/* * Copyright (c) 2001, 2022, Oracle and/or its affiliates. All rights reserved. * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. * * This code is free software; you can redistribute it and/or modify it * under the terms of the GNU General Public License version 2 only, as * published by the Free Software Foundation. * * This code is distributed in the hope that it will be useful, but WITHOUT * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License * version 2 for more details (a copy is included in the LICENSE file that * accompanied this code). * * You should have received a copy of the GNU General Public License version * 2 along with this work; if not, write to the Free Software Foundation, * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. * * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA * or visit www.oracle.com if you need additional information or have any * questions. *
*/
// This file contains copies of the fdlibm routines used by // StrictMath. It turns out that it is almost always required to use // these runtime routines; the Intel CPU doesn't meet the Java // specification for sin/cos outside a certain limited argument range, // and the SPARC CPU doesn't appear to have sin/cos instructions. It // also turns out that avoiding the indirect call through function // pointer out to libjava.so in SharedRuntime speeds these routines up // by roughly 15% on both Win32/x86 and Solaris/SPARC.
/* * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) * double x[],y[]; int e0,nx,prec; int ipio2[]; * * __kernel_rem_pio2 return the last three digits of N with * y = x - N*pi/2 * so that |y| < pi/2. * * The method is to compute the integer (mod 8) and fraction parts of * (2/pi)*x without doing the full multiplication. In general we * skip the part of the product that are known to be a huge integer ( * more accurately, = 0 mod 8 ). Thus the number of operations are * independent of the exponent of the input. * * (2/pi) is represented by an array of 24-bit integers in ipio2[]. * * Input parameters: * x[] The input value (must be positive) is broken into nx * pieces of 24-bit integers in double precision format. * x[i] will be the i-th 24 bit of x. The scaled exponent * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 * match x's up to 24 bits. * * Example of breaking a double positive z into x[0]+x[1]+x[2]: * e0 = ilogb(z)-23 * z = scalbn(z,-e0) * for i = 0,1,2 * x[i] = floor(z) * z = (z-x[i])*2**24 * * * y[] output result in an array of double precision numbers. * The dimension of y[] is: * 24-bit precision 1 * 53-bit precision 2 * 64-bit precision 2 * 113-bit precision 3 * The actual value is the sum of them. Thus for 113-bit * precsion, one may have to do something like: * * long double t,w,r_head, r_tail; * t = (long double)y[2] + (long double)y[1]; * w = (long double)y[0]; * r_head = t+w; * r_tail = w - (r_head - t); * * e0 The exponent of x[0] * * nx dimension of x[] * * prec an integer indicating the precision: * 0 24 bits (single) * 1 53 bits (double) * 2 64 bits (extended) * 3 113 bits (quad) * * ipio2[] * integer array, contains the (24*i)-th to (24*i+23)-th * bit of 2/pi after binary point. The corresponding * floating value is * * ipio2[i] * 2^(-24(i+1)). * * External function: * double scalbn(), floor(); * * * Here is the description of some local variables: * * jk jk+1 is the initial number of terms of ipio2[] needed * in the computation. The recommended value is 2,3,4, * 6 for single, double, extended,and quad. * * jz local integer variable indicating the number of * terms of ipio2[] used. * * jx nx - 1 * * jv index for pointing to the suitable ipio2[] for the * computation. In general, we want * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 * is an integer. Thus * e0-3-24*jv >= 0 or (e0-3)/24 >= jv * Hence jv = max(0,(e0-3)/24). * * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. * * q[] double array with integral value, representing the * 24-bits chunk of the product of x and 2/pi. * * q0 the corresponding exponent of q[0]. Note that the * exponent for q[i] would be q0-24*i. * * PIo2[] double precision array, obtained by cutting pi/2 * into 24 bits chunks. * * f[] ipio2[] in floating point * * iq[] integer array by breaking up q[] in 24-bits chunk. * * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] * * ih integer. If >0 it indicates q[] is >= 0.5, hence * it also indicates the *sign* of the result. *
*/
/* * 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.
