double nexttoward(double x, long double y)
{
union {double f; uint64_t i;} ux = {x};
int e;
if (isnan(x) || isnan(y))
return x + y;
if (x == y)
return y;
if (x == 0) {
ux.i = 1;
if (signbit(y))
ux.i |= 1ULL<<63;
} else if (x < y) {
if (signbit(x))
ux.i--;
else
ux.i++;
} else {
if (signbit(x))
ux.i++;
else
ux.i--;
}
e = ux.i>>52 & 0x7ff;
/* raise overflow if ux.f is infinite and x is finite */
if (e == 0x7ff)
FORCE_EVAL(x+x);
/* raise underflow if ux.f is subnormal or zero */
if (e == 0)
FORCE_EVAL(x*x + ux.f*ux.f);
return ux.f;
}
double exp(double x)
{
double_t hi, lo, c, xx, y;
int k, sign;
uint32_t hx;
GET_HIGH_WORD(hx, x);
sign = hx>>31;
hx &= 0x7fffffff; /* high word of |x| */
/* special cases */
if (hx >= 0x4086232b) { /* if |x| >= 708.39... */
if (isnan(x))
return x;
if (x > 709.782712893383973096) {
/* overflow if x!=inf */
x *= 0x1p1023;
return x;
}
if (x < -708.39641853226410622) {
/* underflow if x!=-inf */
FORCE_EVAL((float)(-0x1p-149/x));
if (x < -745.13321910194110842)
return 0;
}
}
/* argument reduction */
if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
if (hx >= 0x3ff0a2b2) /* if |x| >= 1.5 ln2 */
k = (int)(invln2*x + half[sign]);
else
k = 1 - sign - sign;
hi = x - k*ln2hi; /* k*ln2hi is exact here */
lo = k*ln2lo;
x = hi - lo;
} else if (hx > 0x3e300000) { /* if |x| > 2**-28 */
k = 0;
hi = x;
lo = 0;
} else {
/* inexact if x!=0 */
FORCE_EVAL(0x1p1023 + x);
return 1 + x;
}
/* x is now in primary range */
xx = x*x;
c = x - xx*(__EXP_P1+xx*(__EXP_P2+xx*(__EXP_P3+xx*(__EXP_P4+xx*__EXP_P5))));
y = 1 + (x*c/(2-c) - lo + hi);
if (k == 0)
return y;
return scalbn(y, k);
}
float expf(float x)
{
float_t hi, lo, c, xx, y;
int k, sign;
uint32_t hx;
GET_FLOAT_WORD(hx, x);
sign = hx >> 31; /* sign bit of x */
hx &= 0x7fffffff; /* high word of |x| */
/* special cases */
if (hx >= 0x42aeac50) { /* if |x| >= -87.33655f or NaN */
if (hx >= 0x42b17218 && !sign) { /* x >= 88.722839f */
/* overflow */
x *= 0x1p127f;
return x;
}
if (sign) {
/* underflow */
FORCE_EVAL(-0x1p-149f/x);
if (hx >= 0x42cff1b5) /* x <= -103.972084f */
return 0;
}
}
/* argument reduction */
if (hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */
if (hx > 0x3f851592) /* if |x| > 1.5 ln2 */
k = invln2*x + half[sign];
else
k = 1 - sign - sign;
hi = x - k*ln2hi; /* k*ln2hi is exact here */
lo = k*ln2lo;
x = hi - lo;
} else if (hx > 0x39000000) { /* |x| > 2**-14 */
k = 0;
hi = x;
lo = 0;
} else {
/* raise inexact */
FORCE_EVAL(0x1p127f + x);
return 1 + x;
}
/* x is now in primary range */
xx = x*x;
c = x - xx*(P1+xx*P2);
y = 1 + (x*c/(2-c) - lo + hi);
if (k == 0)
return y;
return scalbnf(y, k);
}
long double tanl(long double x)
{
union IEEEl2bits z;
