[PATCH] log2 and log10 for wordsize-64
Andreas Jaeger
aj@suse.com
Mon May 14 18:17:00 GMT 2012
On 05/14/2012 05:31 PM, Adhemerval Zanella wrote:
> These were in my backlog for while. On x86_64 I observed an improvement of about 7%
> for log10 and about 20% for log2. And on PPC64 I observed an improvement of about
> 20% for log10 and about 30 for log2.
You copied from sysdeps/ieee754/dbl-64 and reformatted the file which
makes a diff difficult (unless using -w). Could you send a separate
patch that reformats the two files, please?
The patch itself is fine, just a few nits below.
> Tested on ppc64 and x86_64.
>
> ---
>
> 2012-05-14 Adhemerval Zanella<azanella@linux.vnet.ibm.com>
>
> * sysdeps/ieee754/dbl-64/wordsize-64/e_log10.c: New file.
> * sysdeps/ieee754/dbl-64/wordsize-64/e_log2.c: New file.
>
>
> diff --git a/sysdeps/ieee754/dbl-64/wordsize-64/e_log10.c b/sysdeps/ieee754/dbl-64/wordsize-64/e_log10.c
> new file mode 100644
> index 0000000..2562a49
> --- /dev/null
> +++ b/sysdeps/ieee754/dbl-64/wordsize-64/e_log10.c
> @@ -0,0 +1,88 @@
> +/* @(#)e_log10.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_log10(x)
> + * Return the base 10 logarithm of x
> + *
> + * Method :
> + * Let log10_2hi = leading 40 bits of log10(2) and
> + * log10_2lo = log10(2) - log10_2hi,
> + * ivln10 = 1/log(10) rounded.
> + * Then
> + * n = ilogb(x),
> + * if(n<0) n = n+1;
> + * x = scalbn(x,-n);
> + * log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
> + *
> + * Note 1:
> + * To guarantee log10(10**n)=n, where 10**n is normal, the rounding
> + * mode must set to Round-to-Nearest.
> + * Note 2:
> + * [1/log(10)] rounded to 53 bits has error .198 ulps;
> + * log10 is monotonic at all binary break points.
> + *
> + * Special cases:
> + * log10(x) is NaN with signal if x< 0;
> + * log10(+INF) is +INF with no signal; log10(0) is -INF with signal;
> + * log10(NaN) is that NaN with no signal;
> + * log10(10**N) = N for N=0,1,...,22.
> + *
> + * Constants:
> + * The hexadecimal values are the intended ones for the following constants.
> + * The decimal values may be used, provided that the compiler will convert
> + * from decimal to binary accurately enough to produce the hexadecimal values
> + * shown.
> + */
> +
> +#include<math.h>
> +#include<math_private.h>
> +
> +static const double two54 = 1.80143985094819840000e+16; /* 0x4350000000000000 */
> +static const double ivln10 = 4.34294481903251816668e-01; /* 0x3FDBCB7B1526E50E */
> +static const double log10_2hi = 3.01029995663611771306e-01; /* 0x3FD34413509F6000 */
> +static const double log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF311F12B36 */
> +
> +static const double zero = 0.0;
zero seems to be unused in this code.
> +
> +double
> +__ieee754_log10 (double x)
> +{
> + double y, z;
> + int64_t i, hx;
> + int32_t k;
> +
> + EXTRACT_WORDS64 (hx, x);
> +
> + k = 0;
> + if (hx< INT64_C(0x0010000000000000))
> + { /* x< 2**-1022 */
> + if (__builtin_expect ((hx& UINT64_C(0x7fffffffffffffff)) == 0, 0))
> + return -two54 / (x - x); /* log(+-0)=-inf */
> + if (__builtin_expect (hx< 0, 0))
> + return (x - x) / (x - x); /* log(-#) = NaN */
> + k -= 54;
> + x *= two54; /* subnormal number, scale up x */
> + EXTRACT_WORDS64 (hx, x);
> + }
> + /* scale up resulted in a NaN number */
> + if (__builtin_expect (hx>= UINT64_C(0x7ff0000000000000), 0))
> + return x + x;
> + k += (hx>> 52) - 1023;
> + i = ((uint64_t) k& UINT64_C(0x8000000000000000))>> 63;
> + hx = (hx& UINT64_C(0x000fffffffffffff)) | ((0x3ff - i)<< 52);
> + y = (double)(k+i);
> + INSERT_WORDS64 (x, hx);
> + z = y * log10_2lo + ivln10 * __ieee754_log (x);
> + return z + y * log10_2hi;
> +}
> +
> +strong_alias (__ieee754_log10, __log10_finite)
> diff --git a/sysdeps/ieee754/dbl-64/wordsize-64/e_log2.c b/sysdeps/ieee754/dbl-64/wordsize-64/e_log2.c
> new file mode 100644
> index 0000000..d532f1f
> --- /dev/null
> +++ b/sysdeps/ieee754/dbl-64/wordsize-64/e_log2.c
> @@ -0,0 +1,129 @@
> +/*
> + * ====================================================
> + * 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_log2(x)
> + * Return the logarithm to base 2 of x
> + *
> + * Method :
> + * 1. Argument Reduction: find k and f such that
> + * x = 2^k * (1+f),
> + * where sqrt(2)/2< 1+f< sqrt(2) .
