/*-*- mode:c;indent-tabs-mode:nil;c-basic-offset:2;tab-width:8;coding:utf-8 -*-│
│vi: set net ft=c ts=2 sts=2 sw=2 fenc=utf-8 :vi│
╞══════════════════════════════════════════════════════════════════════════════╡
│ Copyright 2023 Justine Alexandra Roberts Tunney │
│ │
│ Permission to use, copy, modify, and/or distribute this software for │
│ any purpose with or without fee is hereby granted, provided that the │
│ above copyright notice and this permission notice appear in all copies. │
│ │
│ THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL │
│ WARRANTIES WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED │
│ WARRANTIES OF MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE │
│ AUTHOR BE LIABLE FOR ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL │
│ DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR │
│ PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER │
│ TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR │
│ PERFORMANCE OF THIS SOFTWARE. │
╚─────────────────────────────────────────────────────────────────────────────*/
#include "libc/math.h"
#if LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
#include "libc/tinymath/internal.h"
asm(".ident\t\"\\n\\n\
OpenBSD libm (ISC License)\\n\
Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>\"");
asm(".ident\t\"\\n\\n\
Musl libc (MIT License)\\n\
Copyright 2005-2014 Rich Felker, et. al.\"");
asm(".include \"libc/disclaimer.inc\"");
// clang-format off
/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_expl.c */
/*
* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
*
* Permission to use, copy, modify, and distribute this software for any
* purpose with or without fee is hereby granted, provided that the above
* copyright notice and this permission notice appear in all copies.
*
* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
*/
/*
* Exponential function, long double precision
*
*
* SYNOPSIS:
*
* long double x, y, expl();
*
* y = expl( x );
*
*
* DESCRIPTION:
*
* Returns e (2.71828...) raised to the x power.
*
* Range reduction is accomplished by separating the argument
* into an integer k and fraction f such that
*
* x k f
* e = 2 e.
*
* A Pade' form of degree 5/6 is used to approximate exp(f) - 1
* in the basic range [-0.5 ln 2, 0.5 ln 2].
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-10000 50000 1.12e-19 2.81e-20
*
*
* Error amplification in the exponential function can be
* a serious matter. The error propagation involves
* exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
* which shows that a 1 lsb error in representing X produces
* a relative error of X times 1 lsb in the function.
* While the routine gives an accurate result for arguments
* that are exactly represented by a long double precision
* computer number, the result contains amplified roundoff
* error for large arguments not exactly represented.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* exp underflow x < MINLOG 0.0
* exp overflow x > MAXLOG MAXNUM
*
*/
static const long double P[3] = {
1.2617719307481059087798E-4L,
3.0299440770744196129956E-2L,
9.9999999999999999991025E-1L,
};
static const long double Q[4] = {
3.0019850513866445504159E-6L,
2.5244834034968410419224E-3L,
2.2726554820815502876593E-1L,
2.0000000000000000000897E0L,
};
static const long double
LN2HI = 6.9314575195312500000000E-1L,
LN2LO = 1.4286068203094172321215E-6L,
LOG2E = 1.4426950408889634073599E0L;
/**
* Returns 𝑒ˣ.
*/
long double expl(long double x)
{
long double px, xx;
int k;
if (isnan(x))
return x;
if (x > 11356.5234062941439488L) /* x > ln(2^16384 - 0.5) */
return x * 0x1p16383L;
if (x < -11399.4985314888605581L) /* x < ln(2^-16446) */
return -0x1p-16445L/x;
/* Express e**x = e**f 2**k
* = e**(f + k ln(2))
*/
px = floorl(LOG2E * x + 0.5);
k = px;
x -= px * LN2HI;
x -= px * LN2LO;
/* rational approximation of the fractional part:
* e**x = 1 + 2x P(x**2)/(Q(x**2) - x P(x**2))
*/
xx = x * x;
px = x * __polevll(xx, P, 2);
x = px/(__polevll(xx, Q, 3) - px);
x = 1.0 + 2.0 * x;
return scalbnl(x, k);
}
#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
#include "libc/tinymath/freebsd.internal.h"
asm(".ident\t\"\\n\\n\
FreeBSD libm (BSD-2 License)\\n\
Copyright (c) 2005-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.\"");
asm(".ident\t\"\\n\\n\
fdlibm (fdlibm license)\\n\
Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.\"");
asm(".include \"libc/disclaimer.inc\"");
// clang-format off
/*-
* 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.