long double y[2];
unsigned n;
z.e = x;
z.bits.sign = 0;
/* If x = NaN or Inf, then tan(x) = NaN. */
if (z.bits.exp == 0x7fff)
return (x - x) / (x - x);
/* |x| < (double)pi/4 */
if (z.e < M_PI_4) {
/* |x| < 0x1p-64 */
if (z.bits.exp < 0x3fff - 64) {
/* raise inexact if x!=0 and underflow if subnormal */
FORCE_EVAL(z.bits.exp == 0 ? x/0x1p120f : x+0x1p120f);
return x;
}
return __tanl(x, 0, 0);
}
n = __rem_pio2l(x, y);
return __tanl(y[0], y[1], n&1);
}
/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
float asinhf(float x) {
union {
float f;
uint32_t i;
} u = {.f = x};
uint32_t i = u.i & 0x7fffffff;
unsigned s = u.i >> 31;
/* |x| */
u.i = i;
x = u.f;
if (i >= 0x3f800000 + (12 << 23)) {
/* |x| >= 0x1p12 or inf or nan */
x = logf(x) + 0.693147180559945309417232121458176568f;
} else if (i >= 0x3f800000 + (1 << 23)) {
/* |x| >= 2 */
x = logf(2 * x + 1 / (sqrtf(x * x + 1) + x));
} else if (i >= 0x3f800000 - (12 << 23)) {
/* |x| >= 0x1p-12, up to 1.6ulp error in [0.125,0.5] */
x = log1pf(x + x * x / (sqrtf(x * x + 1) + 1));
} else {
/* |x| < 0x1p-12, raise inexact if x!=0 */
FORCE_EVAL(x + 0x1p120f);
}
return s ? -x : x;
}
long double coshl(long double x)
{
union ldshape u = {x};
unsigned ex = u.i.se & 0x7fff;
uint32_t w;
long double t;
/* |x| */
u.i.se = ex;
x = u.f;
w = u.i.m >> 32;
/* |x| < log(2) */
if (ex < 0x3fff-1 || (ex == 0x3fff-1 && w < 0xb17217f7)) {
if (ex < 0x3fff-32) {
FORCE_EVAL(x + 0x1p120f);
return 1;
}
t = expm1l(x);
return 1 + t*t/(2*(1+t));
}
/* |x| < log(LDBL_MAX) */
if (ex < 0x3fff+13 || (ex == 0x3fff+13 && w < 0xb17217f7)) {
t = expl(x);
return 0.5*(t + 1/t);
}
/* |x| > log(LDBL_MAX) or nan */
t = expl(0.5*x);
return 0.5*t*t;
}
double nexttoward(double x, long double y)
{
union dshape ux;
int e;
if (isnan(x) || isnan(y))
return x + y;
if (x == y)
return y;
ux.value = x;
if (x == 0) {
ux.bits = 1;
if (signbit(y))
ux.bits |= SIGN;
} else if (x < y) {
if (signbit(x))
ux.bits--;
else
ux.bits++;
} else {
if (signbit(x))
ux.bits++;
else
ux.bits--;
}
e = ux.bits>>52 & 0x7ff;
/* raise overflow if ux.value is infinite and x is finite */
if (e == 0x7ff)
return x + x;
/* raise underflow if ux.value is subnormal or zero */
if (e == 0)
FORCE_EVAL(x*x + ux.value*ux.value);
return ux.value;
}
float coshf(float x)
{
union {float f; uint32_t i;} u = {.f = x};
uint32_t w;
float t;
/* |x| */
u.i &= 0x7fffffff;
x = u.f;
w = u.i;
/* |x| < log(2) */
if (w < 0x3f317217) {
if (w < 0x3f800000 - (12<<23)) {
FORCE_EVAL(x + 0x1p120f);
return 1;
}
t = expm1f(x);
return 1 + t*t/(2*(1+t));
}
/* |x| < log(FLT_MAX) */
if (w < 0x42b17217) {
t = expf(x);
return 0.5f*(t + 1/t);
}
/* |x| > log(FLT_MAX) or nan */
t = __expo2f(x);