*/
staticconstint init_jk[] = {2,3,4,6}; /* initial value for jk */
staticint __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, constint *ipio2) { int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; double z,fw,f[20],fq[20],q[20];
/* compute n */
z = scalbnA(z,q0); /* actual value of z */
z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
n = (int) z;
z -= (double)n;
ih = 0; if(q0>0) { /* need iq[jz-1] to determine n */
i = (iq[jz-1]>>(24-q0)); n += i;
iq[jz-1] -= i<<(24-q0);
ih = iq[jz-1]>>(23-q0);
} elseif(q0==0) ih = iq[jz-1]>>23; elseif(z>=0.5) ih=2;
/* * ==================================================== * Copyright (c) 1993 Oracle and/or its affiliates. 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. * ==================================================== *
*/
/* __ieee754_rem_pio2(x,y) * * return the remainder of x rem pi/2 in y[0]+y[1] * use __kernel_rem_pio2()
*/
/* * invpio2: 53 bits of 2/pi * pio2_1: first 33 bit of pi/2 * pio2_1t: pi/2 - pio2_1 * pio2_2: second 33 bit of pi/2 * pio2_2t: pi/2 - (pio2_1+pio2_2) * pio2_3: third 33 bit of pi/2 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
*/
i0 = ((*(int*)&two24A)>>30)^1; /* high word index */
hx = *(i0+(int*)&x); /* high word of x */
ix = hx&0x7fffffff; if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
{y[0] = x; y[1] = 0; return 0;} if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ if(hx>0) {
z = x - pio2_1; if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
y[0] = z - pio2_1t;
y[1] = (z-y[0])-pio2_1t;
} else { /* near pi/2, use 33+33+53 bit pi */
z -= pio2_2;
y[0] = z - pio2_2t;
y[1] = (z-y[0])-pio2_2t;
} return 1;
} else { /* negative x */
z = x + pio2_1; if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
y[0] = z + pio2_1t;
y[1] = (z-y[0])+pio2_1t;
} else { /* near pi/2, use 33+33+53 bit pi */
z += pio2_2;
y[0] = z + pio2_2t;
y[1] = (z-y[0])+pio2_2t;
} return -1;
}
} if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
t = fabsd(x);
n = (int) (t*invpio2+half);
fn = (double)n;
r = t-fn*pio2_1;
w = fn*pio2_1t; /* 1st round good to 85 bit */ if(n<32&&ix!=npio2_hw[n-1]) {
y[0] = r-w; /* quick check no cancellation */
} else {
j = ix>>20;
y[0] = r-w;
i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); if(i>16) { /* 2nd iteration needed, good to 118 */
t = r;
w = fn*pio2_2;
r = t-w;
w = fn*pio2_2t-((t-r)-w);
y[0] = r-w;
i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); if(i>49) { /* 3rd iteration need, 151 bits acc */
t = r; /* will cover all possible cases */
w = fn*pio2_3;
r = t-w;
w = fn*pio2_3t-((t-r)-w);
y[0] = r-w;
}
}
}
y[1] = (r-y[0])-w; if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} elsereturn n;
} /* * all other (large) arguments
*/ if(ix>=0x7ff00000) { /* x is inf or NaN */
y[0]=y[1]=x-x; return 0;
} /* set z = scalbn(|x|,ilogb(x)-23) */
*(1-i0+(int*)&z) = *(1-i0+(int*)&x);
e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
*(i0+(int*)&z) = ix - (e0<<20); for(i=0;i<2;i++) {
tx[i] = (double)((int)(z));
z = (z-tx[i])*two24A;
}
tx[2] = z;
nx = 3; while(tx[nx-1]==zeroA) nx--; /* skip zero term */
n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi); if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} return n;
}
/* __kernel_sin( x, y, iy) * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). * * Algorithm * 1. Since sin(-x) = -sin(x), we need only to consider positive x. * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. * 3. sin(x) is approximated by a polynomial of degree 13 on * [0,pi/4] * 3 13 * sin(x) ~ x + S1*x + ... + S6*x * where * * |sin(x) 2 4 6 8 10 12 | -58 * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 * | x | * * 4. sin(x+y) = sin(x) + sin'(x')*y * ~ sin(x) + (1-x*x/2)*y * For better accuracy, let * 3 2 2 2 2 * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) * then 3 2 * sin(x) = x + (S1*x + (x *(r-y/2)+y))
