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C library: sqrt models must round correctly in all rounding modes - #9244
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IEEE 754 squareRoot is correctly rounded (C11 F.10.4.5), but the sqrtf, sqrt and sqrtl models were not: - They picked lower and upper = nextUp(lower) such that the rounded squares bracket the argument x, and in round-to-nearest returned lower iff x - lower*lower < upper*upper - x, again with rounded squares and differences. Comparing distances between squares does not decide which root is nearer, and the rounding of the squares adds further errors. For example, sqrt(1.0 + DBL_EPSILON) yielded 1.0 + DBL_EPSILON instead of 1.0. Emulating the sqrtf model natively for all 2^24 floats in [1, 4) gives the wrong result for 2754580 of them (16%). - In round-downward and round-toward-zero modes they returned upper unless the root was exact, i.e., almost always the wrong neighbour. - In round-upward mode, exactness was tested via the rounded square. Compute the squares exactly, using double for sqrtf, and a CBMC floating-point type with a 112-bit fraction for sqrt (144 or 230 bits for sqrtl, depending on the precision of long double), so that lower <= sqrt(x) < upper holds exactly. Then return lower when rounding down or toward zero, lower when rounding upward if lower * lower == x, and, when rounding to nearest, lower iff x is less than the square of the midpoint of lower and upper (there are no ties, as that square needs more bits than the format has). The same native emulation, with exact squares and the new round-to-nearest choice, yields no wrong results. In a differential test against glibc of 190 sqrtf and sqrt calls on special and random inputs in all four rounding modes, the previous models gave the wrong result in 119 cases, and the new ones in none. The sqrt (double) model takes about 1.5 times as long to verify as before; sqrtf is as fast as before. Reported by Jeroen Robben (https://reports.robben.io/methodology/). Fixes diffblue#9299 Co-authored-by: Kiro <kiro-agent@users.noreply.github.com>
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IEEE 754 squareRoot is correctly rounded (C11 F.10.4.5), but the
sqrtf, sqrt and sqrtl models were not:
squares bracket the argument x, and in round-to-nearest returned
lower iff x - lowerlower < upperupper - x, again with rounded
squares and differences. Comparing distances between squares does
not decide which root is nearer, and the rounding of the squares
adds further errors. For example, sqrt(1.0 + DBL_EPSILON) yielded
1.0 + DBL_EPSILON instead of 1.0. Emulating the sqrtf model
natively for all 2^24 floats in [1, 4) gives the wrong result for
2754580 of them (16%).
unless the root was exact, i.e., almost always the wrong neighbour.
Compute the squares exactly, using double for sqrtf, and a CBMC
floating-point type with a 112-bit fraction for sqrt (144 or 230
bits for sqrtl, depending on the precision of long double), so that
lower <= sqrt(x) < upper holds exactly. Then return lower when
rounding down or toward zero, lower when rounding upward if
lower * lower == x, and, when rounding to nearest, lower iff x is less
than the square of the midpoint of lower and upper (there are no
ties, as that square needs more bits than the format has).
The same native emulation, with exact squares and the new
round-to-nearest choice, yields no wrong results. In a differential
test against glibc of 190 sqrtf and sqrt calls on special and random
inputs in all four rounding modes, the previous models gave the wrong
result in 119 cases, and the new ones in none. The sqrt (double) model takes about 1.5 times as long
to verify as before; sqrtf is as fast as before.
Fixes #9299 (#9301 is an identical duplicate of #9299.)
Found by Jeroen Robben's specification-conformance pipeline (https://reports.robben.io/methodology/).