return t;
}
/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
long double asinhl(long double x)
{
union ldshape u = {x};
unsigned e = u.i.se & 0x7fff;
unsigned s = u.i.se >> 15;
/* |x| */
u.i.se = e;
x = u.f;
if (e >= 0x3fff + 32) {
/* |x| >= 0x1p32 or inf or nan */
x = logl(x) + 0.693147180559945309417232121458176568L;
} else if (e >= 0x3fff + 1) {
/* |x| >= 2 */
x = logl(2*x + 1/(sqrtl(x*x+1)+x));
} else if (e >= 0x3fff - 32) {
/* |x| >= 0x1p-32 */
x = log1pl(x + x*x/(sqrtl(x*x+1)+1));
} else {
/* |x| < 0x1p-32, raise inexact if x!=0 */
FORCE_EVAL(x + 0x1p120f);
}
return s ? -x : x;
}
double tan(double x) {
double y[2];
uint32_t ix;
unsigned n;
GET_HIGH_WORD(ix, x);
ix &= 0x7fffffff;
/* |x| ~< pi/4 */
if (ix <= 0x3fe921fb) {
if (ix < 0x3e400000) { /* |x| < 2**-27 */
/* raise inexact if x!=0 and underflow if subnormal */
FORCE_EVAL(ix < 0x00100000 ? x / 0x1p120f : x + 0x1p120f);
return x;
}
return __tan(x, 0.0, 0);
}
/* tan(Inf or NaN) is NaN */
if (ix >= 0x7ff00000)
return x - x;
/* argument reduction */
n = __rem_pio2(x, y);
return __tan(y[0], y[1], n & 1);
}
double _mysin(double x)
{
double y[2];
uint32_t ix;
unsigned n;
/* High word of x. */
GET_HIGH_WORD(ix, x);
ix &= 0x7fffffff;
/* |x| ~< pi/4 */
if (ix <= 0x3fe921fb) {
if (ix < 0x3e500000) { /* |x| < 2**-26 */
/* raise inexact if x != 0 and underflow if subnormal*/
FORCE_EVAL(ix < 0x00100000 ? x/get_0x1p120f() : x+get_0x1p120f());
return x;
}
return my__sin(x, 0.0, 0);
}
/* sin(Inf or NaN) is NaN */
if (ix >= 0x7ff00000)
return x - x;
/* argument reduction needed */
n = my__rem_pio2(x, y);
switch (n&3) {
case 0: return my__sin(y[0], y[1], 1);
case 1: return my__cos(y[0], y[1]);
case 2: return -my__sin(y[0], y[1], 1);
default:
return -my__cos(y[0], y[1]);
}
}
long double sinl(long double x)
{
union ldshape u = {x};
unsigned n;
long double y[2], hi, lo;
u.i.se &= 0x7fff;
if (u.i.se == 0x7fff)
return x - x;
if (u.f < M_PI_4) {
if (u.i.se < 0x3fff - LDBL_MANT_DIG/2) {
/* raise inexact if x!=0 and underflow if subnormal */
FORCE_EVAL(u.i.se == 0 ? x*0x1p-120f : x+0x1p120f);
return x;
}
return __sinl(x, 0.0, 0);
}
n = __rem_pio2l(x, y);
hi = y[0];
lo = y[1];
switch (n & 3) {
case 0:
return __sinl(hi, lo, 1);
case 1:
return __cosl(hi, lo);
case 2:
return -__sinl(hi, lo, 1);
case 3:
default:
return -__cosl(hi, lo);
}
}
/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */
double atanh(double x)
{
union {double f; uint64_t i;} u = {.f = x};
unsigned e = u.i >> 52 & 0x7ff;
unsigned s = u.i >> 63;
double_t y;
/* |x| */
u.i &= (uint64_t)-1/2;
y = u.f;
if (e < 0x3ff - 1) {
if (e < 0x3ff - 32) {
/* handle underflow */
if (e == 0)
FORCE_EVAL((float)y);
} else {
