long double
__cbrtl (long double x)
{
long double xm, u;
int xe;
/* Reduce X. XM now is an range 1.0 to 0.5. */
xm = __frexpl (fabsl (x), &xe);
/* If X is not finite or is null return it (with raising exceptions
if necessary.
Note: *Our* version of `frexp' sets XE to zero if the argument is
Inf or NaN. This is not portable but faster. */
if (xe == 0 && fpclassify (x) <= FP_ZERO)
return x + x;
u = (((-1.34661104733595206551E-1 * xm
+ 5.46646013663955245034E-1) * xm
- 9.54382247715094465250E-1) * xm
+ 1.13999833547172932737E0) * xm
+ 4.02389795645447521269E-1;
u *= factor[2 + xe % 3];
u = __ldexpl (x > 0.0 ? u : -u, xe / 3);
u -= (u - (x / (u * u))) * third;
u -= (u - (x / (u * u))) * third;
return u;
}
long double
__expm1l (long double x)
{
long double px, qx, xx;
int32_t ix, sign;
ieee854_long_double_shape_type u;
int k;
/* Detect infinity and NaN. */
u.value = x;
ix = u.parts32.w0;
sign = ix & 0x80000000;
ix &= 0x7fffffff;
if (ix >= 0x7ff00000)
{
/* Infinity. */
if (((ix & 0xfffff) | u.parts32.w1 | (u.parts32.w2&0x7fffffff) | u.parts32.w3) == 0)
{
if (sign)
return -1.0L;
else
return x;
}
/* NaN. No invalid exception. */
return x;
}
/* expm1(+- 0) = +- 0. */
if ((ix == 0) && (u.parts32.w1 | (u.parts32.w2&0x7fffffff) | u.parts32.w3) == 0)
return x;
/* Overflow. */
if (x > maxlog)
{
__set_errno (ERANGE);
return (big * big);
}
/* Minimum value. */
if (x < minarg)
return (4.0/big - 1.0L);
/* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
xx = C1 + C2; /* ln 2. */
px = __floorl (0.5 + x / xx);
k = px;
/* remainder times ln 2 */
x -= px * C1;
x -= px * C2;
/* Approximate exp(remainder ln 2). */
px = (((((((P7 * x
+ P6) * x
+ P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
qx = (((((((x
+ Q7) * x
+ Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
xx = x * x;
qx = x + (0.5 * xx + xx * px / qx);
/* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
We have qx = exp(remainder ln 2) - 1, so
exp(x) - 1 = 2^k (qx + 1) - 1
= 2^k qx + 2^k - 1. */
px = __ldexpl (1.0L, k);
x = px * qx + (px - 1.0);
return x;
}
long double
__expm1l (long double x)
{
long double px, qx, xx;
int32_t ix, sign;
ieee854_long_double_shape_type u;
int k;
/* Detect infinity and NaN. */
u.value = x;
ix = u.parts32.w0;
sign = ix & 0x80000000;
ix &= 0x7fffffff;
if (!sign && ix >= 0x40060000)
{
/* If num is positive and exp >= 6 use plain exp. */
return __expl (x);
}
if (ix >= 0x7fff0000)
{
/* Infinity (which must be negative infinity). */
if (((ix & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
return -1.0L;
/* NaN. No invalid exception. */
return x;
}
/* expm1(+- 0) = +- 0. */
if ((ix == 0) && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
return x;
/* Minimum value. */
if (x < minarg)
return (4.0/big - 1.0L);
/* Avoid internal underflow when result does not underflow, while
ensuring underflow (without returning a zero of the wrong sign)
when the result does underflow. */
if (fabsl (x) < 0x1p-113L)
{
math_check_force_underflow (x);
return x;
}
/* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
xx = C1 + C2; /* ln 2. */
px = __floorl (0.5 + x / xx);
k = px;
/* remainder times ln 2 */
x -= px * C1;
x -= px * C2;
/* Approximate exp(remainder ln 2). */
px = (((((((P7 * x
+ P6) * x
+ P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
qx = (((((((x
+ Q7) * x
+ Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
xx = x * x;
qx = x + (0.5 * xx + xx * px / qx);
/* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
We have qx = exp(remainder ln 2) - 1, so
exp(x) - 1 = 2^k (qx + 1) - 1
= 2^k qx + 2^k - 1. */
px = __ldexpl (1.0L, k);
x = px * qx + (px - 1.0);
return x;
}
long double
__cbrtl (long double x)
{
int e, rem, sign;
long double z;
if (!__finitel (x))
return x + x;
if (x == 0)
return (x);
if (x > 0)
sign = 1;
else
{
sign = -1;
x = -x;
}
z = x;
/* extract power of 2, leaving mantissa between 0.5 and 1 */
x = __frexpl (x, &e);
/* Approximate cube root of number between .5 and 1,
peak relative error = 1.2e-6 */
x = ((((1.3584464340920900529734e-1L * x
- 6.3986917220457538402318e-1L) * x
+ 1.2875551670318751538055e0L) * x
- 1.4897083391357284957891e0L) * x
+ 1.3304961236013647092521e0L) * x + 3.7568280825958912391243e-1L;
/* exponent divided by 3 */
if (e >= 0)
{
rem = e;
e /= 3;
rem -= 3 * e;
if (rem == 1)
x *= CBRT2;
else if (rem == 2)
x *= CBRT4;
}
else
{ /* argument less than 1 */
e = -e;
rem = e;
e /= 3;
rem -= 3 * e;
if (rem == 1)
x *= CBRT2I;
else if (rem == 2)
x *= CBRT4I;
e = -e;
}
/* multiply by power of 2 */
x = __ldexpl (x, e);
/* Newton iteration */
x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333L;
x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333L;
x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333L;
if (sign < 0)
x = -x;
return (x);
}
