Fix and improve some magic numbers for the "medium size" case.
e_rem_pio2.c: This case goes up to about 2**20pi/2, but the comment about it said that it goes up to about 2**19pi/2. It went too far above 2**pi/2, giving a multiplier fn with 21 significant bits in some cases. This would be harmful except for a numerical accident. It happens that the terms of the approximation to pi/2, when rounded to 33 bits so that multiplications by 20-bit fn's are exact, happen to be rounded to 32 bits so multiplications by 21-bit fn's are exact too, so the bug only complicates the error analysis (we might lose a bit of accuracy but have bits to spare). e_rem_pio2f.c: The bogus comment in e_rem_pio2.c was copied and the code was changed to be bug-for-bug compatible with it, except the limit was made 90 ulps smaller than necessary. The approximation to pi/2 was not modified except for discarding some of it. The same rough error analysis that justifies the limit of 2**20pi/2 for double precision only justifies a limit of 2**18pi/2 for float precision. We depended on exhaustive testing to check the magic numbers for float precision. More exaustive testing shows that we can go up to 2**28pi/2 using a 53+25 bit approximation to pi/2 for float precision, with a the maximum error for cosf() and sinf() unchanged at 0.5009 ulps despite the maximum error in rem_pio2f being ~0.25 ulps. Implement this.
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svn2git
2020-12-20 02:59:44 +00:00
svn path=/head/; revision=176640
@ -126,7 +126,7 @@ __ieee754_rem_pio2(double x, double *y)
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}
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}
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}
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if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
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if(ix<0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
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medium:
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/* Use a specialized rint() to get fn. Assume round-to-nearest. */
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STRICT_ASSIGN(double,fn,x*invpio2+0x1.8p52);
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@ -38,8 +38,8 @@ __FBSDID("$FreeBSD$");
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static const double
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half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
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invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
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pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
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pio2_1t = 6.07710050650619224932e-11; /* 0x3DD0B461, 0x1A626331 */
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pio2_1 = 1.57079631090164184570e+00, /* 0x3FF921FB, 0x50000000 */
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pio2_1t = 1.58932547735281966916e-08; /* 0x3E5110b4, 0x611A6263 */
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#ifdef INLINE_REM_PIO2F
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extern inline
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@ -55,7 +55,7 @@ __ieee754_rem_pio2f(float x, double *y)
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GET_FLOAT_WORD(hx,x);
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ix = hx&0x7fffffff;
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/* 33+53 bit pi is good enough for medium size */
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if(ix<=0x49490f80) { /* |x| ~<= 2^19*(pi/2), medium size */
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if(ix<0x4dc90fdb) { /* |x| ~< 2^28*(pi/2), medium size */
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/* Use a specialized rint() to get fn. Assume round-to-nearest. */
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STRICT_ASSIGN(double,fn,x*invpio2+0x1.8p52);
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fn = fn-0x1.8p52;
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