*/
staticdouble __kernel_sin(double x, double y, int iy)
{ double z,r,v; int ix;
ix = high(x)&0x7fffffff; /* high word of x */ if(ix<0x3e400000) /* |x| < 2**-27 */
{if((int)x==0) return x;} /* generate inexact */
z = x*x;
v = z*x;
r = S2+z*(S3+z*(S4+z*(S5+z*S6))); if(iy==0) return x+v*(S1+z*r); elsereturn x-((z*(half*y-v*r)-y)-v*S1);
}
/* * __kernel_cos( x, y ) * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * * Algorithm * 1. Since cos(-x) = cos(x), we need only to consider positive x. * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. * 3. cos(x) is approximated by a polynomial of degree 14 on * [0,pi/4] * 4 14 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x * where the remez error is * * | 2 4 6 8 10 12 14 | -58 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 * | | * * 4 6 8 10 12 14 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then * cos(x) = 1 - x*x/2 + r * since cos(x+y) ~ cos(x) - sin(x)*y * ~ 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.
*/
staticdouble __kernel_cos(double x, double y)
{ double a,h,z,r,qx=0; int ix;
ix = high(x)&0x7fffffff; /* ix = |x|'s high word*/ if(ix<0x3e400000) { /* if x < 2**27 */ if(((int)x)==0) return one; /* generate inexact */
}
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 {
set_high(&qx, ix-0x00200000); /* x/4 */
set_low(&qx, 0);
}
h = 0.5*z-qx;
a = one-qx; return a - (h - (z*r-x*y));
}
}
/* __kernel_tan( x, y, k ) * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * Input k indicates whether tan (if k=1) or * -1/tan (if k= -1) is returned. * * Algorithm * 1. Since tan(-x) = -tan(x), we need only to consider positive x. * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. * 3. tan(x) is approximated by a odd polynomial of degree 27 on * [0,0.67434] * 3 27 * tan(x) ~ x + T1*x + ... + T13*x * where * * |tan(x) 2 4 26 | -59.2 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 * | x | * * Note: tan(x+y) = tan(x) + tan'(x)*y * ~ tan(x) + (1+x*x)*y * Therefore, for better accuracy in computing tan(x+y), let * 3 2 2 2 2 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) * then * 3 2 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) * * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
*/
staticdouble __kernel_tan(double x, double y, int iy)
{ double z,r,v,w,s; int ix,hx;
hx = high(x); /* high word of x */
ix = hx&0x7fffffff; /* high word of |x| */ if(ix<0x3e300000) { /* x < 2**-28 */ if((int)x==0) { /* generate inexact */ if (((ix | low(x)) | (iy + 1)) == 0) return one / fabsd(x); else { if (iy == 1) return x; else { /* compute -1 / (x+y) carefully */ double a, t;
z = w = x + y;
set_low(&z, 0);
v = y - (z - x);
t = a = -one / w;
set_low(&t, 0);
s = one + t * z; return t + a * (s + t * v);
}
}
}
} if(ix>=0x3FE59428) { /* |x|>=0.6744 */ if(hx<0) {x = -x; y = -y;}
z = pio4-x;
w = pio4lo-y;
x = z+w; y = 0.0;
}
z = x*x;
w = z*z; /* Break x^5*(T[1]+x^2*T[2]+...) into * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
*/
r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
s = z*x;
r = y + z*(s*(r+v)+y);
r += T[0]*s;
w = x+r; if(ix>=0x3FE59428) {
v = (double)iy; return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
} if(iy==1) return w; else { /* if allow error up to 2 ulp,
simply return -1.0/(x+r) here */ /* compute -1.0/(x+r) accurately */ double a,t;
