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Import of the watch repository from Pebble
This commit is contained in:
Matthieu Jeanson
Katharine Berry
commit
3b92768480
10334 changed files with 2564465 additions and 0 deletions
6
third_party/math/LICENSE
vendored
Normal file
6
third_party/math/LICENSE
vendored
Normal file
|
|
@ -0,0 +1,6 @@
|
|||
Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
|
||||
Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
Permission to use, copy, modify, and distribute this
|
||||
software is freely granted, provided that this notice
|
||||
is preserved.
|
||||
173
third_party/math/log.c
vendored
Normal file
173
third_party/math/log.c
vendored
Normal file
|
|
@ -0,0 +1,173 @@
|
|||
///////////////////////////////////////
|
||||
// Implements:
|
||||
// double log(double x);
|
||||
///////////////////////////////////////
|
||||
// Notes:
|
||||
// This is taken from newlib.
|
||||
// @nolint
|
||||
|
||||
#include <pblibc_private.h>
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <stdint.h>
|
||||
|
||||
/* __ieee754_log(x)
|
||||
* Return the logrithm of x
|
||||
*
|
||||
* Method :
|
||||
* 1. Argument Reduction: find k and f such that
|
||||
* x = 2^k * (1+f),
|
||||
* where sqrt(2)/2 < 1+f < sqrt(2) .
|
||||
*
|
||||
* 2. Approximation of log(1+f).
|
||||
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
|
||||
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
|
||||
* = 2s + s*R
|
||||
* We use a special Reme algorithm on [0,0.1716] to generate
|
||||
* a polynomial of degree 14 to approximate R The maximum error
|
||||
* of this polynomial approximation is bounded by 2**-58.45. In
|
||||
* other words,
|
||||
* 2 4 6 8 10 12 14
|
||||
* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
|
||||
* (the values of Lg1 to Lg7 are listed in the program)
|
||||
* and
|
||||
* | 2 14 | -58.45
|
||||
* | Lg1*s +...+Lg7*s - R(z) | <= 2
|
||||
* | |
|
||||
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
|
||||
* In order to guarantee error in log below 1ulp, we compute log
|
||||
* by
|
||||
* log(1+f) = f - s*(f - R) (if f is not too large)
|
||||
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
|
||||
*
|
||||
* 3. Finally, log(x) = k*ln2 + log(1+f).
|
||||
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
|
||||
* Here ln2 is split into two floating point number:
|
||||
* ln2_hi + ln2_lo,
|
||||
* where n*ln2_hi is always exact for |n| < 2000.
|
||||
*
|
||||
* Special cases:
|
||||
* log(x) is NaN with signal if x < 0 (including -INF) ;
|
||||
* log(+INF) is +INF; log(0) is -INF with signal;
|
||||
* log(NaN) is that NaN with no signal.
|
||||
*
|
||||
* Accuracy:
|
||||
* according to an error analysis, the error is always less than
|
||||
* 1 ulp (unit in the last place).
|
||||
*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
* constants. The decimal values may be used, provided that the
|
||||
* compiler will convert from decimal to binary accurately enough
|
||||
* to produce the hexadecimal values shown.
|
||||
*/
|
||||
|
||||
static const double
|
||||
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
|
||||
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
|
||||
two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
|
||||
Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
|
||||
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
|
||||
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
|
||||
Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
|
||||
Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
|
||||
Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
|
||||
Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
|
||||
|
||||
static const double zero = 0.0;
|
||||
|
||||
typedef union
|
||||
{
|
||||
double value;
|
||||
struct
|
||||
{
|
||||
uint32_t lsw;
|
||||
uint32_t msw;
|
||||
} parts;
|
||||
} ieee_double_shape_type;
|
||||
|
||||
#define EXTRACT_WORDS(ix0,ix1,d) \
|
||||
do { \
|
||||
ieee_double_shape_type ew_u; \
|
||||
ew_u.value = (d); \
|
||||
(ix0) = ew_u.parts.msw; \
|
||||
(ix1) = ew_u.parts.lsw; \
|
||||
