Tcl Source Code

'make check' produces -

==== expr-47.12 isqrt of various sizes of integer FAILED
==== Contents of test case:

    set faults 0
    for {set i 0} {$faults < 10 && $i <= 1024} {incr i} {
        set root [expr {1 << $i}]
        set rm1 [expr {$root - 1}]
        set arg [expr {1 << (2 * $i)}]
        set tval [expr {isqrt($arg-1)}]
        if {$tval != $rm1} {
            append trouble "i = " $i ": isqrt(" $arg "-1) == " $tval  " != " $rm1 "\n"
            incr faults
        }
        set tval [expr {isqrt($arg)}]
        if {$tval != $root} {
            append trouble "i = " $i ": isqrt(" $arg ") == " $tval  " != " $root "\n"
            incr faults
        }
        set tval [expr {isqrt($arg+1)}]
        if {$tval != $root} {
            append trouble "i = " $i ": isqrt(" $arg "+1) == " $tval  " != " $root "\n"
            incr faults
        }
    }
    set trouble

---- Result was:
i = 28: isqrt(72057594037927936-1) == 279620267 != 268435455
i = 28: isqrt(72057594037927936+1) == 279620267 != 268435456
i = 29: isqrt(288230376151711744-1) == 545818760 != 536870911
i = 29: isqrt(288230376151711744+1) == 545818760 != 536870912
i = 30: isqrt(1152921504606846976-1) == 1079707056 != 1073741823
i = 30: isqrt(1152921504606846976+1) == 1079707056 != 1073741824
i = 31: isqrt(4611686018427387904-1) == 2150992608 != 2147483647
i = 31: isqrt(4611686018427387904+1) == 2150992608 != 2147483648
i = 32: isqrt(18446744073709551616-1) == 4296881274 != 4294967295
i = 41: isqrt(4835703278458516698824704-1) == 2199023259647 != 2199023255551

---- Result should have been (exact matching):

==== expr-47.12 FAILED

This is with gcc-4.2.1, -O2.  Without -O2, the test passes.

I believe the cause is in bn_mp_sqrt.c; the following patch fixes it.

--- tcl8.6a2/libtommath/bn_mp_sqrt.c    2006-12-01 05:48:23.000000000 +0000
+++ tcl8.6a2-patched/libtommath/bn_mp_sqrt.c    2008-10-01 00:02:55.000000000 +0100
@@ -69,7 +69,7 @@
   if (dig) {
       t1.used = i+2;
       d -= ldexp((double) dig, DIGIT_BIT);
-      if (d != 0.0) {
+      if (d >= 1.0) {
          t1.dp[i+1] = dig;
          t1.dp[i] = ((mp_digit) d) - 1;
       } else {

The reason is that d can become small, much less than 1.0 but not exactly 0.0 - in that case t1 is not set to a valid representation.
t1.dp[i] is -1, outside the range of a 28bit mp_digit.
Without -O2, d is exactly 0.0 for the first failing case, hence t1 is valid and the final result is correct.
This could be due to intermediate results being rounded to 64 bit double, but with -O2 everything is kept as 80 bit long double.

With attached code extracted from bn_mp_isqrt.c, I get
% make CC=/usr/local/gcc-4.2.1/bin/gcc CFLAGS=-O2 LDFLAGS=-lm isqrt && ./isqrt
/usr/local/gcc-4.2.1/bin/gcc -O2  -lm  isqrt.c   -o isqrt
d=-1.86265e-09=-0x1p-29 dig=1


System is
Linux piglet 2.4.36 #1 Tue Jan 22 01:12:23 GMT 2008 i686 unknown unknown GNU/Linux


Malc.
[email protected]

Your analysis appears correct. Thanks for the patch!

I see that this issue has come up before - there are several closed tickets that look like exactly the
same issue (2030480  1609936 (which in turn mentions 1735583)).
Compiling without optimisation (or debugging prints) gets expr.test to pass, however there are
still values that fail, for example (1<<56) + (1<<5) -

malc@piglet:~/src/tcl/tcl8.6a2/unix> ./tcltest ../tests/expr.test 
expr.test:      Total   1749    Passed  1743    Skipped 6       Failed  0
Number of tests skipped for each constraint:
        5       !ieeeFloatingPoint
        1       longIs64bit
malc@piglet:~/src/tcl/tcl8.6a2/unix> ./tcltest ~/isqrt.tcl
d=100000000000020 q=10aaaaab qq=115c71c7ce38e39
malc@piglet:~/src/tcl/tcl8.6a2/unix> tclsh8.6 ~/isqrt.tcl
d=100000000000020 q=10000000 qq=100000000000000

(where ./tcltest is unmodified compiled with -O0, tclsh8.6 has the above path applied).
isqrt.tcl -

set d [expr (1<<56) + (1<<5)]
set q [expr isqrt($d)]
set qq [expr $q*$q]
puts [format "d=%llx q=%llx qq=%llx" $d $q $qq]

File Added - 295837: isqrt.c