*/
/*
* ld128 version of s_expl.c. See ../ld80/s_expl.c for most comments.
*/
/* 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 long double
/* log(2**16384 - 0.5) rounded towards zero: */
/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
o_threshold = 11356.523406294143949491931077970763428L,
/* log(2**(-16381-64-1)) rounded towards zero: */
u_threshold = -11433.462743336297878837243843452621503L;
static const double
/*
* ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must
* have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest
* bits zero so that multiplication of it by n is exact.
*/
INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */
L2 = -1.0253670638894731e-29; /* -0x1.9ff0342542fc3p-97 */
static const long double
/* 0x1.62e42fefa39ef35793c768000000p-8 */
L1 = 5.41521234812457272982212595914567508e-3L;
/*
* XXX values in hex in comments have been lost (or were never present)
* from here.
*/
static const long double
/*
* Domain [-0.002708, 0.002708], range ~[-2.4021e-38, 2.4234e-38]:
* |exp(x) - p(x)| < 2**-124.9
* (0.002708 is ln2/(2*INTERVALS) rounded up a little).
*
* XXX the coeffs aren't very carefully rounded, and I get 3.6 more bits.
*/
A2 = 0.5,
A3 = 1.66666666666666666666666666651085500e-1L,
A4 = 4.16666666666666666666666666425885320e-2L,
A5 = 8.33333333333333333334522877160175842e-3L,
A6 = 1.38888888888888888889971139751596836e-3L;
static const double
A7 = 1.9841269841269470e-4, /* 0x1.a01a01a019f91p-13 */
A8 = 2.4801587301585286e-5, /* 0x1.71de3ec75a967p-19 */
A9 = 2.7557324277411235e-6, /* 0x1.71de3ec75a967p-19 */
A10 = 2.7557333722375069e-7; /* 0x1.27e505ab56259p-22 */
/**
* Returns 𝑒ˣ.
*/
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 or -NaN */
RETURNP(-1 / x);
RETURNP(x + x); /* x is +Inf or +NaN */
}
if (x > o_threshold)
RETURNP(huge * huge);
if (x < u_threshold)
RETURNP(tiny * tiny);
} else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */
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. */
/*
* XXX sparc64 multiplication was so slow that scalbnl() is faster,
* but performance on aarch64 and riscv hasn't yet been quantified.
*/
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);
}
}
/*
* 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.
*
* Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2].
* Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear
* in both subintervals, so set T3 = 2**-5, which places the condition
* into the [T1, T3] interval.
*
* XXX we now do this more to (partially) balance the number of terms
* in the C and D polys than to avoid checking the condition in both
* intervals.
*
* XXX these micro-optimizations are excessive.
*/
static const double
T1 = -0.1659, /* ~-30.625/128 * log(2) */
T2 = 0.1659, /* ~30.625/128 * log(2) */
T3 = 0.03125;
/*
* Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]:
* |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03
*
* XXX none of the long double C or D coeffs except C10 is correctly printed.
* If you re-print their values in %.35Le format, the result is always
* different. For example, the last 2 digits in C3 should be 59, not 67.
* 67 is apparently from rounding an extra-precision value to 36 decimal
* places.
*/
static const long double
C3 = 1.66666666666666666666666666666666667e-1L,
C4 = 4.16666666666666666666666666666666645e-2L,
C5 = 8.33333333333333333333333333333371638e-3L,
C6 = 1.38888888888888888888888888891188658e-3L,
C7 = 1.98412698412698412698412697235950394e-4L,
C8 = 2.48015873015873015873015112487849040e-5L,
C9 = 2.75573192239858906525606685484412005e-6L,
C10 = 2.75573192239858906612966093057020362e-7L,
C11 = 2.50521083854417203619031960151253944e-8L,
C12 = 2.08767569878679576457272282566520649e-9L,
C13 = 1.60590438367252471783548748824255707e-10L;
/*
* XXX this has 1 more coeff than needed.
* XXX can start the double coeffs but not the double mults at C10.
