diff options
| -rw-r--r-- | lib/libm/src/k_cos.c | 45 | ||||
| -rw-r--r-- | lib/libm/src/s_cos.c | 6 |
2 files changed, 23 insertions, 28 deletions
diff --git a/lib/libm/src/k_cos.c b/lib/libm/src/k_cos.c index 8f3882b6a00..0839243e90c 100644 --- a/lib/libm/src/k_cos.c +++ b/lib/libm/src/k_cos.c @@ -36,13 +36,17 @@ * ~ cos(x) - x*y, * a correction term is necessary in cos(x) and hence * cos(x+y) = 1 - (x*x/2 - (r - x*y)) - * For better accuracy when x > 0.3, let qx = |x|/4 with - * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. - * Then - * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). - * Note that 1-qx and (x*x/2-qx) is EXACT here, and the - * magnitude of the latter is at least a quarter of x*x/2, - * thus, reducing the rounding error in the subtraction. + * For better accuracy, rearrange to + * cos(x+y) ~ w + (tmp + (r-x*y)) + * where w = 1 - x*x/2 and tmp is a tiny correction term + * (1 - x*x/2 == w + tmp exactly in infinite precision). + * The exactness of w + tmp in infinite precision depends on w + * and tmp having the same precision as x. If they have extra + * precision due to compiler bugs, then the extra precision is + * only good provided it is retained in all terms of the final + * expression for cos(). Retention happens in all cases tested + * under FreeBSD, so don't pessimize things by forcibly clipping + * any extra precision in w. */ #include "math.h" @@ -60,25 +64,12 @@ C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ double __kernel_cos(double x, double y) { - double a,hz,z,r,qx; - int32_t ix; - GET_HIGH_WORD(ix,x); - ix &= 0x7fffffff; /* ix = |x|'s high word*/ - if(ix<0x3e400000) { /* if x < 2**27 */ - if(((int)x)==0) return one; /* generate inexact */ - } + double hz,z,r,w; + z = x*x; - r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); - if(ix < 0x3FD33333) /* if |x| < 0.3 */ - return one - (0.5*z - (z*r - x*y)); - else { - if(ix > 0x3fe90000) { /* x > 0.78125 */ - qx = 0.28125; - } else { - INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */ - } - hz = 0.5*z-qx; - a = one-qx; - return a - (hz - (z*r-x*y)); - } + w = z*z; + r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6)); + hz = 0.5*z; + w = one-hz; + return w + (((one-w)-hz) + (z*r-x*y)); } diff --git a/lib/libm/src/s_cos.c b/lib/libm/src/s_cos.c index 8b923d5fe61..1406504e9ab 100644 --- a/lib/libm/src/s_cos.c +++ b/lib/libm/src/s_cos.c @@ -57,7 +57,11 @@ cos(double x) /* |x| ~< pi/4 */ ix &= 0x7fffffff; - if(ix <= 0x3fe921fb) return __kernel_cos(x,z); + if(ix <= 0x3fe921fb) { + if(ix<0x3e46a09e) /* if x < 2**-27 * sqrt(2) */ + if(((int)x)==0) return 1.0; /* generate inexact */ + return __kernel_cos(x,z); + } /* cos(Inf or NaN) is NaN */ else if (ix>=0x7ff00000) return x-x; |