} while (0)
|
||||
|
||||
/* Get the more significant 32 bit int from a double. */
|
||||
|
||||
#define GET_HIGH_WORD(i,d) \
|
||||
do { \
|
||||
ieee_double_shape_type gh_u; \
|
||||
gh_u.value = (d); \
|
||||
(i) = gh_u.parts.msw; \
|
||||
} while (0)
|
||||
|
||||
/* Set the more significant 32 bits of a double from an int. */
|
||||
|
||||
#define SET_HIGH_WORD(d,v) \
|
||||
do { \
|
||||
ieee_double_shape_type sh_u; \
|
||||
sh_u.value = (d); \
|
||||
sh_u.parts.msw = (v); \
|
||||
(d) = sh_u.value; \
|
||||
} while (0)
|
||||
|
||||
double log(double x)
|
||||
{
|
||||
double hfsq,f,s,z,R,w,t1,t2,dk;
|
||||
int32_t k,hx,i,j;
|
||||
uint32_t lx;
|
||||
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
|
||||
k=0;
|
||||
if (hx < 0x00100000) { /* x < 2**-1022 */
|
||||
if (((hx&0x7fffffff)|lx)==0)
|
||||
return -two54/zero; /* log(+-0)=-inf */
|
||||
if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
|
||||
k -= 54; x *= two54; /* subnormal number, scale up x */
|
||||
GET_HIGH_WORD(hx,x);
|
||||
}
|
||||
if (hx >= 0x7ff00000) return x+x;
|
||||
k += (hx>>20)-1023;
|
||||
hx &= 0x000fffff;
|
||||
i = (hx+0x95f64)&0x100000;
|
||||
SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
|
||||
k += (i>>20);
|
||||
f = x-1.0;
|
||||
if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
|
||||
if(f==zero) { if(k==0) return zero; else {dk=(double)k;
|
||||
return dk*ln2_hi+dk*ln2_lo;}}
|
||||
R = f*f*(0.5-0.33333333333333333*f);
|
||||
if(k==0) return f-R; else {dk=(double)k;
|
||||
return dk*ln2_hi-((R-dk*ln2_lo)-f);}
|
||||
}
|
||||
s = f/(2.0+f);
|
||||
dk = (double)k;
|
||||
z = s*s;
|
||||
i = hx-0x6147a;
|
||||
w = z*z;
|
||||
j = 0x6b851-hx;
|
||||
t1= w*(Lg2+w*(Lg4+w*Lg6));
|
||||
t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
|
||||
i |= j;
|
||||
R = t2+t1;
|
||||
if(i>0) {
|
||||
hfsq=0.5*f*f;
|
||||
if(k==0) return f-(hfsq-s*(hfsq+R)); else
|
||||
return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
|
||||
} else {
|
||||
if(k==0) return f-s*(f-R); else
|
||||
return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
|
||||
}
|
||||
}
|
||||
364
third_party/math/pow.c
vendored
Normal file
364
third_party/math/pow.c
vendored
Normal file
|
|
@ -0,0 +1,364 @@
|
|||
///////////////////////////////////////
|
||||
// Implements:
|
||||
// double pow(double x, double y);
|
||||
///////////////////////////////////////
|
||||
// Notes:
|
||||
// This is taken from newlib.
|
||||
// @nolint
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <stdint.h>
|
||||
#include <pblibc_private.h>
|
||||
|
||||
/* __ieee754_pow(x,y) return x**y
|
||||
*
|
||||
* n
|
||||
* Method: Let x = 2 * (1+f)
|
||||
* 1. Compute and return log2(x) in two pieces:
|
||||
* log2(x) = w1 + w2,
|
||||
* where w1 has 53-24 = 29 bit trailing zeros.
|
||||
* 2. Perform y*log2(x) = n+y' by simulating multi-precision
|
||||
* arithmetic, where |y'|<=0.5.
|
||||
* 3. Return x**y = 2**n*exp(y'*log2)
|
||||
*
|
||||
* Special cases:
|
||||
* 1. (anything) ** 0 is 1
|
||||
* 2. (anything) ** 1 is itself
|
||||
* 3a. (anything) ** NAN is NAN except
|
||||
* 3b. +1 ** NAN is 1
|
||||
* 4. NAN ** (anything except 0) is NAN
|
||||
* 5. +-(|x| > 1) ** +INF is +INF
|
||||
* 6. +-(|x| > 1) ** -INF is +0
|
||||
* 7. +-(|x| < 1) ** +INF is +0
|
||||
* 8. +-(|x| < 1) ** -INF is +INF
|
||||
* 9. +-1 ** +-INF is 1
|
||||
* 10. +0 ** (+anything except 0, NAN) is +0
|
||||
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
|
||||
* 12. +0 ** (-anything except 0, NAN) is +INF
|
||||
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
|
||||
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
|
||||
* 15. +INF ** (+anything except 0,NAN) is +INF
|
||||
* 16. +INF ** (-anything except 0,NAN) is +0
|
||||
* 17. -INF ** (anything) = -0 ** (-anything)
|
||||
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
|
||||
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
|
||||
*
|
||||
* Accuracy:
|
||||
* pow(x,y) returns x**y nearly rounded. In particular
|
||||
* pow(integer,integer)
|
||||
* always returns the correct integer provided it is
|
||||
* representable.
