[v4,02/12] math: Use an improved algorithm for hypot (dbl-64)
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From: Wilco Dijkstra <Wilco.Dijkstra@arm.com>
This implementation is based on the 'An Improved Algorithm for
hypot(a,b)' by Carlos F. Borges [1] using the MyHypot3 with the
following changes:
- Handle qNaN and sNaN.
- Tune the 'widely varying operands' to avoid spurious underflow
due the multiplication and fix the return value for upwards
rounding mode.
- Handle required underflow exception for denormal results.
The main advantage of the new algorithm is its precision: with a
random 1e9 input pairs in the range of [DBL_MIN, DBL_MAX], glibc
current implementation shows around 0.34% results with an error of
1 ulp (3424869 results) while the new implementation only shows
0.002% of total (18851).
The performance result are also only slight worse than current
implementation. On x86_64 (Ryzen 5900X) with gcc 12:
Before:
"hypot": {
"workload-random": {
"duration": 3.73319e+09,
"iterations": 1.12e+08,
"reciprocal-throughput": 22.8737,
"latency": 43.7904,
"max-throughput": 4.37184e+07,
"min-throughput": 2.28361e+07
}
}
After:
"hypot": {
"workload-random": {
"duration": 3.7597e+09,
"iterations": 9.8e+07,
"reciprocal-throughput": 23.7547,
"latency": 52.9739,
"max-throughput": 4.2097e+07,
"min-throughput": 1.88772e+07
}
}
Co-Authored-By: Adhemerval Zanella <adhemerval.zanella@linaro.org>
Checked on x86_64-linux-gnu and aarch64-linux-gnu.
[1] https://arxiv.org/pdf/1904.09481.pdf
---
sysdeps/ieee754/dbl-64/e_hypot.c | 235 ++++++++++++-------------------
1 file changed, 92 insertions(+), 143 deletions(-)
Comments
Hi Adhemeval,
+ /* If ax is huge, scale both inputs down. */
+ if (__glibc_unlikely (ax > LARGE_VAL))
{
+ if (__glibc_unlikely (ay <= ax * EPS))
+ return ax + ay;
+
+ return kernel (ax * SCALE, ay * SCALE) / SCALE;
}
Both of these returns can overflow and thus should have a math_narrow_eval
(only the case where we can't overflow or underflow doesn't). OK with that change.
Cheers,
Wilco
On 03/12/2021 10:41, Wilco Dijkstra wrote:
> Hi Adhemeval,
>
> + /* If ax is huge, scale both inputs down. */
> + if (__glibc_unlikely (ax > LARGE_VAL))
> {
> + if (__glibc_unlikely (ay <= ax * EPS))
> + return ax + ay;
> +
> + return kernel (ax * SCALE, ay * SCALE) / SCALE;
> }
>
> Both of these returns can overflow and thus should have a math_narrow_eval
> (only the case where we can't overflow or underflow doesn't). OK with that change.
Ack.
@@ -1,164 +1,113 @@
-/* @(#)e_hypot.c 5.1 93/09/24 */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-
-/* __ieee754_hypot(x,y)
- *
- * Method :
- * If (assume round-to-nearest) z=x*x+y*y
- * has error less than sqrt(2)/2 ulp, than
- * sqrt(z) has error less than 1 ulp (exercise).
- *
- * So, compute sqrt(x*x+y*y) with some care as
- * follows to get the error below 1 ulp:
- *
- * Assume x>y>0;
- * (if possible, set rounding to round-to-nearest)
- * 1. if x > 2y use
- * x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
- * where x1 = x with lower 32 bits cleared, x2 = x-x1; else
- * 2. if x <= 2y use
- * t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
- * where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1,
- * y1= y with lower 32 bits chopped, y2 = y-y1.
- *
- * NOTE: scaling may be necessary if some argument is too
- * large or too tiny
- *
- * Special cases:
- * hypot(x,y) is INF if x or y is +INF or -INF; else
- * hypot(x,y) is NAN if x or y is NAN.
- *
- * Accuracy:
- * hypot(x,y) returns sqrt(x^2+y^2) with error less
- * than 1 ulps (units in the last place)
- */
+/* Euclidean distance function. Double/Binary64 version.
+ Copyright (C) 2021 Free Software Foundation, Inc.
+ This file is part of the GNU C Library.
+
+ The GNU C Library is free software; you can redistribute it and/or
+ modify it under the terms of the GNU Lesser General Public
+ License as published by the Free Software Foundation; either
+ version 2.1 of the License, or (at your option) any later version.
+
+ The GNU C Library is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ Lesser General Public License for more details.
+
+ You should have received a copy of the GNU Lesser General Public
+ License along with the GNU C Library; if not, see
+ <https://www.gnu.org/licenses/>. */
+
+/* The implementation uses a correction based on 'An Improved Algorithm for
+ hypot(a,b)' by Carlos F. Borges [1] usingthe MyHypot3 with the following
+ changes:
+
+ - Handle qNaN and sNaN.
+ - Tune the 'widely varying operands' to avoid spurious underflow
+ due the multiplication and fix the return value for upwards
+ rounding mode.
+ - Handle required underflow exception for subnormal results.
+
+ The expected ULP is ~0.792.
