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Speed-up and improve accuracy with Rump Algorithms (3.1) and (5.10) #101366

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Jan 27, 2023
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Speed-up and improve accuracy with Rump Algorithms (3.1) and (5.10) #101366

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37 changes: 16 additions & 21 deletions Modules/mathmodule.c
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Original file line number Diff line number Diff line change
Expand Up @@ -2851,17 +2851,17 @@ typedef struct{ double hi; double lo; } DoubleLength;
static inline DoubleLength
twosum(double a, double b)
{
double s = a + b;
double ap = s - b;
double bp = s - a;
double da = a - ap;
double db = b - bp;
double t = da + db;
return (DoubleLength) {s, t};
// Rump 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;
Expand All @@ -2871,7 +2871,7 @@ dl_split(double x) {
static inline DoubleLength
dl_mul(double x, double y)
{
/* Dekker mul12(). Section (5.12) */
// Dekker (5.12) and mul12()
DoubleLength xx = dl_split(x);
DoubleLength yy = dl_split(y);
double p = xx.hi * yy.hi;
Expand All @@ -2881,24 +2881,19 @@ dl_mul(double x, double y)
return (DoubleLength) {z, zz};
}

typedef struct{ double hi; double lo; double tiny; } TripleLength;
typedef struct { double hi; double lo; double tiny; } TripleLength;

static const TripleLength tl_zero = {0.0, 0.0, 0.0};

static inline TripleLength
tl_add(TripleLength total, double x)
tl_fma(TripleLength total, double x, double y)
{
DoubleLength s = twosum(x, total.hi);
DoubleLength t = twosum(s.lo, total.lo);
return (TripleLength) {s.hi, t.hi, t.lo + total.tiny};
}

static inline TripleLength
tl_fma(TripleLength total, double p, double q)
{
DoubleLength product = dl_mul(p, q);
total = tl_add(total, product.hi);
return tl_add(total, product.lo);
// Rump Algorithm 5.10 with K=3 and using SumKVert
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);
return (TripleLength) {sm.hi, r2.hi, total.tiny + r1.lo + r2.lo};
}

static inline double
Expand Down