137 lines
4.1 KiB
C
137 lines
4.1 KiB
C
/*
|
|
* Double-precision log(1+x) function.
|
|
*
|
|
* Copyright (c) 2022-2023, Arm Limited.
|
|
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
|
|
*/
|
|
|
|
#include "estrin.h"
|
|
#include "math_config.h"
|
|
#include "pl_sig.h"
|
|
#include "pl_test.h"
|
|
|
|
#define Ln2Hi 0x1.62e42fefa3800p-1
|
|
#define Ln2Lo 0x1.ef35793c76730p-45
|
|
#define HfRt2Top 0x3fe6a09e /* top32(asuint64(sqrt(2)/2)). */
|
|
#define OneMHfRt2Top \
|
|
0x00095f62 /* top32(asuint64(1)) - top32(asuint64(sqrt(2)/2)). */
|
|
#define OneTop12 0x3ff
|
|
#define BottomMask 0xffffffff
|
|
#define OneMHfRt2 0x3fd2bec333018866
|
|
#define Rt2MOne 0x3fda827999fcef32
|
|
#define AbsMask 0x7fffffffffffffff
|
|
#define ExpM63 0x3c00
|
|
#define C(i) __log1p_data.coeffs[i]
|
|
|
|
static inline double
|
|
eval_poly (double f)
|
|
{
|
|
double f2 = f * f;
|
|
double f4 = f2 * f2;
|
|
double f8 = f4 * f4;
|
|
return ESTRIN_18 (f, f2, f4, f8, f8 * f8, C);
|
|
}
|
|
|
|
/* log1p approximation using polynomial on reduced interval. Largest
|
|
observed errors are near the lower boundary of the region where k
|
|
is 0.
|
|
Maximum measured error: 1.75ULP.
|
|
log1p(-0x1.2e1aea97b3e5cp-2) got -0x1.65fb8659a2f9p-2
|
|
want -0x1.65fb8659a2f92p-2. */
|
|
double
|
|
log1p (double x)
|
|
{
|
|
uint64_t ix = asuint64 (x);
|
|
uint64_t ia = ix & AbsMask;
|
|
uint32_t ia16 = ia >> 48;
|
|
|
|
/* Handle special cases first. */
|
|
if (unlikely (ia16 >= 0x7ff0 || ix >= 0xbff0000000000000
|
|
|| ix == 0x8000000000000000))
|
|
{
|
|
if (ix == 0x8000000000000000 || ix == 0x7ff0000000000000)
|
|
{
|
|
/* x == -0 => log1p(x) = -0.
|
|
x == Inf => log1p(x) = Inf. */
|
|
return x;
|
|
}
|
|
if (ix == 0xbff0000000000000)
|
|
{
|
|
/* x == -1 => log1p(x) = -Inf. */
|
|
return __math_divzero (-1);
|
|
;
|
|
}
|
|
if (ia16 >= 0x7ff0)
|
|
{
|
|
/* x == +/-NaN => log1p(x) = NaN. */
|
|
return __math_invalid (asdouble (ia));
|
|
}
|
|
/* x < -1 => log1p(x) = NaN.
|
|
x == -Inf => log1p(x) = NaN. */
|
|
return __math_invalid (x);
|
|
}
|
|
|
|
/* With x + 1 = t * 2^k (where t = f + 1 and k is chosen such that f
|
|
is in [sqrt(2)/2, sqrt(2)]):
|
|
log1p(x) = k*log(2) + log1p(f).
|
|
|
|
f may not be representable exactly, so we need a correction term:
|
|
let m = round(1 + x), c = (1 + x) - m.
|
|
c << m: at very small x, log1p(x) ~ x, hence:
|
|
log(1+x) - log(m) ~ c/m.
|
|
|
|
We therefore calculate log1p(x) by k*log2 + log1p(f) + c/m. */
|
|
|
|
uint64_t sign = ix & ~AbsMask;
|
|
if (ia <= OneMHfRt2 || (!sign && ia <= Rt2MOne))
|
|
{
|
|
if (unlikely (ia16 <= ExpM63))
|
|
{
|
|
/* If exponent of x <= -63 then shortcut the polynomial and avoid
|
|
underflow by just returning x, which is exactly rounded in this
|
|
region. */
|
|
return x;
|
|
}
|
|
/* If x is in [sqrt(2)/2 - 1, sqrt(2) - 1] then we can shortcut all the
|
|
logic below, as k = 0 and f = x and therefore representable exactly.
|
|
All we need is to return the polynomial. */
|
|
return fma (x, eval_poly (x) * x, x);
|
|
}
|
|
|
|
/* Obtain correctly scaled k by manipulation in the exponent. */
|
|
double m = x + 1;
|
|
uint64_t mi = asuint64 (m);
|
|
uint32_t u = (mi >> 32) + OneMHfRt2Top;
|
|
int32_t k = (int32_t) (u >> 20) - OneTop12;
|
|
|
|
/* Correction term c/m. */
|
|
double cm = (x - (m - 1)) / m;
|
|
|
|
/* Reduce x to f in [sqrt(2)/2, sqrt(2)]. */
|
|
uint32_t utop = (u & 0x000fffff) + HfRt2Top;
|
|
uint64_t u_red = ((uint64_t) utop << 32) | (mi & BottomMask);
|
|
double f = asdouble (u_red) - 1;
|
|
|
|
/* Approximate log1p(x) on the reduced input using a polynomial. Because
|
|
log1p(0)=0 we choose an approximation of the form:
|
|
x + C0*x^2 + C1*x^3 + C2x^4 + ...
|
|
Hence approximation has the form f + f^2 * P(f)
|
|
where P(x) = C0 + C1*x + C2x^2 + ... */
|
|
double p = fma (f, eval_poly (f) * f, f);
|
|
|
|
double kd = k;
|
|
double y = fma (Ln2Lo, kd, cm);
|
|
return y + fma (Ln2Hi, kd, p);
|
|
}
|
|
|
|
PL_SIG (S, D, 1, log1p, -0.9, 10.0)
|
|
PL_TEST_ULP (log1p, 1.26)
|
|
PL_TEST_INTERVAL (log1p, -10.0, 10.0, 10000)
|
|
PL_TEST_INTERVAL (log1p, 0.0, 0x1p-23, 50000)
|
|
PL_TEST_INTERVAL (log1p, 0x1p-23, 0.001, 50000)
|
|
PL_TEST_INTERVAL (log1p, 0.001, 1.0, 50000)
|
|
PL_TEST_INTERVAL (log1p, 0.0, -0x1p-23, 50000)
|
|
PL_TEST_INTERVAL (log1p, -0x1p-23, -0.001, 50000)
|
|
PL_TEST_INTERVAL (log1p, -0.001, -1.0, 50000)
|
|
PL_TEST_INTERVAL (log1p, -1.0, inf, 5000)
|