+
+ [1] https://arxiv.org/pdf/1904.09481.pdf */
#include <math.h>
#include <math_private.h>
#include <math-underflow.h>
+#include <math-narrow-eval.h>
#include <libm-alias-finite.h>
+#include "math_config.h"
-double
-__ieee754_hypot (double x, double y)
+#define SCALE 0x1p-600
+#define LARGE_VAL 0x1p+511
+#define TINY_VAL 0x1p-459
+#define EPS 0x1p-54
+
+/* Hypot kernel. The inputs must be adjusted so that ax >= ay >= 0
+ and squaring ax, ay and (ax - ay) does not overflow or underflow. */
+static inline double
+kernel (double ax, double ay)
{
- double a, b, t1, t2, y1, y2, w;
- int32_t j, k, ha, hb;
-
- GET_HIGH_WORD (ha, x);
- ha &= 0x7fffffff;
- GET_HIGH_WORD (hb, y);
- hb &= 0x7fffffff;
- if (hb > ha)
+ double t1, t2;
+ double h = sqrt (ax * ax + ay * ay);
+ if (h <= 2.0 * ay)
{
- a = y; b = x; j = ha; ha = hb; hb = j;
+ double delta = h - ay;
+ t1 = ax * (2.0 * delta - ax);
+ t2 = (delta - 2.0 * (ax - ay)) * delta;
}
else
{
- a = x; b = y;
- }
- SET_HIGH_WORD (a, ha); /* a <- |a| */
- SET_HIGH_WORD (b, hb); /* b <- |b| */
- if ((ha - hb) > 0x3c00000)
- {
- return a + b;
- } /* x/y > 2**60 */
- k = 0;
- if (__glibc_unlikely (ha > 0x5f300000)) /* a>2**500 */
- {
- if (ha >= 0x7ff00000) /* Inf or NaN */
- {
- uint32_t low;
- w = a + b; /* for sNaN */
- if (issignaling (a) || issignaling (b))
- return w;
- GET_LOW_WORD (low, a);
- if (((ha & 0xfffff) | low) == 0)
- w = a;
- GET_LOW_WORD (low, b);
- if (((hb ^ 0x7ff00000) | low) == 0)
- w = b;
- return w;
- }
- /* scale a and b by 2**-600 */
- ha -= 0x25800000; hb -= 0x25800000; k += 600;
- SET_HIGH_WORD (a, ha);
- SET_HIGH_WORD (b, hb);
- }
- if (__builtin_expect (hb < 0x23d00000, 0)) /* b < 2**-450 */
- {
- if (hb <= 0x000fffff) /* subnormal b or 0 */
- {
- uint32_t low;
- GET_LOW_WORD (low, b);
- if ((hb | low) == 0)
- return a;
- t1 = 0;
- SET_HIGH_WORD (t1, 0x7fd00000); /* t1=2^1022 */
- b *= t1;
- a *= t1;
- k -= 1022;
- GET_HIGH_WORD (ha, a);
- GET_HIGH_WORD (hb, b);
- if (hb > ha)
- {
- t1 = a;
- a = b;
- b = t1;
- j = ha;
- ha = hb;
- hb = j;
- }
- }
- else /* scale a and b by 2^600 */
- {
- ha += 0x25800000; /* a *= 2^600 */
- hb += 0x25800000; /* b *= 2^600 */
- k -= 600;
- SET_HIGH_WORD (a, ha);
- SET_HIGH_WORD (b, hb);
- }
+ double delta = h - ax;
+ t1 = 2.0 * delta * (ax - 2.0 * ay);
+ t2 = (4.0 * delta - ay) * ay + delta * delta;
}
- /* medium size a and b */
- w = a - b;
- if (w > b)
+
+ h -= (t1 + t2) / (2.0 * h);
+ return h;
+}
+
+double
+__ieee754_hypot (double x, double y)
+{
+ if (!isfinite(x) || !isfinite(y))
{
- t1 = 0;
- SET_HIGH_WORD (t1, ha);
- t2 = a - t1;
- w = sqrt (t1 * t1 - (b * (-b) - t2 * (a + t1)));
+ if ((isinf (x) || isinf (y))
+ && !issignaling_inline (x) && !issignaling_inline (y))
+ return INFINITY;
+ return x + y;
}
- else
+
+ x = fabs (x);
+ y = fabs (y);
+
+ double ax = x < y ? y : x;
+ double ay = x < y ? x : y;
+
+ /* If ax is huge, scale both inputs down. */
+ if (__glibc_unlikely (ax > LARGE_VAL))
{
- a = a + a;
- y1 = 0;
- SET_HIGH_WORD (y1, hb);
- y2 = b - y1;
- t1 = 0;
- SET_HIGH_WORD (t1, ha + 0x00100000);
- t2 = a - t1;
- w = sqrt (t1 * y1 - (w * (-w) - (t1 * y2 + t2 * b)));
+ if (__glibc_unlikely (ay <= ax * EPS))
+ return ax + ay;
+
+ return kernel (ax * SCALE, ay * SCALE) / SCALE;
}
- if (k != 0)
+
+ /* If ay is tiny, scale both inputs up. */
+ if (__glibc_unlikely (ay < TINY_VAL))
{
- uint32_t high;
- t1 = 1.0;
- GET_HIGH_WORD (high, t1);
- SET_HIGH_WORD (t1, high + (k << 20));
- w *= t1;
- math_check_force_underflow_nonneg (w);
- return w;
+ if (__glibc_unlikely (ax >= ay / EPS))
+ return math_narrow_eval (ax + ay);
+
+ ax = math_narrow_eval (kernel (ax / SCALE, ay / SCALE) * SCALE);
+ math_check_force_underflow_nonneg (ax);
+ return ax;
}
- else
- return w;
+
+ /* Common case: ax is not huge and ay is not tiny. */
+ if (__glibc_unlikely (ay <= ax * EPS))
+ return ax + ay;
+
+ return kernel (ax, ay);
}
#ifndef __ieee754_hypot
libm_alias_finite (__ieee754_hypot, __hypot)