/*-
* SPDX-License-Identifier: BSD-2-Clause-FreeBSD
*
* Copyright (c) 2009-2013 Steven G. Kargl
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice unmodified, this list of conditions, and the following
* disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*
* Optimized by Bruce D. Evans.
*/
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
/**
* Compute the exponential of x for Intel 80-bit format. This is based on:
*
* PTP Tang, "Table-driven implementation of the exponential function
* in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15,
* 144-157 (1989).
*
* where the 32 table entries have been expanded to INTERVALS (see below).
*/
#include <float.h>
#ifdef __i386__
#include <ieeefp.h>
#endif
#include "fpmath.h"
#include "math.h"
#include "math_private.h"
#include "k_expl.h"
/* XXX Prevent compilers from erroneously constant folding these: */
static const volatile long double
huge = 0x1p10000L,
tiny = 0x1p-10000L;
static const long double
twom10000 = 0x1p-10000L;
static const union IEEEl2bits
/* log(2**16384 - 0.5) rounded towards zero: */
/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L),
#define o_threshold (o_thresholdu.e)
/* log(2**(-16381-64-1)) rounded towards zero: */
u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L);
#define u_threshold (u_thresholdu.e)
long double
expl(long double x)
{
union IEEEl2bits u;
long double hi, lo, t, twopk;
int k;
uint16_t hx, ix;
DOPRINT_START(&x);
/* Filter out exceptional cases. */
u.e = x;
hx = u.xbits.expsign;
ix = hx & 0x7fff;
if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */
if (ix == BIAS + LDBL_MAX_EXP) {
if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */
RETURNP(-1 / x);
RETURNP(x + x); /* x is +Inf, +NaN or unsupported */
}
if (x > o_threshold)
RETURNP(huge * huge);
if (x < u_threshold)
RETURNP(tiny * tiny);
} else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */
RETURN2P(1, x); /* 1 with inexact iff x != 0 */
}
ENTERI();
twopk = 1;
__k_expl(x, &hi, &lo, &k);
t = SUM2P(hi, lo);
/* Scale by 2**k. */
if (k >= LDBL_MIN_EXP) {
if (k == LDBL_MAX_EXP)
RETURNI(t * 2 * 0x1p16383L);
SET_LDBL_EXPSIGN(twopk, BIAS + k);
RETURNI(t * twopk);
} else {
SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000);
RETURNI(t * twopk * twom10000);
}
}
/**
* Compute expm1l(x) for Intel 80-bit format. This is based on:
*
* PTP Tang, "Table-driven implementation of the Expm1 function
* in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18,
* 211-222 (1992).
*/
/*
* Our T1 and T2 are chosen to be approximately the points where method
* A and method B have the same accuracy. Tang's T1 and T2 are the
* points where method A's accuracy changes by a full bit. For Tang,
* this drop in accuracy makes method A immediately less accurate than
* method B, but our larger INTERVALS makes method A 2 bits more
* accurate so it remains the most accurate method significantly
* closer to the origin despite losing the full bit in our extended
* range for it.
*/
static const double
T1 = -0.1659, /* ~-30.625/128 * log(2) */
T2 = 0.1659; /* ~30.625/128 * log(2) */
/*
* Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]:
* |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6
*
* XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits,
* but unlike for ld128 we can't drop any terms.
*/
static const union IEEEl2bits
B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L),
B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L);
static const double
B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */
B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */
B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */
B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */
B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */
B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */
B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */
B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */
long double
expm1l(long double x)
{
union IEEEl2bits u, v;
long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi;
long double x_lo, x2, z;
long double x4;
int k, n, n2;
uint16_t hx, ix;
DOPRINT_START(&x);
/* Filter out exceptional cases. */
u.e = x;
hx = u.xbits.expsign;
ix = hx & 0x7fff;
if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */
if (ix == BIAS + LDBL_MAX_EXP) {
if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */
RETURNP(-1 / x - 1);
RETURNP(x + x); /* x is +Inf, +NaN or unsupported */
}
if (x > o_threshold)
RETURNP(huge * huge);
/*
* expm1l() never underflows, but it must avoid
* unrepresentable large negative exponents. We used a
* much smaller threshold for large |x| above than in
* expl() so as to handle not so large negative exponents
* in the same way as large ones here.
*/
if (hx & 0x8000) /* x <= -64 */
RETURN2P(tiny, -1); /* good for x < -65ln2 - eps */
}
ENTERI();
if (T1 < x && x < T2) {
if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */
/* x (rounded) with inexact if x != 0: */
RETURNPI(x == 0 ? x :
(0x1p100 * x + fabsl(x)) * 0x1p-100);
}
x2 = x * x;
x4 = x2 * x2;
q = x4 * (x2 * (x4 *
/*
* XXX the number of terms is no longer good for
* pairwise grouping of all except B3, and the
* grouping is no longer from highest down.
*/
(x2 * B12 + (x * B11 + B10)) +
(x2 * (x * B9 + B8) + (x * B7 + B6))) +
(x * B5 + B4.e)) + x2 * x * B3.e;
x_hi = (float)x;
x_lo = x - x_hi;
hx2_hi = x_hi * x_hi / 2;
hx2_lo = x_lo * (x + x_hi) / 2;
if (ix >= BIAS - 7)
RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q);
else
RETURN2PI(x, hx2_lo + q + hx2_hi);
}
/* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
fn = rnintl(x * INV_L);
n = irint(fn);
n2 = (unsigned)n % INTERVALS;
k = n >> LOG2_INTERVALS;
r1 = x - fn * L1;
r2 = fn * -L2;
r = r1 + r2;
/* Prepare scale factor. */
v.e = 1;
v.xbits.expsign = BIAS + k;
twopk = v.e;
/*
* Evaluate lower terms of
* expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
*/
z = r * r;
q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
t = (long double)tbl[n2].lo + tbl[n2].hi;
if (k == 0) {
t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q +
tbl[n2].hi * r1);
RETURNI(t);
}
if (k == -1) {
t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q +
tbl[n2].hi * r1);
RETURNI(t / 2);
}
if (k < -7) {
t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
RETURNI(t * twopk - 1);
}
if (k > 2 * LDBL_MANT_DIG - 1) {
t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
if (k == LDBL_MAX_EXP)
RETURNI(t * 2 * 0x1p16383L - 1);
RETURNI(t * twopk - 1);
}
v.xbits.expsign = BIAS - k;
twomk = v.e;
if (k > LDBL_MANT_DIG - 1)
t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1));
else
t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1));
RETURNI(t * twopk);
}