|
||||
*
|
||||
* Constants :
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
* constants. The decimal values may be used, provided that the
|
||||
* compiler will convert from decimal to binary accurately enough
|
||||
* to produce the hexadecimal values shown.
|
||||
*/
|
||||
|
||||
static const double
|
||||
bp[] = {1.0, 1.5,},
|
||||
dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
|
||||
dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
|
||||
zero = 0.0,
|
||||
one = 1.0,
|
||||
two = 2.0,
|
||||
two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
|
||||
huge = 1.0e300,
|
||||
tiny = 1.0e-300,
|
||||
/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
|
||||
L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
|
||||
L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
|
||||
L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
|
||||
L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
|
||||
L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
|
||||
L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
|
||||
P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
|
||||
P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
|
||||
P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
|
||||
P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
|
||||
P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
|
||||
lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
|
||||
lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
|
||||
lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
|
||||
ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
|
||||
cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
|
||||
cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
|
||||
cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
|
||||
ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
|
||||
ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
|
||||
ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
|
||||
|
||||
typedef union
|
||||
{
|
||||
double value;
|
||||
struct
|
||||
{
|
||||
uint32_t lsw;
|
||||
uint32_t msw;
|
||||
} parts;
|
||||
} ieee_double_shape_type;
|
||||
|
||||
#define EXTRACT_WORDS(ix0,ix1,d) \
|
||||
do { \
|
||||
ieee_double_shape_type ew_u; \
|
||||
ew_u.value = (d); \
|
||||
(ix0) = ew_u.parts.msw; \
|
||||
(ix1) = ew_u.parts.lsw; \
|
||||
} while (0)
|
||||
|
||||
/* Get the more significant 32 bit int from a double. */
|
||||
|
||||
#define GET_HIGH_WORD(i,d) \
|
||||
do { \
|
||||
ieee_double_shape_type gh_u; \
|
||||
gh_u.value = (d); \
|
||||
(i) = gh_u.parts.msw; \
|
||||
} while (0)
|
||||
|
||||
/* Get the less significant 32 bit int from a double. */
|
||||
|
||||
#define GET_LOW_WORD(i,d) \
|
||||
do { \
|
||||
ieee_double_shape_type gl_u; \
|
||||
gl_u.value = (d); \
|
||||
(i) = gl_u.parts.lsw; \
|
||||
} while (0)
|
||||
|
||||
/* Set the more significant 32 bits of a double from an int. */
|
||||
|
||||
#define SET_HIGH_WORD(d,v) \
|
||||
do { \
|
||||
ieee_double_shape_type sh_u; \
|
||||
sh_u.value = (d); \
|
||||
sh_u.parts.msw = (v); \
|
||||
(d) = sh_u.value; \
|
||||
} while (0)
|
||||
|
||||
/* Set the less significant 32 bits of a double from an int. */
|
||||
|
||||
#define SET_LOW_WORD(d,v) \
|
||||
do { \
|
||||
ieee_double_shape_type sl_u; \
|
||||
sl_u.value = (d); \
|
||||
sl_u.parts.lsw = (v); \
|
||||
(d) = sl_u.value; \
|
||||
} while (0)
|
||||
|
||||
double pow(double x, double y)
|
||||
{
|
||||
double z,ax,z_h,z_l,p_h,p_l;
|
||||
double y1,t1,t2,r,s,t,u,v,w;
|
||||
int32_t i,j,k,yisint,n;
|
||||
int32_t hx,hy,ix,iy;
|
||||
uint32_t lx,ly;
|
||||
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