/* |x| < 0.5, up to 1.7ulp error */
y = 0.5*log1p(2*y + 2*y*y/(1-y));
}
} else {
/* avoid overflow */
y = 0.5*log1p(2*(y/(1-y)));
}
return s ? -y : y;
}
/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */
float atanhf(float x)
{
union {float f; uint32_t i;} u = {.f = x};
unsigned s = u.i >> 31;
float_t y;
/* |x| */
u.i &= 0x7fffffff;
y = u.f;
if (u.i < 0x3f800000 - (1<<23)) {
if (u.i < 0x3f800000 - (32<<23)) {
/* handle underflow */
if (u.i < (1<<23))
FORCE_EVAL((float)(y*y));
} else {
/* |x| < 0.5, up to 1.7ulp error */
y = 0.5f*log1pf(2*y + 2*y*y/(1-y));
}
} else {
/* avoid overflow */
y = 0.5f*log1pf(2*(y/(1-y)));
}
return s ? -y : y;
}
long double nextafterl(long double x, long double y)
{
union ldshape ux, uy;
if (isnan(x) || isnan(y))
return x + y;
if (x == y)
return y;
ux.f = x;
if (x == 0) {
uy.f = y;
ux.i.m = 1;
ux.i.se = uy.i.se & 0x8000;
} else if ((x < y) == !(ux.i.se & 0x8000)) {
ux.i.m++;
if (ux.i.m << 1 == 0) {
ux.i.m = 1ULL << 63;
ux.i.se++;
}
} else {
if (ux.i.m << 1 == 0) {
ux.i.se--;
if (ux.i.se)
ux.i.m = 0;
}
ux.i.m--;
}
/* raise overflow if ux is infinite and x is finite */
if ((ux.i.se & 0x7fff) == 0x7fff)
return x + x;
/* raise underflow if ux is subnormal or zero */
if ((ux.i.se & 0x7fff) == 0)
FORCE_EVAL(x*x + ux.f*ux.f);
return ux.f;
}
/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
double asinh(double x)
{
union {double f; uint64_t i;} u = {.f = x};
unsigned e = u.i >> 52 & 0x7ff;
unsigned s = u.i >> 63;
/* |x| */
u.i &= (uint64_t)-1/2;
x = u.f;
if (e >= 0x3ff + 26) {
/* |x| >= 0x1p26 or inf or nan */
x = log(x) + 0.693147180559945309417232121458176568;
} else if (e >= 0x3ff + 1) {
/* |x| >= 2 */
x = log(2*x + 1/(sqrt(x*x+1)+x));
} else if (e >= 0x3ff - 26) {
/* |x| >= 0x1p-26, up to 1.6ulp error in [0.125,0.5] */
x = log1p(x + x*x/(sqrt(x*x+1)+1));
} else {
/* |x| < 0x1p-26, raise inexact if x != 0 */
FORCE_EVAL(x + 0x1p120f);
}
return s ? -x : x;
}
long double atanl(long double x)
{
union ldshape u = {x};
long double w, s1, s2, z;
int id;
unsigned e = u.i.se & 0x7fff;
unsigned sign = u.i.se >> 15;
unsigned expman;
if (e >= 0x3fff + LDBL_MANT_DIG + 1) { /* if |x| is large, atan(x)~=pi/2 */
if (isnan(x))
return x;
return sign ? -atanhi[3] : atanhi[3];
}
/* Extract the exponent and the first few bits of the mantissa. */
expman = EXPMAN(u);
if (expman < ((0x3fff - 2) << 8) + 0xc0) { /* |x| < 0.4375 */
if (e < 0x3fff - (LDBL_MANT_DIG+1)/2) { /* if |x| is small, atanl(x)~=x */
/* raise underflow if subnormal */
if (e == 0)
FORCE_EVAL((float)x);
return x;
}
id = -1;
} else {
x = fabsl(x);
if (expman < (0x3fff << 8) + 0x30) { /* |x| < 1.1875 */
if (expman < ((0x3fff - 1) << 8) + 0x60) { /* 7/16 <= |x| < 11/16 */
