Bruce Evans 74bbe8ed42 Fixed range reduction for large multiples of pi/2 on systems with ... broken assignment to floats (e.g., i386 with gcc -O, but not amd64 or ia64; i386 with gcc -O0 worked accidentally). Use an unnamed volatile temporary variable to trick gcc -O into clipping extra precision on assignment. It's surprising that only 1 place needed to be changed. For tanf() on i386 with gcc -O, the bug caused errors > 1 ulp with a density of 2.3% for args larger in magnitude than 128*pi/2, with a maximum error of 1.624 ulps. After this fix, exhaustive testing shows that range reduction for floats works as intended assuming that it is in within a factor of about 2^16 of working as intended for doubles. It provides >= 8 extra bits of precision for all ranges. On i386: range max error in double/single ulps extra precision ----- ------------------------------- --------------- 0 to 3*pi/4 0x000d3132 / 0.0016 9+ bits 3*pi/4 to 128*pi/2 0x00160445 / 0.0027 8+ 128*pi/2 to +Inf 0x00000030 / 0.00000009 23+ 128*pi/2 up, -O0 before fix 0x00000030 / 0.00000009 23+ 128*pi/2 up, -O1 before fix 0x10000000 / 0.5 1 The 23+ bits of extra precision for large multiples corresponds to almost perfect reduction to a pair of floats (24 extra would be perfect). After this fix, the maximum relative error (relative to the corresponding fdlibm double precision function) is < 1 ulp for all basic trig functions on all 2^32 float args on all machines tested: amd64 ia64 i386-O0 i386-O1 ------ ------ ------ ------ cosf: 0.8681 0.8681 0.7927 0.5650 sinf: 0.8733 0.8610 0.7849 0.5651 tanf: 0.9708 0.9329 0.9329 0.7035		2005-10-11 07:56:05 +00:00
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alpha
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bsdsrc	Fixed aliasing bugs in TRUNC() by using the fdlibm macros for access	2005-09-19 11:28:19 +00:00
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ia64
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src	Fixed range reduction for large multiples of pi/2 on systems with	2005-10-11 07:56:05 +00:00
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