EXTRACT_WORDS(hy,ly,y);
|
||||
ix = hx&0x7fffffff; iy = hy&0x7fffffff;
|
||||
|
||||
/* y==zero: x**0 = 1 */
|
||||
if((iy|ly)==0) return one;
|
||||
|
||||
/* x|y==NaN return NaN unless x==1 then return 1 */
|
||||
if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
|
||||
iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) {
|
||||
if(((ix-0x3ff00000)|lx)==0) return one;
|
||||
else return __builtin_nan("");
|
||||
}
|
||||
|
||||
/* determine if y is an odd int when x < 0
|
||||
* yisint = 0 ... y is not an integer
|
||||
* yisint = 1 ... y is an odd int
|
||||
* yisint = 2 ... y is an even int
|
||||
*/
|
||||
yisint = 0;
|
||||
if(hx<0) {
|
||||
if(iy>=0x43400000) yisint = 2; /* even integer y */
|
||||
else if(iy>=0x3ff00000) {
|
||||
k = (iy>>20)-0x3ff; /* exponent */
|
||||
if(k>20) {
|
||||
j = ly>>(52-k);
|
||||
if((uint32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
|
||||
} else if(ly==0) {
|
||||
j = iy>>(20-k);
|
||||
if((j<<(20-k))==iy) yisint = 2-(j&1);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
/* special value of y */
|
||||
if(ly==0) {
|
||||
if (iy==0x7ff00000) { /* y is +-inf */
|
||||
if(((ix-0x3ff00000)|lx)==0)
|
||||
return one; /* +-1**+-inf = 1 */
|
||||
else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
|
||||
return (hy>=0)? y: zero;
|
||||
else /* (|x|<1)**-,+inf = inf,0 */
|
||||
return (hy<0)?-y: zero;
|
||||
}
|
||||
if(iy==0x3ff00000) { /* y is +-1 */
|
||||
if(hy<0) return one/x; else return x;
|
||||
}
|
||||
if(hy==0x40000000) return x*x; /* y is 2 */
|
||||
if(hy==0x3fe00000) { /* y is 0.5 */
|
||||
if(hx>=0) /* x >= +0 */
|
||||
return __builtin_sqrt(x);
|
||||
}
|
||||
}
|
||||
|
||||
ax = __builtin_fabs(x);
|
||||
/* special value of x */
|
||||
if(lx==0) {
|
||||
if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
|
||||
z = ax; /*x is +-0,+-inf,+-1*/
|
||||
if(hy<0) z = one/z; /* z = (1/|x|) */
|
||||
if(hx<0) {
|
||||
if(((ix-0x3ff00000)|yisint)==0) {
|
||||
z = (z-z)/(z-z); /* (-1)**non-int is NaN */
|
||||
} else if(yisint==1)
|
||||
z = -z; /* (x<0)**odd = -(|x|**odd) */
|
||||
}
|
||||
return z;
|
||||
}
|
||||
}
|
||||
|
||||
/* (x<0)**(non-int) is NaN */
|
||||
/* REDHAT LOCAL: This used to be
|
||||
if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x);
|
||||
but ANSI C says a right shift of a signed negative quantity is
|
||||
implementation defined. */
|
||||
if(((((uint32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
|
||||
|
||||
/* |y| is huge */
|
||||
if(iy>0x41e00000) { /* if |y| > 2**31 */
|
||||
if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
|
||||
if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
|
||||
if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
|
||||
}
|
||||
/* over/underflow if x is not close to one */
|
||||
if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
|
||||
if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
|
||||
/* now |1-x| is tiny <= 2**-20, suffice to compute
|
||||
log(x) by x-x^2/2+x^3/3-x^4/4 */
|
||||
t = ax-1; /* t has 20 trailing zeros */
|
||||
w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
|
||||
u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
|
||||
v = t*ivln2_l-w*ivln2;
|
||||
t1 = u+v;
|
||||
SET_LOW_WORD(t1,0);
|
||||
t2 = v-(t1-u);
|
||||
} else {
|
||||
double s2,s_h,s_l,t_h,t_l;
|
||||
n = 0;
|
||||
/* take care subnormal number */
|
||||
if(ix<0x00100000)
|
||||
{ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
|
||||
n += ((ix)>>20)-0x3ff;
|
||||
j = ix&0x000fffff;
|
||||
/* determine interval */
|
||||
ix = j|0x3ff00000; /* normalize ix */
|
||||
if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
|
||||