id = 0;
x = (2.0*x-1.0)/(2.0+x);
} else { /* 11/16 <= |x| < 19/16 */
id = 1;
x = (x-1.0)/(x+1.0);
}
} else {
if (expman < ((0x3fff + 1) << 8) + 0x38) { /* |x| < 2.4375 */
id = 2;
x = (x-1.5)/(1.0+1.5*x);
} else { /* 2.4375 <= |x| */
id = 3;
x = -1.0/x;
}
}
}
/* end of argument reduction */
z = x*x;
w = z*z;
/* break sum aT[i]z**(i+1) into odd and even poly */
s1 = z*T_even(w);
s2 = w*T_odd(w);
if (id < 0)
return x - x*(s1+s2);
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
return sign ? -z : z;
}
float cosf(float x) {
double y;
uint32_t ix;
unsigned n, sign;
GET_FLOAT_WORD(ix, x);
sign = ix >> 31;
ix &= 0x7fffffff;
if (ix <= 0x3f490fda) { /* |x| ~<= pi/4 */
if (ix < 0x39800000) { /* |x| < 2**-12 */
/* raise inexact if x != 0 */
FORCE_EVAL(x + 0x1p120f);
return 1.0f;
}
return __cosdf(x);
}
if (ix <= 0x407b53d1) { /* |x| ~<= 5*pi/4 */
if (ix > 0x4016cbe3) /* |x| ~> 3*pi/4 */
return -__cosdf(sign ? x + c2pio2 : x - c2pio2);
else {
if (sign)
return __sindf(x + c1pio2);
else
return __sindf(c1pio2 - x);
}
}
if (ix <= 0x40e231d5) { /* |x| ~<= 9*pi/4 */
if (ix > 0x40afeddf) /* |x| ~> 7*pi/4 */
return __cosdf(sign ? x + c4pio2 : x - c4pio2);
else {
if (sign)
return __sindf(-x - c3pio2);
else
return __sindf(x - c3pio2);
}
}
/* cos(Inf or NaN) is NaN */
if (ix >= 0x7f800000)
return x - x;
/* general argument reduction needed */
n = __rem_pio2f(x, &y);
switch (n & 3) {
case 0:
return __cosdf(y);
case 1:
return __sindf(-y);
case 2:
return -__cosdf(y);
default:
return __sindf(y);
}
}
void sincos(double x, double *sin, double *cos)
{
double y[2], s, c;
uint32_t ix;
unsigned n;
GET_HIGH_WORD(ix, x);
ix &= 0x7fffffff;
/* |x| ~< pi/4 */
if (ix <= 0x3fe921fb) {
/* if |x| < 2**-27 * sqrt(2) */
if (ix < 0x3e46a09e) {
/* raise inexact if x!=0 and underflow if subnormal */
FORCE_EVAL(ix < 0x00100000 ? x/0x1p120f : x+0x1p120f);
*sin = x;
*cos = 1.0;
return;
}
*sin = __sin(x, 0.0, 0);
*cos = __cos(x, 0.0);
return;
}
/* sincos(Inf or NaN) is NaN */
if (ix >= 0x7ff00000) {
*sin = *cos = x - x;
return;
}
/* argument reduction needed */
n = __rem_pio2(x, y);
s = __sin(y[0], y[1], 1);
c = __cos(y[0], y[1]);
switch (n&3) {
case 0:
*sin = s;
*cos = c;
break;
case 1:
*sin = c;
*cos = -s;
break;
case 2:
*sin = -s;
*cos = -c;
break;
case 3:
default:
*sin = -c;
*cos = s;
break;
}
}
void sincosl(long double x, long double *sin, long double *cos)
{
union IEEEl2bits u;
unsigned n;
long double y[2], s, c;
u.e = x;
u.bits.sign = 0;
/* x = nan or inf */
if (u.bits.exp == 0x7fff) {
*sin = *cos = x - x;
return;
}
/* |x| < (double)pi/4 */
if (u.e < M_PI_4) {
/* |x| < 0x1p-64 */
if (u.bits.exp < 0x3fff - 64) {
/* raise underflow if subnormal */
