Fix some off-by-1 errors.
e_rem_pio2.c: Float and double precision didn't work because init_jk[] was 1 too small. It needs to be 2 larger than you might expect, and 1 larger than it was for these precisions, since its test for recomputing needs a margin of 47 bits (almost 2 24-bit units). init_jk[] seems to be barely enough for extended and quad precisions. This hasn't been completely verified. Callers now get about 24 bits of extra precision for float, and about 19 for double, but only about 8 for extended and quad. 8 is not enough for callers that want to produce extra-precision results, but current callers have rounding errors of at least 0.8 ulps, so another 1/2**8 ulps of error from the reduction won't affect them much. Add a comment about some of the magic for init_jk[]. e_rem_pio2.c: Double precision worked in practice because of a compensating off-by-1 error here. Extended precision was asked for, and it executed exactly the same code as the unbroken double precision. e_rem_pio2f.c: Float precision worked in practice because of a compensating off-by-1 error here. Double precision was asked for, and was almost needed, since the cosf() and sinf() callers want to produce extra-precision results, at least internally so that their error is only 0.5009 ulps. However, the extra precision provided by unbroken float precision is enough, and the double-precision code has extra overheads, so the off-by-1 error cost about 5% in efficiency on amd64 and i386.
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Notes:
svn2git
2020-12-20 02:59:44 +00:00
svn path=/head/; revision=176550
@ -182,7 +182,7 @@ __ieee754_rem_pio2(double x, double *y)
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tx[2] = z;
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nx = 3;
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while(tx[nx-1]==zero) nx--; /* skip zero term */
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n = __kernel_rem_pio2(tx,y,e0,nx,2);
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n = __kernel_rem_pio2(tx,y,e0,nx,1);
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if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
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return n;
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}
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@ -45,7 +45,7 @@ int
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__ieee754_rem_pio2f(float x, float *y)
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{
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double w,r,fn;
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double tx[1],ty[2];
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double tx[1],ty[1];
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float z;
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int32_t e0,n,ix,hx;
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@ -77,7 +77,7 @@ __ieee754_rem_pio2f(float x, float *y)
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e0 = (ix>>23)-150; /* e0 = ilogb(|x|)-23; */
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SET_FLOAT_WORD(z, ix - ((int32_t)(e0<<23)));
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tx[0] = z;
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n = __kernel_rem_pio2(tx,ty,e0,1,1);
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n = __kernel_rem_pio2(tx,ty,e0,1,0);
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y[0] = ty[0];
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y[1] = ty[0] - y[0];
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if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
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@ -78,8 +78,13 @@ __FBSDID("$FreeBSD$");
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* Here is the description of some local variables:
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*
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* jk jk+1 is the initial number of terms of ipio2[] needed
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* in the computation. The recommended value is 2,3,4,
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* 6 for single, double, extended,and quad.
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* in the computation. The minimum and recommended value
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* for jk is 3,4,4,6 for single, double, extended, and quad.
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* jk+1 must be 2 larger than you might expect so that our
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* recomputation test works. (Up to 24 bits in the integer
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* part (the 24 bits of it that we compute) and 23 bits in
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* the fraction part may be lost to cancelation before we
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* recompute.)
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*
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* jz local integer variable indicating the number of
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* terms of ipio2[] used.
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@ -129,7 +134,7 @@ __FBSDID("$FreeBSD$");
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#include "math.h"
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#include "math_private.h"
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static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
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static const int init_jk[] = {3,4,4,6}; /* initial value for jk */
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/*
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* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
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