* With my coeffs (C10-C17 double; s = best_s):
* Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]:
* |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
*/
static const double
C14 = 1.1470745580491932e-11, /* 0x1.93974a81dae30p-37 */
C15 = 7.6471620181090468e-13, /* 0x1.ae7f3820adab1p-41 */
C16 = 4.7793721460260450e-14, /* 0x1.ae7cd18a18eacp-45 */
C17 = 2.8074757356658877e-15, /* 0x1.949992a1937d9p-49 */
C18 = 1.4760610323699476e-16; /* 0x1.545b43aabfbcdp-53 */
/*
* Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]:
* |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44
*/
static const long double
D3 = 1.66666666666666666666666666666682245e-1L,
D4 = 4.16666666666666666666666666634228324e-2L,
D5 = 8.33333333333333333333333364022244481e-3L,
D6 = 1.38888888888888888888887138722762072e-3L,
D7 = 1.98412698412698412699085805424661471e-4L,
D8 = 2.48015873015873015687993712101479612e-5L,
D9 = 2.75573192239858944101036288338208042e-6L,
D10 = 2.75573192239853161148064676533754048e-7L,
D11 = 2.50521083855084570046480450935267433e-8L,
D12 = 2.08767569819738524488686318024854942e-9L,
D13 = 1.60590442297008495301927448122499313e-10L;
/*
* XXX this has 1 more coeff than needed.
* XXX can start the double coeffs but not the double mults at D11.
* With my coeffs (D11-D16 double):
* Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]:
* |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
*/
static const double
D14 = 1.1470726176204336e-11, /* 0x1.93971dc395d9ep-37 */
D15 = 7.6478532249581686e-13, /* 0x1.ae892e3D16fcep-41 */
D16 = 4.7628892832607741e-14, /* 0x1.ad00Dfe41feccp-45 */
D17 = 3.0524857220358650e-15; /* 0x1.D7e8d886Df921p-49 */
/**
* Returns 𝑒ˣ-1.
*/
long double
expm1l(long double x)
{
union IEEEl2bits u, v;
long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi;
long double x_lo, x2;
double dr, dx, fn, r2;
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 + 7) { /* |x| >= 128 or x is NaN */
if (ix == BIAS + LDBL_MAX_EXP) {
if (hx & 0x8000) /* x is -Inf or -NaN */
RETURNP(-1 / x - 1);
RETURNP(x + x); /* x is +Inf or +NaN */
}
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 <= -128 */
RETURN2P(tiny, -1); /* good for x < -114ln2 - eps */
}
ENTERI();
if (T1 < x && x < T2) {
x2 = x * x;
dx = x;
if (x < T3) {
if (ix < BIAS - 113) { /* |x| < 0x1p-113 */
/* x (rounded) with inexact if x != 0: */
RETURNPI(x == 0 ? x :
(0x1p200 * x + fabsl(x)) * 0x1p-200);
}
q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 +
x * (C7 + x * (C8 + x * (C9 + x * (C10 +
x * (C11 + x * (C12 + x * (C13 +
dx * (C14 + dx * (C15 + dx * (C16 +
dx * (C17 + dx * C18))))))))))))));
} else {
q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 +
x * (D7 + x * (D8 + x * (D9 + x * (D10 +
x * (D11 + x * (D12 + x * (D13 +
dx * (D14 + dx * (D15 + dx * (D16 +
dx * D17)))))))))))));
}
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 = rnint((double)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) = kExplData[n2] * expl(r1 + r2).
*/
dr = r;
q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 +
dr * (A7 + dr * (A8 + dr * (A9 + dr * A10))))))));
t = kExplData[n2].lo + kExplData[n2].hi;
if (k == 0) {
t = SUM2P(kExplData[n2].hi - 1, kExplData[n2].lo * (r1 + 1) + t * q +
kExplData[n2].hi * r1);
RETURNI(t);
}
if (k == -1) {
t = SUM2P(kExplData[n2].hi - 2, kExplData[n2].lo * (r1 + 1) + t * q +
kExplData[n2].hi * r1);
RETURNI(t / 2);
}
if (k < -7) {
t = SUM2P(kExplData[n2].hi, kExplData[n2].lo + t * (q + r1));
RETURNI(t * twopk - 1);
}
if (k > 2 * LDBL_MANT_DIG - 1) {
t = SUM2P(kExplData[n2].hi, kExplData[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(kExplData[n2].hi, kExplData[n2].lo - twomk + t * (q + r1));
else
t = SUM2P(kExplData[n2].hi - twomk, kExplData[n2].lo + t * (q + r1));
RETURNI(t * twopk);
}
#endif