if (u.bits.exp == 0) FORCE_EVAL(x*0x1p-120f);
*sin = x;
/* raise inexact if x!=0 */
*cos = 1.0 + x;
return;
}
*sin = __sinl(x, 0, 0);
*cos = __cosl(x, 0);
return;
}
n = __rem_pio2l(x, y);
s = __sinl(y[0], y[1], 1);
c = __cosl(y[0], y[1]);
switch (n & 3) {
case 0:
*sin = s;
*cos = c;
break;
case 1:
*sin = c;
*cos = -s;
break;
case 2:
*sin = -s;
*cos = -c;
break;
case 3:
default:
*sin = -c;
*cos = s;
break;
}
}
double atan(double x)
{
double_t w,s1,s2,z;
uint32_t ix,sign;
int id;
GET_HIGH_WORD(ix, x);
sign = ix >> 31;
ix &= 0x7fffffff;
if (ix >= 0x44100000) { /* if |x| >= 2^66 */
if (isnan(x))
return x;
z = atanhi[3] + 0x1p-120f;
return sign ? -z : z;
}
if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
if (ix < 0x3e400000) { /* |x| < 2^-27 */
if (ix < 0x00100000)
/* raise underflow for subnormal x */
FORCE_EVAL((float)x);
return x;
}
id = -1;
} else {
x = fabs(x);
if (ix < 0x3ff30000) { /* |x| < 1.1875 */
if (ix < 0x3fe60000) { /* 7/16 <= |x| < 11/16 */
id = 0;
x = (2.0*x-1.0)/(2.0+x);
} else { /* 11/16 <= |x| < 19/16 */
id = 1;
x = (x-1.0)/(x+1.0);
}
} else {
if (ix < 0x40038000) { /* |x| < 2.4375 */
id = 2;
x = (x-1.5)/(1.0+1.5*x);
} else { /* 2.4375 <= |x| < 2^66 */
id = 3;
x = -1.0/x;
}
}
}
/* end of argument reduction */
z = x*x;
w = z*z;
/* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
if (id < 0)
return x - x*(s1+s2);
z = atanhi[id] - (x*(s1+s2) - atanlo[id] - x);
return sign ? -z : z;
}
float atanf(float x)
{
float_t w,s1,s2,z;
uint32_t ix,sign;
int id;
GET_FLOAT_WORD(ix, x);
sign = ix>>31;
ix &= 0x7fffffff;
if (ix >= 0x4c800000) { /* if |x| >= 2**26 */
if (isnan(x))
return x;
z = atanhi[3] + 0x1p-120f;
return sign ? -z : z;
}
if (ix < 0x3ee00000) { /* |x| < 0.4375 */
if (ix < 0x39800000) { /* |x| < 2**-12 */
if (ix < 0x00800000)
/* raise underflow for subnormal x */
FORCE_EVAL(x*x);
return x;
}
id = -1;
} else {
x = fabsf(x);
if (ix < 0x3f980000) { /* |x| < 1.1875 */
if (ix < 0x3f300000) { /* 7/16 <= |x| < 11/16 */
id = 0;
x = (2.0f*x - 1.0f)/(2.0f + x);
} else { /* 11/16 <= |x| < 19/16 */
id = 1;
x = (x - 1.0f)/(x + 1.0f);
}
} else {
if (ix < 0x401c0000) { /* |x| < 2.4375 */
id = 2;
x = (x - 1.5f)/(1.0f + 1.5f*x);
} else { /* 2.4375 <= |x| < 2**26 */
id = 3;
x = -1.0f/x;
}
}
}
/* end of argument reduction */
z = x*x;
w = z*z;
/* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
s1 = z*(aT[0]+w*(aT[2]+w*aT[4]));
s2 = w*(aT[1]+w*aT[3]);
if (id < 0)
return x - x*(s1+s2);
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
return sign ? -z : z;
}
int ilogb(double x)
{
union dshape u = {x};
int e = u.bits>>52 & 0x7ff;
if (!e) {
u.bits <<= 12;
if (u.bits == 0) {
FORCE_EVAL(0/0.0f);
return FP_ILOGB0;