else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
|
||||
else {k=0;n+=1;ix -= 0x00100000;}
|
||||
SET_HIGH_WORD(ax,ix);
|
||||
|
||||
/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
|
||||
u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
|
||||
v = one/(ax+bp[k]);
|
||||
s = u*v;
|
||||
s_h = s;
|
||||
SET_LOW_WORD(s_h,0);
|
||||
/* t_h=ax+bp[k] High */
|
||||
t_h = zero;
|
||||
SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
|
||||
t_l = ax - (t_h-bp[k]);
|
||||
s_l = v*((u-s_h*t_h)-s_h*t_l);
|
||||
/* compute log(ax) */
|
||||
s2 = s*s;
|
||||
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
|
||||
r += s_l*(s_h+s);
|
||||
s2 = s_h*s_h;
|
||||
t_h = 3.0+s2+r;
|
||||
SET_LOW_WORD(t_h,0);
|
||||
t_l = r-((t_h-3.0)-s2);
|
||||
/* u+v = s*(1+...) */
|
||||
u = s_h*t_h;
|
||||
v = s_l*t_h+t_l*s;
|
||||
/* 2/(3log2)*(s+...) */
|
||||
p_h = u+v;
|
||||
SET_LOW_WORD(p_h,0);
|
||||
p_l = v-(p_h-u);
|
||||
z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
|
||||
z_l = cp_l*p_h+p_l*cp+dp_l[k];
|
||||
/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
|
||||
t = (double)n;
|
||||
t1 = (((z_h+z_l)+dp_h[k])+t);
|
||||
SET_LOW_WORD(t1,0);
|
||||
t2 = z_l-(((t1-t)-dp_h[k])-z_h);
|
||||
}
|
||||
|
||||
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
|
||||
if(((((uint32_t)hx>>31)-1)|(yisint-1))==0)
|
||||
s = -one;/* (-ve)**(odd int) */
|
||||
|
||||
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
|
||||
y1 = y;
|
||||
SET_LOW_WORD(y1,0);
|
||||
p_l = (y-y1)*t1+y*t2;
|
||||
p_h = y1*t1;
|
||||
z = p_l+p_h;
|
||||
EXTRACT_WORDS(j,i,z);
|
||||
if (j>=0x40900000) { /* z >= 1024 */
|
||||
if(((j-0x40900000)|i)!=0) /* if z > 1024 */
|
||||
return s*huge*huge; /* overflow */
|
||||
else {
|
||||
if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
|
||||
}
|
||||
} else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
|
||||
if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
|
||||
return s*tiny*tiny; /* underflow */
|
||||
else {
|
||||
if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
|
||||
}
|
||||
}
|
||||
/*
|
||||
* compute 2**(p_h+p_l)
|
||||
*/
|
||||
i = j&0x7fffffff;
|
||||
k = (i>>20)-0x3ff;
|
||||
n = 0;
|
||||
if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
|
||||
n = j+(0x00100000>>(k+1));
|
||||
k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
|
||||
t = zero;
|
||||
SET_HIGH_WORD(t,n&~(0x000fffff>>k));
|
||||
n = ((n&0x000fffff)|0x00100000)>>(20-k);
|
||||
if(j<0) n = -n;
|
||||
p_h -= t;
|
||||
}
|
||||
t = p_l+p_h;
|
||||
SET_LOW_WORD(t,0);
|
||||
u = t*lg2_h;
|
||||
v = (p_l-(t-p_h))*lg2+t*lg2_l;
|
||||
z = u+v;
|
||||
w = v-(z-u);
|
||||
t = z*z;
|
||||
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
|
||||
r = (z*t1)/(t1-two)-(w+z*w);
|
||||
z = one-(r-z);
|
||||
GET_HIGH_WORD(j,z);
|
||||
j += (n<<20);
|
||||
if((j>>20)<=0) z = __builtin_scalbn(z,(int)n); /* subnormal output */
|
||||
else SET_HIGH_WORD(z,j);
|
||||
return s*z;
|
||||
}
|
||||
127
third_party/math/scalbn.c
vendored
Normal file
127
third_party/math/scalbn.c
vendored
Normal file
|
|
@ -0,0 +1,127 @@
|
|||
///////////////////////////////////////
|
||||
// Implements:
|
||||
// double scalbn(double x, int n);
|
||||
///////////////////////////////////////
|
||||
// Notes:
|
||||
// This is taken from newlib.
|
||||
// Only included because pow() needs it.
|
||||
// @nolint
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <stdint.h>
|
||||
#include <pblibc_private.h>
|
||||
|
||||
/*
|
||||
* scalbn (double x, int n)
|
||||
* scalbn(x,n) returns x* 2**n computed by exponent
|
||||
* manipulation rather than by actually performing an
|
||||
* exponentiation or a multiplication.
