, -RS
;# alternating sequences of {condition} {result}
;# which can in Tcl (and many other languages)
;# be implemented as variables & conditionals
;# if \\$\A then \\$\B elseif \\$\C then \\$\D ...
;# or more compactly with expr's ?: ternary operator:
;# \\$\A\? \\$\B\: \\$\C\? \\$\D\: ...       -RS
;# using expr in these experiments only for the ?: operator
;# shorthand >> !! lists namespace commands
;# shorthand >> \$\$ invoke namespace liner::
;# shorthand >> && invoke liner::demo
;# enter  liner::func_name to invoke single func
;# invoke liner name  either by liner:: or ::liner::
;# here's a tiny multi-case tester:  -RS
;# \demo   added puts result to original proc test1
;# demo integer functions  from RS, Tcl Wiki "
tk_messageBox -title "self_helpxx" -message \$msg } }
;# end of file```

### Console Program for step_function Table

```;# pretty print from autoindent and ased editor
;# step_function_table V2
;# testbed for One Liners Program
;# written on Windows 10 on Tcl 8.6
;# working under Tcl version 8.6
;# gold on Tcl Club , 2020-10-18
;# Ref. WIKI BOOKS, Tcl_Programming_Introduction
;# Book Section  contrasts one liners program
;# below contains redundant procs
package require Tk
package require math::numtheory
package require math::constants
package require math::trig
package require math
namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory math::trig math::constants }
;# set tcl_precision 17
set tcl_precision 3
proc pie {} {return [expr acos(-1)]}
console show
console eval {.console config -bg palegreen}
console eval {.console config -font {fixed 20 bold}}
console eval {wm geometry . 40x20}
;# model if procedure from RS derived
;# for conditional if_elseif_else statement
;# reasonably fast too for Tcl script
;# merely string text assignments,
;# no extra math calculation after then
;# from Modeling COND with expr on Tcl Wiki
;# alternates after then is either return \$sgn ,
;#  return 1 , or return 0
;# vis set statement
proc model_proc_sgn nn {
# standard multi-test if from RS derived
if {\$nn>0} {
set sgn  1
} elseif {\$nn<0} {
set sgn -1
} else {
set sgn  0
}
;# one liner program  step_function on extended line
proc step_function nn { if {\$nn<-1} { return  0 } elseif {\$nn<0 } /
{ return 1 } elseif {\$nn==0} { return 1 } elseif { \$nn<1} /
{ return 1 } elseif {\$nn>1 } { return 0 } else { return  0 }}
set limit 20
puts "%|table| | printed in|Tcl format |% "
puts "&| session| proc &  mean value|  comment, if any|& "
for { set i -10 } { \$i <= \$limit }  { incr i } {
;# set lister { 1 2 4 5 6 7 8 9 10 }
;# lappend lister  [* \$i [pie]]
puts "&| [* \$i .2 ]   |step_function [ step_function [* \$i .2 ] ]  |   |&"
}
;# end of file     ```

Please place any comments here with your moniker, Thanks. gold 11/2/2020. expr sqrt ( 1.618033988749895**2 - .618033988749895*2.) -> 1.175570504584946 ref 1.2 factor in steve's formula.

test! seems to work

** Hidden Change Log Section **

gold10/25/2020. Reloaded version 102 proof of PYK, page was overwritten, omitting items as requested.

gold 10/27/2020. Typos, added positive step function, added tables with some figures. Did not want to break up prose more or type in too much inside "proofed" prose

gold 10/28/2020. added testcase 4 for phi and testcase 5 for pi approximation

gold 11/13/2020. added Table for McCarthy Logic and Table of Procs and funcs.

gold 11/20/2020. added Triangular Number Multiplication.

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