# following statements can be pasted into eTCL console
set project 1.0
set  approx_legendre_primes2  [/ 100. [- [log 100.] 1.] ] #  answer 27.73794157864211
set  approx_gauss_number_primes   [/ 100. [log 100.] ]   # answer   21.71472409516259

        # pretty print from autoindent and ased editor
        # Gauss approximate number of primes calculator
        # written on Windows XP on eTCL
        # working under TCL version 8.5.6 and eTCL 1.0.1
        # gold on TCL WIKI, 15may2016
        package require Tk
        package require math::numtheory
        #namespace path {math::numtheory}
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        frame .frame -relief flat -bg aquamarine4
        pack .frame -side top -fill y -anchor center
        set names {{} {interval number L1  :} }
        lappend names {end number L2  :}
        lappend names {optional, if one & L2 < 190,  invoke exact solution (? long!!): }
        lappend names {answers: gauss function ~number_primes for N1 :}
        lappend names {legendre function ~number_primes for N1 :}
        lappend names {gauss approx number of primes from zero: }
        lappend names {gauss approx number of primes from zero to interval start: }
        lappend names {gauss number of primes over interval:} 
        foreach i {1 2 3 4 5 6 7 8} {
            label .frame.label$i -text [lindex $names $i] -anchor e
            entry .frame.entry$i -width 35 -textvariable side$i
            grid .frame.label$i .frame.entry$i -sticky ew -pady 2 -padx 1 }
        proc about {} {
            set msg "Calculator for Gauss Approximate Number of Primes 
            from TCL WIKI,
            written on eTCL "
            tk_messageBox -title "About" -message $msg } 
            # ref. TCLLIB::math::numtheory::prime_counting_functionx 
            # ref. alternate TCLLIB::math::numtheory::isprime 
             proc prime_counting_function { start  end } {
             set start [int $start ]
             set end [int $end ]
             set counter 0
             if { $start > 1  } { set counter  $start  } 
             set end  $end 
             set prime_counting_function 0
             while { $counter < $end  } {
             incr counter 
             incr  prime_counting_function  [isprime $counter ] 
             }
             return $prime_counting_function }  
       proc calculate {     } {
            global answer2
            global side1 side2 side3 side4 side5
            global side6 side7 side8 
            global switch_factor big_giant_list legendre_function
            global testcase_number 
            incr testcase_number 
            set side1 [* $side1 1. ]
            set side2 [* $side2 1. ]
            set side3 [* $side3 1. ]
            set side4 [* $side4 1. ]
            set side5 [* $side5 1. ]
            set side6 [* $side6 1. ]
            set side7 [* $side7 1. ]
            set side8 [* $side8 1. ]
            set L1 $side1  
            set L2 $side2  
            set N1 $side2 
            set switch_factor $side3 
            set approx_number_primes [/ $N1   [log $N1 ] ] 
            set legendre_function  [/ $N1   [- [log $N1 ] 1.08366 ] ] 
            # conditions, all numbers real and positive. 
            # conditions L1 =! 1 and L2 > L1, otherwise division by zero and undefined Q's.
            # if { $L1 == 1.0 } { set $L1 0.0 } 
            set approx_number_primes_interval_start [/ $L1   [log $L1 ] ]
            set approx_number_primes_over_interval [- [/ $L2   [log $L2 ] ] [/ $L1   [log $L1 ] ] ]
            set side4 $approx_number_primes
            set side5 $legendre_function
            set side6 $approx_number_primes 
            set side7 $approx_number_primes_interval_start
            set side8 $approx_number_primes_over_interval
            if { $switch_factor > 0 && $L2 < 200 } { exact_solution } 
                    }
            proc listprimes { aa bb} {lappend booboo 2.0; for {set i 1} {$i<=$bb} {incr i 2} { if {[isprime $i] } {lappend booboo [expr 1.*$i]   } };return $booboo}
            #returns list of prime numbers  from aa to bb, usage [listprimes 0 25],
            # answer is 2.0 3.0 5.0 7.0 11.0 13.0 17.0 19.0 23.0, 
            # list returned as real numbers, not counting 1.0 unity
               proc exact_solution {} {
            global side1 side2 side3 side4 side5
            global side6 side7 side8 big_giant_list prime_counting 
            set big_giant_list  [ listprimes $side1 $side2 ]           
            set side8 [ llength $big_giant_list ] 
             }
        proc fillup {aa bb cc dd ee ff gg hh} {
            .frame.entry1 insert 0 "$aa"
            .frame.entry2 insert 0 "$bb"
            .frame.entry3 insert 0 "$cc"
            .frame.entry4 insert 0 "$dd"
            .frame.entry5 insert 0 "$ee"
            .frame.entry6 insert 0 "$ff" 
            .frame.entry7 insert 0 "$gg"
            .frame.entry8 insert 0 "$hh" 
             }
        proc clearx {} {
            foreach i {1 2 3 4 5 6 7 8 } {
                .frame.entry$i delete 0 end } }
        proc reportx {} {
            global side1 side2 side3 side4 side5
            global side6 side7 side8
            global switch_factor big_giant_list legendre_function
            global testcase_number
            console show;
            puts "%|table $testcase_number|printed in| tcl wiki format|% "
            puts "&| quantity| value| comment, if any|& "
            puts "&| testcase number:|$testcase_number | |&"
            puts "&| $side1 :|interval L1   |   |&"
            puts "&| $side2 :|end number L2  | |& "  
            puts "&| $side3 :|optional,if one,  invoke exact solution (? long) | |& "
            puts "&| $side4 :|answers: gauss function ~number_primes| |&"
            puts "&| $side5 :|legendre function ~number_primes| |&"
            puts "&| $side6 :|gauss approx_number_primes  |  |&"
            puts "&| $side7 :|gauss approx_number_primes_interval_start |  |&"
            puts "&| $side8 :|gauss number_primes_over_interval |  |&" 
            puts "&| $legendre_function :|legendre_function number_primes_N1|  |&"
            if {$switch_factor > 0.0} { puts "&| [prime_counting_function $side1 $side2  ] :|prime_counting_function over interval|  |&" }
            if {$switch_factor > 0.0 && $side2 < 190.} {puts "&| $big_giant_list :|list number_primes_over_interval |  |&" }
            }
        frame .buttons -bg aquamarine4
        ::ttk::button .calculator -text "Solve" -command { calculate   }
        ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 0.0 1E1 1.0 4.3  8.20  4.3 0.00 4.30}
        ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 0.0 5.0 1.0 3.1  9.50  3.1 0.00 3.10 }
        ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 1E3 1E6 1.0 7E5  7E5 72382.0 0.0 72382.0 }
        ::ttk::button .clearallx -text clear -command {clearx }
        ::ttk::button .about -text about -command {about}
        ::ttk::button .cons -text report -command { reportx }
        ::ttk::button .exit -text exit -command {exit}
        pack .calculator  -in .buttons -side top -padx 10 -pady 5
        pack  .clearallx .cons .about .exit .test4 .test3 .test2   -side bottom -in .buttons
        grid .frame .buttons -sticky ns -pady {0 10}
               . configure -background aquamarine4 -highlightcolor brown -relief raised -border 30
        wm title . "Gauss Approximate Number of Primes Calculator"