|
||||
*/
|
||||
|
||||
static const double
|
||||
two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
|
||||
twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */
|
||||
huge = 1.0e+300,
|
||||
tiny = 1.0e-300;
|
||||
|
||||
typedef union
|
||||
{
|
||||
double value;
|
||||
struct
|
||||
{
|
||||
uint32_t lsw;
|
||||
uint32_t msw;
|
||||
} parts;
|
||||
} ieee_double_shape_type;
|
||||
|
||||
#define EXTRACT_WORDS(ix0,ix1,d) \
|
||||
do { \
|
||||
ieee_double_shape_type ew_u; \
|
||||
ew_u.value = (d); \
|
||||
(ix0) = ew_u.parts.msw; \
|
||||
(ix1) = ew_u.parts.lsw; \
|
||||
} while (0)
|
||||
|
||||
/* Get the more significant 32 bit int from a double. */
|
||||
|
||||
#define GET_HIGH_WORD(i,d) \
|
||||
do { \
|
||||
ieee_double_shape_type gh_u; \
|
||||
gh_u.value = (d); \
|
||||
(i) = gh_u.parts.msw; \
|
||||
} while (0)
|
||||
|
||||
/* Get the less significant 32 bit int from a double. */
|
||||
|
||||
#define GET_LOW_WORD(i,d) \
|
||||
do { \
|
||||
ieee_double_shape_type gl_u; \
|
||||
gl_u.value = (d); \
|
||||
(i) = gl_u.parts.lsw; \
|
||||
} while (0)
|
||||
|
||||
/* Set the more significant 32 bits of a double from an int. */
|
||||
|
||||
#define SET_HIGH_WORD(d,v) \
|
||||
do { \
|
||||
ieee_double_shape_type sh_u; \
|
||||
sh_u.value = (d); \
|
||||
sh_u.parts.msw = (v); \
|
||||
(d) = sh_u.value; \
|
||||
} while (0)
|
||||
|
||||
/* Set the less significant 32 bits of a double from an int. */
|
||||
|
||||
#define SET_LOW_WORD(d,v) \
|
||||
do { \
|
||||
ieee_double_shape_type sl_u; \
|
||||
sl_u.value = (d); \
|
||||
sl_u.parts.lsw = (v); \
|
||||
(d) = sl_u.value; \
|
||||
} while (0)
|
||||
|
||||
static double copysign(double x, double y)
|
||||
{
|
||||
uint32_t hx,hy;
|
||||
GET_HIGH_WORD(hx,x);
|
||||
GET_HIGH_WORD(hy,y);
|
||||
SET_HIGH_WORD(x,(hx&0x7fffffff)|(hy&0x80000000));
|
||||
return x;
|
||||
}
|
||||
|
||||
double scalbn(double x, int n)
|
||||
{
|
||||
int32_t k,hx,lx;
|
||||
EXTRACT_WORDS(hx,lx,x);
|
||||
k = (hx&0x7ff00000)>>20; /* extract exponent */
|
||||
if (k==0) { /* 0 or subnormal x */
|
||||
if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */
|
||||
x *= two54;
|
||||
GET_HIGH_WORD(hx,x);
|
||||
k = ((hx&0x7ff00000)>>20) - 54;
|
||||
if (n< -50000) return tiny*x; /*underflow*/
|
||||
}
|
||||
if (k==0x7ff) return x+x; /* NaN or Inf */
|
||||
k = k+n;
|
||||
if (k > 0x7fe) return huge*copysign(huge,x); /* overflow */
|
||||
if (k > 0) /* normal result */
|
||||
{SET_HIGH_WORD(x,(hx&0x800fffff)|(k<<20)); return x;}
|
||||
if (k <= -54) {
|
||||
if (n > 50000) /* in case integer overflow in n+k */
|
||||
return huge*copysign(huge,x); /*overflow*/
|
||||
else return tiny*copysign(tiny,x); /*underflow*/
|
||||
}
|
||||
k += 54; /* subnormal result */
|
||||
SET_HIGH_WORD(x,(hx&0x800fffff)|(k<<20));
|
||||
return x*twom54;
|
||||
}
|
||||
254
third_party/math/sqrt.c
vendored
Normal file
254
third_party/math/sqrt.c
vendored
Normal file
|
|
@ -0,0 +1,254 @@
|
|||
///////////////////////////////////////
|
||||
// Implements:
|
||||
// double sqrt(double x);
|
||||
///////////////////////////////////////
|
||||
// Notes:
|
||||
// This is taken from newlib.
|
||||
// Only included because pow() needs it.
