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GH-100485: Add extended accuracy test. Switch to faster fma() based variant. GH-101383)

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Raymond Hettinger 2023-01-28 06:29:21 -06:00 • committed by GitHub
parent db757f0e44
commit 84483aacc0
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2 changed files with 100 additions and 36 deletions
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53
Modules/mathmodule.c
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@ -2832,12 +2832,7 @@ long_add_would_overflow(long a, long b)
}
/*
Double and triple length extended precision floating point arithmetic
based on:
A Floating-Point Technique for Extending the Available Precision
by T. J. Dekker
https://csclub.uwaterloo.ca/~pbarfuss/dekker1971.pdf
Double and triple length extended precision algorithms from:
Accurate Sum and Dot Product
by Takeshi Ogita, Siegfried M. Rump, and Shin’Ichi Oishi
@ -2848,36 +2843,22 @@ based on:
typedef struct{ double hi; double lo; } DoubleLength;
static inline DoubleLength
twosum(double a, double b)
static DoubleLength
dl_sum(double a, double b)
{
// Rump Algorithm 3.1 Error-free transformation of the sum
/* Algorithm 3.1 Error-free transformation of the sum */
double x = a + b;
double z = x - a;
double y = (a - (x - z)) + (b - z);
return (DoubleLength) {x, y};
}
static inline DoubleLength
dl_split(double x) {
// Rump Algorithm 3.2 Error-free splitting of a floating point number
// Dekker (5.5) and (5.6).
double t = x * 134217729.0; // Veltkamp constant = 2.0 ** 27 + 1
double hi = t - (t - x);
double lo = x - hi;
return (DoubleLength) {hi, lo};
}
static inline DoubleLength
static DoubleLength
dl_mul(double x, double y)
{
// Dekker (5.12) and mul12()
DoubleLength xx = dl_split(x);
DoubleLength yy = dl_split(y);
double p = xx.hi * yy.hi;
double q = xx.hi * yy.lo + xx.lo * yy.hi;
double z = p + q;
double zz = p - z + q + xx.lo * yy.lo;
/* Algorithm 3.5. Error-free transformation of a product */
double z = x * y;
double zz = fma(x, y, -z);
return (DoubleLength) {z, zz};
}
@ -2885,21 +2866,21 @@ typedef struct { double hi; double lo; double tiny; } TripleLength;
static const TripleLength tl_zero = {0.0, 0.0, 0.0};
static inline TripleLength
tl_fma(TripleLength total, double x, double y)
static TripleLength
tl_fma(double x, double y, TripleLength total)
{
// Rump Algorithm 5.10 with K=3 and using SumKVert
/* Algorithm 5.10 with SumKVert for K=3 */
DoubleLength pr = dl_mul(x, y);
DoubleLength sm = twosum(total.hi, pr.hi);
DoubleLength r1 = twosum(total.lo, pr.lo);
DoubleLength r2 = twosum(r1.hi, sm.lo);
DoubleLength sm = dl_sum(total.hi, pr.hi);
DoubleLength r1 = dl_sum(total.lo, pr.lo);
DoubleLength r2 = dl_sum(r1.hi, sm.lo);
return (TripleLength) {sm.hi, r2.hi, total.tiny + r1.lo + r2.lo};
}
static inline double
static double
tl_to_d(TripleLength total)
{
DoubleLength last = twosum(total.lo, total.hi);
DoubleLength last = dl_sum(total.lo, total.hi);
return total.tiny + last.lo + last.hi;
}
@ -3066,7 +3047,7 @@ math_sumprod_impl(PyObject *module, PyObject *p, PyObject *q)
} else {
goto finalize_flt_path;
}
TripleLength new_flt_total = tl_fma(flt_total, flt_p, flt_q);
TripleLength new_flt_total = tl_fma(flt_p, flt_q, flt_total);
if (isfinite(new_flt_total.hi)) {
flt_total = new_flt_total;
flt_total_in_use = true;