  puts " %| testcase $testcase_number | value| units |comment |%"
  puts " &| volume| $volume| cubic meters |based on length $side1 and width $side2   |&"  

                  # autoindent from ased editor
                  # console program for graph numbers 
                  # written on Windows XP on eTCL
                  # working under TCL version 8.5.6 and eTCL 1.0.1
                  # TCL WIKI , 12dec2016
             console show
             package require math::numtheory
             #namespace path {math::numtheory}
             namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
             set tcl_precision 17
             proc factors {} {
             set counter 0
             set prime_counting_function 0
             while {$counter < 500} {
             puts " [* $counter 1.] ,  [/ [* $counter 1.] [log [* $counter 1.]] ],  [isprime $counter ],  $prime_counting_function   "
             incr counter 
             incr  prime_counting_function  [isprime $counter ] 
             }
             return }
             factors

 2.0 ,  2.8853900817779268,  1,  1   
 3.0 ,  2.7307176798805122,  1,  2   
 4.0 ,  2.8853900817779268,  0,  2   
 5.0 ,  3.1066746727980594,  1,  3   
 6.0 ,  3.3486637593074837,  0,  3   
 7.0 ,  3.5972883965882549,  1,  4   
 8.0 ,  3.8471867757039027,  0,  4   
 9.0 ,  4.0960765198207678,  0,  4   
 10.0 ,  4.3429448190325175,  0,  4   

             # under test, biasing approx. from interval to save computer time
       proc prime_counting_function { start  end } {
             set counter 0
             if { $start > 1  } { set counter [int $start ]  } 
             set end [int $end ] 
             set prime_counting_function 0
             while { $counter < $end  } {
             incr counter 
             incr  prime_counting_function  [isprime $counter ] 
             }
             return $prime_counting_function }  
        proc biased_count { N1    } {
             set N1 [int $N1 ]
             set bias1 [* 1. [ prime_counting_function [int  [* $N1 .99] ] $N1 ]]
             set bias2  [* 1. [ prime_counting_function $N1 [int [* $N1 1.01]]   ]]
             puts " $bias1 "
             puts " $bias2 "
             set bias 1.
             set approx_number_primes 1.
             set bias [expr  $bias2/ double($bias1)  ] 
             puts " $bias $bias2   $bias1 "
             set approx_number_primes [* [/ $N1   [log $N1 ] ] $bias ]
             return $approx_number_primes
             }