|
||||
// @nolint
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
#include <stdint.h>
|
||||
#include <pblibc_private.h>
|
||||
|
||||
typedef union
|
||||
{
|
||||
double value;
|
||||
struct
|
||||
{
|
||||
uint32_t lsw;
|
||||
uint32_t msw;
|
||||
} parts;
|
||||
} ieee_double_shape_type;
|
||||
|
||||
#define EXTRACT_WORDS(ix0,ix1,d) \
|
||||
do { \
|
||||
ieee_double_shape_type ew_u; \
|
||||
ew_u.value = (d); \
|
||||
(ix0) = ew_u.parts.msw; \
|
||||
(ix1) = ew_u.parts.lsw; \
|
||||
} while (0)
|
||||
|
||||
/* Get the more significant 32 bit int from a double. */
|
||||
|
||||
#define GET_HIGH_WORD(i,d) \
|
||||
do { \
|
||||
ieee_double_shape_type gh_u; \
|
||||
gh_u.value = (d); \
|
||||
(i) = gh_u.parts.msw; \
|
||||
} while (0)
|
||||
|
||||
/* Get the less significant 32 bit int from a double. */
|
||||
|
||||
#define GET_LOW_WORD(i,d) \
|
||||
do { \
|
||||
ieee_double_shape_type gl_u; \
|
||||
gl_u.value = (d); \
|
||||
(i) = gl_u.parts.lsw; \
|
||||
} while (0)
|
||||
|
||||
#define INSERT_WORDS(d,ix0,ix1) \
|
||||
do { \
|
||||
ieee_double_shape_type iw_u; \
|
||||
iw_u.parts.msw = (ix0); \
|
||||
iw_u.parts.lsw = (ix1); \
|
||||
(d) = iw_u.value; \
|
||||
} while (0)
|
||||
|
||||
/* Set the more significant 32 bits of a double from an int. */
|
||||
|
||||
#define SET_HIGH_WORD(d,v) \
|
||||
do { \
|
||||
ieee_double_shape_type sh_u; \
|
||||
sh_u.value = (d); \
|
||||
sh_u.parts.msw = (v); \
|
||||
(d) = sh_u.value; \
|
||||
} while (0)
|
||||
|
||||
/* Set the less significant 32 bits of a double from an int. */
|
||||
|
||||
#define SET_LOW_WORD(d,v) \
|
||||
do { \
|
||||
ieee_double_shape_type sl_u; \
|
||||
sl_u.value = (d); \
|
||||
sl_u.parts.lsw = (v); \
|
||||
(d) = sl_u.value; \
|
||||
} while (0)
|
||||
|
||||
/* __ieee754_sqrt(x)
|
||||
* Return correctly rounded sqrt.
|
||||
* ------------------------------------------
|
||||
* | Use the hardware sqrt if you have one |
|
||||
* ------------------------------------------
|
||||
* Method:
|
||||
* Bit by bit method using integer arithmetic. (Slow, but portable)
|
||||
* 1. Normalization
|
||||
* Scale x to y in [1,4) with even powers of 2:
|
||||
* find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
|
||||
* sqrt(x) = 2^k * sqrt(y)
|
||||
* 2. Bit by bit computation
|
||||
* Let q = sqrt(y) truncated to i bit after binary point (q = 1),
|
||||
* i 0
|
||||
* i+1 2
|
||||
* s = 2*q , and y = 2 * ( y - q ). (1)
|
||||
* i i i i
|
||||
*
|
||||
* To compute q from q , one checks whether
|
||||
* i+1 i
|
||||
*
|
||||
* -(i+1) 2
|
||||
* (q + 2 ) <= y. (2)
|
||||
* i
|
||||
* -(i+1)
|
||||
* If (2) is false, then q = q ; otherwise q = q + 2 .
|
||||
* i+1 i i+1 i
|
||||
*
|
||||
* With some algebric manipulation, it is not difficult to see
|
||||
* that (2) is equivalent to
|
||||
* -(i+1)
|
||||
* s + 2 <= y (3)
|
||||
* i i
|
||||
*
|
||||
* The advantage of (3) is that s and y can be computed by
|
||||
* i i
|
||||
* the following recurrence formula:
|
||||
* if (3) is false
|
||||
*
|
||||
* s = s , y = y ; (4)
|
||||
* i+1 i i+1 i
|
||||
*
|
||||
* otherwise,
|
||||
* -i -(i+1)
|
||||
* s = s + 2 , y = y - s - 2 (5)
|
||||
* i+1 i i+1 i i
|
||||
*
|
||||
* One may easily use induction to prove (4) and (5).
|
||||
* Note. Since the left hand side of (3) contain only i+2 bits,
|
||||
* it does not necessary to do a full (53-bit) comparison
|
||||
* in (3).