z = w;
set_low(&z, 0);
v = r-(z - x); /* z+v = r+x */
t = a = -1.0/w; /* a = -1.0/w */
set_low(&t, 0);
s = 1.0+t*z; return t+a*(s+t*v);
}
}
//---------------------------------------------------------------------- // // Routines for new sin/cos implementation // //----------------------------------------------------------------------
/* sin(x) * Return sine function of x. * * kernel function: * __kernel_sin ... sine function on [-pi/4,pi/4] * __kernel_cos ... cose function on [-pi/4,pi/4] * __ieee754_rem_pio2 ... argument reduction routine * * Method. * Let S,C and T denote the sin, cos and tan respectively on * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 * in [-pi/4 , +pi/4], and let n = k mod 4. * We have * * n sin(x) cos(x) tan(x) * ---------------------------------------------------------- * 0 S C T * 1 C -S -1/T * 2 -S -C T * 3 -C S -1/T * ---------------------------------------------------------- * * Special cases: * Let trig be any of sin, cos, or tan. * trig(+-INF) is NaN, with signals; * trig(NaN) is that NaN; * * Accuracy: * TRIG(x) returns trig(x) nearly rounded
*/
JRT_LEAF(jdouble, SharedRuntime::dsin(jdouble x)) double y[2],z=0.0; int n, ix;
/* sin(Inf or NaN) is NaN */ elseif (ix>=0x7ff00000) return x-x;
/* argument reduction needed */ else {
n = __ieee754_rem_pio2(x,y); switch(n&3) { case 0: return __kernel_sin(y[0],y[1],1); case 1: return __kernel_cos(y[0],y[1]); case 2: return -__kernel_sin(y[0],y[1],1); default: return -__kernel_cos(y[0],y[1]);
}
}
JRT_END
/* cos(x) * Return cosine function of x. * * kernel function: * __kernel_sin ... sine function on [-pi/4,pi/4] * __kernel_cos ... cosine function on [-pi/4,pi/4] * __ieee754_rem_pio2 ... argument reduction routine * * Method. * Let S,C and T denote the sin, cos and tan respectively on * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 * in [-pi/4 , +pi/4], and let n = k mod 4. * We have * * n sin(x) cos(x) tan(x) * ---------------------------------------------------------- * 0 S C T * 1 C -S -1/T * 2 -S -C T * 3 -C S -1/T * ---------------------------------------------------------- * * Special cases: * Let trig be any of sin, cos, or tan. * trig(+-INF) is NaN, with signals; * trig(NaN) is that NaN; * * Accuracy: * TRIG(x) returns trig(x) nearly rounded
*/
JRT_LEAF(jdouble, SharedRuntime::dcos(jdouble x)) double y[2],z=0.0; int n, ix;
/* cos(Inf or NaN) is NaN */ elseif (ix>=0x7ff00000) return x-x;
/* argument reduction needed */ else {
n = __ieee754_rem_pio2(x,y); switch(n&3) { case 0: return __kernel_cos(y[0],y[1]); case 1: return -__kernel_sin(y[0],y[1],1); case 2: return -__kernel_cos(y[0],y[1]); default: return __kernel_sin(y[0],y[1],1);
}
}
JRT_END
/* tan(x) * Return tangent function of x. * * kernel function: * __kernel_tan ... tangent function on [-pi/4,pi/4] * __ieee754_rem_pio2 ... argument reduction routine * * Method. * Let S,C and T denote the sin, cos and tan respectively on * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 * in [-pi/4 , +pi/4], and let n = k mod 4. * We have * * n sin(x) cos(x) tan(x) * ---------------------------------------------------------- * 0 S C T * 1 C -S -1/T * 2 -S -C T * 3 -C S -1/T * ---------------------------------------------------------- * * Special cases: * Let trig be any of sin, cos, or tan. * trig(+-INF) is NaN, with signals; * trig(NaN) is that NaN; * * Accuracy: * TRIG(x) returns trig(x) nearly rounded
*/
JRT_LEAF(jdouble, SharedRuntime::dtan(jdouble x)) double y[2],z=0.0; int n, ix;
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