}
/* subnormal x */
for (e = -0x3ff; u.bits < (uint64_t)1<<63; e--, u.bits<<=1);
return e;
}
if (e == 0x7ff) {
FORCE_EVAL(0/0.0f);
return u.bits<<12 ? FP_ILOGBNAN : INT_MAX;
}
return e - 0x3ff;
}
int ilogbl(long double x) {
PRAGMA_STDC_FENV_ACCESS_ON
union ldshape u = {x};
int e = u.i.se & 0x7fff;
if (!e) {
if (x == 0) {
FORCE_EVAL(0 / 0.0f);
return FP_ILOGB0;
}
/* subnormal x */
x *= 0x1p120;
return ilogbl(x) - 120;
}
if (e == 0x7fff) {
FORCE_EVAL(0 / 0.0f);
u.i.se = 0;
return u.f ? FP_ILOGBNAN : INT_MAX;
}
return e - 0x3fff;
}
int ilogbl(long double x) {
PRAGMA_STDC_FENV_ACCESS_ON
union ldshape u = {x};
uint64_t m = u.i.m;
int e = u.i.se & 0x7fff;
if (!e) {
if (m == 0) {
FORCE_EVAL(0 / 0.0f);
return FP_ILOGB0;
}
/* subnormal x */
for (e = -0x3fff + 1; m >> 63 == 0; e--, m <<= 1)
;
return e;
}
if (e == 0x7fff) {
FORCE_EVAL(0 / 0.0f);
return m << 1 ? FP_ILOGBNAN : INT_MAX;
}
return e - 0x3fff;
}
float sinf(float x)
{
double y;
uint32_t ix;
int n, sign;
GET_FLOAT_WORD(ix, x);
sign = ix >> 31;
ix &= 0x7fffffff;
if (ix <= 0x3f490fda) { /* |x| ~<= pi/4 */
if (ix < 0x39800000) { /* |x| < 2**-12 */
/* raise inexact if x!=0 and underflow if subnormal */
FORCE_EVAL(ix < 0x00800000 ? x/0x1p120f : x+0x1p120f);
return x;
}
return __sindf(x);
}
if (ix <= 0x407b53d1) { /* |x| ~<= 5*pi/4 */
if (ix <= 0x4016cbe3) { /* |x| ~<= 3pi/4 */
if (sign)
return -__cosdf(x + s1pio2);
else
return __cosdf(x - s1pio2);
}
return __sindf(sign ? -(x + s2pio2) : -(x - s2pio2));
}
if (ix <= 0x40e231d5) { /* |x| ~<= 9*pi/4 */
if (ix <= 0x40afeddf) { /* |x| ~<= 7*pi/4 */
if (sign)
return __cosdf(x + s3pio2);
else
return -__cosdf(x - s3pio2);
}
return __sindf(sign ? x + s4pio2 : x - s4pio2);
}
/* sin(Inf or NaN) is NaN */
if (ix >= 0x7f800000)
return x - x;
/* general argument reduction needed */
n = __rem_pio2f(x, &y);
switch (n&3) {
case 0: return __sindf(y);
case 1: return __cosdf(y);
case 2: return __sindf(-y);
default:
return -__cosdf(y);
}
}
long double tanl(long double x) {
union ldshape u = {x};
long double y[2];
unsigned n;
u.i.se &= 0x7fff;
if (u.i.se == 0x7fff)
return x - x;
if (u.f < M_PI_4) {
if (u.i.se < 0x3fff - LDBL_MANT_DIG / 2) {
/* raise inexact if x!=0 and underflow if subnormal */
FORCE_EVAL(u.i.se == 0 ? x * 0x1p-120f : x + 0x1p120f);
return x;
}
return __tanl(x, 0, 0);
}
n = __rem_pio2l(x, y);
return __tanl(y[0], y[1], n & 1);
}
int ilogb(double x)
{
PRAGMA_STDC_FENV_ACCESS_ON
union {
double f;
uint64_t i;
} u = {x};
uint64_t i = u.i;
int e = i>>52 & 0x7ff;
if (!e) {
i <<= 12;
if (i == 0) {
FORCE_EVAL(0/0.0f);
return FP_ILOGB0;
}
/* subnormal x */
for (e = -0x3ff; i>>63 == 0; e--, i<<=1);
return e;
}