|
||||
* 3. Final rounding
|
||||
* After generating the 53 bits result, we compute one more bit.
|
||||
* Together with the remainder, we can decide whether the
|
||||
* result is exact, bigger than 1/2ulp, or less than 1/2ulp
|
||||
* (it will never equal to 1/2ulp).
|
||||
* The rounding mode can be detected by checking whether
|
||||
* huge + tiny is equal to huge, and whether huge - tiny is
|
||||
* equal to huge for some floating point number "huge" and "tiny".
|
||||
*
|
||||
* Special cases:
|
||||
* sqrt(+-0) = +-0 ... exact
|
||||
* sqrt(inf) = inf
|
||||
* sqrt(-ve) = NaN ... with invalid signal
|
||||
* sqrt(NaN) = NaN ... with invalid signal for signaling NaN
|
||||
*
|
||||
* Other methods : see the appended file at the end of the program below.
|
||||
*---------------
|
||||
*/
|
||||
|
||||
static const double one = 1.0, tiny=1.0e-300;
|
||||
|
||||
double sqrt(double x)
|
||||
{
|
||||
double z;
|
||||
int32_t sign = 0x80000000;
|
||||
uint32_t r,t1,s1,ix1,q1;
|
||||
int32_t ix0,s0,q,m,t,i;
|
||||
|
||||
EXTRACT_WORDS(ix0,ix1,x);
|
||||
|
||||
/* take care of Inf and NaN */
|
||||
if((ix0&0x7ff00000)==0x7ff00000) {
|
||||
return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
|
||||
sqrt(-inf)=sNaN */
|
||||
}
|
||||
/* take care of zero */
|
||||
if(ix0<=0) {
|
||||
if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
|
||||
else if(ix0<0)
|
||||
return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
|
||||
}
|
||||
/* normalize x */
|
||||
m = (ix0>>20);
|
||||
if(m==0) { /* subnormal x */
|
||||
while(ix0==0) {
|
||||
m -= 21;
|
||||
ix0 |= (ix1>>11); ix1 <<= 21;
|
||||
}
|
||||
for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
|
||||
m -= i-1;
|
||||
ix0 |= (ix1>>(32-i));
|
||||
ix1 <<= i;
|
||||
}
|
||||
m -= 1023; /* unbias exponent */
|
||||
ix0 = (ix0&0x000fffff)|0x00100000;
|
||||
if(m&1){ /* odd m, double x to make it even */
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
}
|
||||
m >>= 1; /* m = [m/2] */
|
||||
|
||||
/* generate sqrt(x) bit by bit */
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
|
||||
r = 0x00200000; /* r = moving bit from right to left */
|
||||
|
||||
while(r!=0) {
|
||||
t = s0+r;
|
||||
if(t<=ix0) {
|
||||
s0 = t+r;
|
||||
ix0 -= t;
|
||||
q += r;
|
||||
}
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
r>>=1;
|
||||
}
|
||||
|
||||
r = sign;
|
||||
while(r!=0) {
|
||||
t1 = s1+r;
|
||||
t = s0;
|
||||
if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
|
||||
s1 = t1+r;
|
||||
if(((t1&sign)==(uint32_t)sign)&&(s1&sign)==0) s0 += 1;
|
||||
ix0 -= t;
|
||||
if (ix1 < t1) ix0 -= 1;
|
||||
ix1 -= t1;
|
||||
q1 += r;
|
||||
}
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
r>>=1;
|
||||
}
|
||||
|
||||
/* use floating add to find out rounding direction */
|
||||
if((ix0|ix1)!=0) {
|
||||
z = one-tiny; /* trigger inexact flag */
|
||||
if (z>=one) {
|
||||
z = one+tiny;
|
||||
if (q1==(uint32_t)0xffffffff) { q1=0; q += 1;}
|
||||
else if (z>one) {
|
||||
if (q1==(uint32_t)0xfffffffe) q+=1;
|
||||
q1+=2;
|
||||
} else
|
||||
q1 += (q1&1);
|
||||
}
|
||||
}
|
||||
ix0 = (q>>1)+0x3fe00000;
|
||||
ix1 = q1>>1;
|
||||
if ((q&1)==1) ix1 |= sign;
|
||||
ix0 += (m <<20);
|
||||
INSERT_WORDS(z,ix0,ix1);
|
||||
return z;
|
||||
}
|
||||
Loading…
Add table
Add a link
Reference in a new issue