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ld80 and ld128 implementations of expm1l(). This code started life

Browse Source 
as a fairly faithful implementation of the algorithm found in

PTP Tang, "Table-driven implementation of the Expm1 function
in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18,
211-222 (1992).

Over the last 18-24 months, the code has under gone significant
optimization and testing.

Reviewed by:	bde
Obtained from:	bde (most of the optimizations)
This commit is contained in:
Steve Kargl 2013-06-03 19:51:32 +00:00
parent 904e385eba
commit 3ffff4bad5
6 changed files with 373 additions and 4 deletions
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1
lib/msun/Symbol.map
View File

@ -262,6 +262,7 @@ FBSD_1.3 {
ctanh;
ctanhf;
expl;
expm1l;
log10l;
log1pl;
log2l;

195
lib/msun/ld128/s_expl.c
View File

@ -298,3 +298,198 @@ expl(long double x)
RETURNI(t * twopkp10000 * 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.
*/
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
*/
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;
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;
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 */
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;
/* 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 */
return (-1 / x - 1);
return (x + x); /* x is +Inf or +NaN */
}
if (x > o_threshold)
return (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 */
return (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: */
RETURNI(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)
RETURNI(hx2_lo + x_lo + q + (hx2_hi + x_hi));
else
RETURNI(hx2_lo + q + hx2_hi + x);
}
/* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
/* Use a specialized rint() to get fn. Assume round-to-nearest. */
fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52;
#if defined(HAVE_EFFICIENT_IRINT)
n = irint(fn);
#else
n = (int)fn;
#endif
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).
*/
dr = r;
q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 +
dr * (A7 + dr * (A8 + dr * (A9 + dr * A10))))))));
t = tbl[n2].lo + tbl[n2].hi;
if (k == 0) {
t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 +
(tbl[n2].hi - 1);
RETURNI(t);
}
if (k == -1) {
t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 +
(tbl[n2].hi - 2);
RETURNI(t / 2);
}
if (k < -7) {
t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi;
RETURNI(t * twopk - 1);
}
if (k > 2 * LDBL_MANT_DIG - 1) {
t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi;
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 = tbl[n2].lo - twomk + t * (q + r1) + tbl[n2].hi;
else
t = tbl[n2].lo + t * (q + r1) + (tbl[n2].hi - twomk);
RETURNI(t * twopk);
}

165
lib/msun/ld80/s_expl.c
View File

@ -302,3 +302,168 @@ expl(long double x)
RETURNI(t * twopkp10000 * 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 ~[-1.2027e-22, 3.4417e-22]:
* |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.2
*/
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;
/* 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 */
return (-1 / x - 1);
return (x + x); /* x is +Inf, +NaN or unsupported */
}
if (x > o_threshold)
return (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 */
return (tiny - 1); /* good for x < -65ln2 - eps */
}
ENTERI();
if (T1 < x && x < T2) {
if (ix < BIAS - 64) { /* |x| < 0x1p-64 (includes pseudos) */
/* x (rounded) with inexact if x != 0: */
RETURNI(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)
RETURNI(hx2_lo + x_lo + q + (hx2_hi + x_hi));
else
RETURNI(hx2_lo + q + hx2_hi + x);
}
/* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
/* Use a specialized rint() to get fn. Assume round-to-nearest. */
fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
#if defined(HAVE_EFFICIENT_IRINTL)
n = irintl(fn);
#elif defined(HAVE_EFFICIENT_IRINT)
n = irint(fn);
#else
n = (int)fn;
#endif
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 = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 +
(tbl[n2].hi - 1);
RETURNI(t);
}
if (k == -1) {
t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 +
(tbl[n2].hi - 2);
RETURNI(t / 2);
}
if (k < -7) {
t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi;
RETURNI(t * twopk - 1);
}
if (k > 2 * LDBL_MANT_DIG - 1) {
t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi;
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 = tbl[n2].lo - twomk + t * (q + r1) + tbl[n2].hi;
else
t = tbl[n2].lo + t * (q + r1) + (tbl[n2].hi - twomk);
RETURNI(t * twopk);
}

10
lib/msun/man/exp.3
View File

@ -28,7 +28,7 @@
.\" from: @(#)exp.3 6.12 (Berkeley) 7/31/91
.\" $FreeBSD$
.\"
.Dd July 10, 2012
.Dd June 3, 2013
.Dt EXP 3
.Os
.Sh NAME
@ -41,6 +41,7 @@
.Nm exp2l ,
.Nm expm1 ,
.Nm expm1f ,
.Nm expm1l ,
.Nm pow ,
.Nm powf
.Nd exponential and power functions
@ -64,6 +65,8 @@
.Fn expm1 "double x"
.Ft float
.Fn expm1f "float x"
.Ft long double
.Fn expm1l "long double x"
.Ft double
.Fn pow "double x" "double y"
.Ft float
@ -88,9 +91,10 @@ functions compute the base 2 exponential of the given argument
.Fa x .
.Pp
The
.Fn expm1
.Fn expm1 ,
.Fn expm1f ,
and the
.Fn expm1f
.Fn expm1l
functions compute the value exp(x)\-1 accurately even for tiny argument
.Fa x .
.Pp

2
lib/msun/src/math.h
View File

@ -405,6 +405,7 @@ long double copysignl(long double, long double) __pure2;
long double cosl(long double);
long double exp2l(long double);
long double expl(long double);
long double expm1l(long double);
long double fabsl(long double) __pure2;
long double fdiml(long double, long double);
long double floorl(long double);
@ -466,7 +467,6 @@ long double atanhl(long double);
long double coshl(long double);
long double erfcl(long double);
long double erfl(long double);
long double expm1l(long double);
long double lgammal(long double);
long double powl(long double, long double);
long double sinhl(long double);

4
lib/msun/src/s_expm1.c
View File

@ -216,3 +216,7 @@ expm1(double x)
}
return y;
}
#if (LDBL_MANT_DIG == 53)
__weak_reference(expm1, expm1l);
#endif