/* tanh(x) = (exp(x) - exp(-x))/(exp(x) + exp(-x))
* = (exp(2*x) - 1)/(exp(2*x) - 1 + 2)
* = (1 - exp(-2*x))/(exp(-2*x) - 1 + 2)
*/
double tanh(double x) {
union {
double f;
uint64_t i;
} u = {.f = x};
uint32_t w;
int sign;
double_t t;
/* x = |x| */
sign = u.i >> 63;
u.i &= (uint64_t)-1 / 2;
x = u.f;
w = u.i >> 32;
if (w > 0x3fe193ea) {
/* |x| > log(3)/2 ~= 0.5493 or nan */
if (w > 0x40340000) {
/* |x| > 20 or nan */
/* note: this branch avoids raising overflow */
t = 1 - 0 / x;
} else {
t = expm1(2 * x);
t = 1 - 2 / (t + 2);
}
} else if (w > 0x3fd058ae) {
/* |x| > log(5/3)/2 ~= 0.2554 */
t = expm1(2 * x);
t = t / (t + 2);
} else if (w >= 0x00100000) {
/* |x| >= 0x1p-1022, up to 2ulp error in [0.1,0.2554] */
t = expm1(-2 * x);
t = -t / (t + 2);
} else {
/* |x| is subnormal */
/* note: the branch above would not raise underflow in [0x1p-1023,0x1p-1022) */
FORCE_EVAL((float)x);
t = x;
}
return sign ? -t : t;
}
/*
* exp2f(x): compute the base 2 exponential of x
*
* Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
*
* Method: (equally-spaced tables)
*
* Reduce x:
* x = k + y, for integer k and |y| <= 1/2.
* Thus we have exp2f(x) = 2**k * exp2(y).
*
* Reduce y:
* y = i/TBLSIZE + z for integer i near y * TBLSIZE.
* Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
* with |z| <= 2**-(TBLSIZE+1).
*
* We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
* degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
* Using double precision for everything except the reduction makes
* roundoff error insignificant and simplifies the scaling step.
*
* This method is due to Tang, but I do not use his suggested parameters:
*
* Tang, P. Table-driven Implementation of the Exponential Function
* in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
*/
float exp2f(float x)
{
double_t t, r, z;
union {float f; uint32_t i;} u = {x};
union {double f; uint64_t i;} uk;
uint32_t ix, i0, k;
/* Filter out exceptional cases. */
ix = u.i & 0x7fffffff;
if (ix > 0x42fc0000) { /* |x| > 126 */
if (u.i >= 0x43000000 && u.i < 0x80000000) { /* x >= 128 */
x *= 0x1p127f;
return x;
}
if (u.i >= 0x80000000) { /* x < -126 */
if (u.i >= 0xc3160000 || (u.i & 0x0000ffff))
FORCE_EVAL(-0x1p-149f/x);
if (u.i >= 0xc3160000) /* x <= -150 */
return 0;
}
} else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
return 1.0f + x;
}
/* Reduce x, computing z, i0, and k. */
u.f = x + redux;
i0 = u.i;
i0 += TBLSIZE / 2;
k = i0 / TBLSIZE;
uk.i = (uint64_t)(0x3ff + k)<<52;
i0 &= TBLSIZE - 1;
u.f -= redux;
z = x - u.f;
/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
r = exp2ft[i0];
t = r * z;
r = r + t * (P1 + z * P2) + t * (z * z) * (P3 + z * P4);
/* Scale by 2**k */
return r * uk.f;
}