> + *
> + * 2. Approximation of log(1+f).
> + * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
> + * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
> + * = 2s + s*R
> + * We use a special Reme algorithm on [0,0.1716] to generate
> + * a polynomial of degree 14 to approximate R The maximum error
> + * of this polynomial approximation is bounded by 2**-58.45. In
> + * other words,
> + * 2 4 6 8 10 12 14
> + * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
> + * (the values of Lg1 to Lg7 are listed in the program)
> + * and
> + * | 2 14 | -58.45
> + * | Lg1*s +...+Lg7*s - R(z) |<= 2
> + * | |
> + * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
> + * In order to guarantee error in log below 1ulp, we compute log
> + * by
> + * log(1+f) = f - s*(f - R) (if f is not too large)
> + * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
> + *
> + * 3. Finally, log(x) = k + log(1+f).
> + * = k+(f-(hfsq-(s*(hfsq+R))))
> + *
> + * Special cases:
> + * log2(x) is NaN with signal if x< 0 (including -INF) ;
> + * log2(+INF) is +INF; log(0) is -INF with signal;
> + * log2(NaN) is that NaN with no signal.
> + *
> + * Constants:
> + * The hexadecimal values are the intended ones for the following
> + * constants. The decimal values may be used, provided that the
> + * compiler will convert from decimal to binary accurately enough
> + * to produce the hexadecimal values shown.
> + */
> +
> +#include<math.h>
> +#include<math_private.h>
> +
> +static const double ln2 = 0.69314718055994530942;
> +static const double two1ln2 = 1.4426950408889636;
Please add as comment: two1ln2 = 1/ln2
> +static const double two54 = 1.80143985094819840000e+16; /* 4350000000000000 */
> +static const double Lg1 = 6.666666666666735130e-01; /* 3FE5555555555593 */
> +static const double Lg2 = 3.999999999940941908e-01; /* 3FD999999997FA04 */
> +static const double Lg3 = 2.857142874366239149e-01; /* 3FD2492494229359 */
> +static const double Lg4 = 2.222219843214978396e-01; /* 3FCC71C51D8E78AF */
> +static const double Lg5 = 1.818357216161805012e-01; /* 3FC7466496CB03DE */
> +static const double Lg6 = 1.531383769920937332e-01; /* 3FC39A09D078C69F */
> +static const double Lg7 = 1.479819860511658591e-01; /* 3FC2F112DF3E5244 */
> +
> +static const double zero = 0.0;
> +
> +double
> +__ieee754_log2 (double x)
> +{
> + double hfsq, f, s, z, R, w, t1, t2, dk;
> + int64_t hx, i, j;
> + int32_t k;
> +
> + EXTRACT_WORDS64 (hx, x);
> +
> + k = 0;
> + if (hx< INT64_C(0x0010000000000000))
> + { /* x< 2**-1022 */
> + if (__builtin_expect ((hx& UINT64_C(0x7fffffffffffffff)) == 0, 0))
> + return -two54 / (x - x); /* log(+-0)=-inf */
> + if (__builtin_expect (hx< 0, 0))
> + return (x - x) / (x - x); /* log(-#) = NaN */
> + k -= 54;
> + x *= two54; /* subnormal number, scale up x */
> + EXTRACT_WORDS64 (hx, x);
> + }
> + if (__builtin_expect (hx>= UINT64_C(0x7ff0000000000000), 0))
> + return x + x;
> + k += (hx>> 52) - 1023;
> + hx&= UINT64_C(0x000fffffffffffff);
> + i = (hx + UINT64_C(0x95f6400000000))& UINT64_C(0x10000000000000);
> + /* normalize x or x/2 */
> + INSERT_WORDS64 (x, hx | (i ^ UINT64_C(0x3ff0000000000000)));
> + k += (i>> 52);
> + dk = (double) k;
> + f = x - 1.0;
> + if ((UINT64_C(0x000fffffffffffff)& (2 + hx))< 3)
> + { /* |f|< 2**-20 */
> + if (f == zero)
> + return dk;
> + R = f * f * (0.5 - 0.33333333333333333 * f);
> + return dk - (R - f) * two1ln2;
> + }
> + s = f / (2.0 + f);
> + z = s * s;
> + i = hx - UINT64_C(0x6147a00000000);
> + w = z * z;
> + j = UINT64_C(0x6b85100000000) - hx;
> + t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
> + t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
> + i |= j;
> + R = t2 + t1;
> + if (i> 0)
> + {
> + hfsq = 0.5 * f * f;
> + return dk - ((hfsq - (s * (hfsq + R))) - f) * two1ln2;
> + }
> + else
> + {
> + return dk - ((s * (f - R)) - f) * two1ln2;
> + }
> +}
> +
> +strong_alias (__ieee754_log2, __log2_finite)
Andreas
--
Andreas Jaeger aj@{suse.com,opensuse.org} Twitter/Identica: jaegerandi
SUSE LINUX Products GmbH, Maxfeldstr. 5, 90409 Nürnberg, Germany
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