--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/hotspot/cpu/aarch64/macroAssembler_aarch64_trig.cpp Mon Jun 25 16:30:49 2018 +0300
@@ -0,0 +1,1488 @@
+/* Copyright (c) 2018, Oracle and/or its affiliates. All rights reserved.
+ * Copyright (c) 2018, Cavium. All rights reserved. (By BELLSOFT)
+ * 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.
+ *
+ */
+
+#include "precompiled.hpp"
+#include "asm/assembler.hpp"
+#include "asm/assembler.inline.hpp"
+#include "runtime/stubRoutines.hpp"
+#include "macroAssembler_aarch64.hpp"
+
+// The following code is a optimized version of fdlibm sin/cos implementation
+// (C code is in share/runtime/sharedRuntimeTrig.cpp) adapted for AARCH64.
+
+// Please refer to sin/cos approximation via polynomial and
+// trigonometric argument reduction techniques to the following literature:
+//
+// [1] Muller, Jean-Michel, Nicolas Brisebarre, Florent De Dinechin,
+// Claude-Pierre Jeannerod, Vincent Lefevre, Guillaume Melquiond,
+// Nathalie Revol, Damien Stehlé, and Serge Torres:
+// Handbook of floating-point arithmetic.
+// Springer Science & Business Media, 2009.
+// [2] K. C. Ng
+// Argument Reduction for Huge Arguments: Good to the Last Bit
+// July 13, 1992, SunPro
+//
+// HOW TO READ THIS CODE:
+// This code consists of several functions. Each function has following header:
+// 1) Description
+// 2) C-pseudo code with differences from fdlibm marked by comments starting
+// with "NOTE". Check unmodified fdlibm code in
+// share/runtime/SharedRuntimeTrig.cpp
+// 3) Brief textual description of changes between fdlibm and current
+// implementation along with optimization notes (if applicable)
+// 4) Assumptions, input and output
+// 5) (Optional) additional notes about intrinsic implementation
+// Each function is separated in blocks which follow the pseudo-code structure
+//
+// HIGH-LEVEL ALGORITHM DESCRIPTION:
+// - entry point: generate_dsin_dcos(...);
+// - check corner cases: NaN, INF, tiny argument.
+// - check if |x| < Pi/4. Then approximate sin/cos via polynomial (kernel_sin/kernel_cos)
+// -- else proceed to argument reduction routine (__ieee754_rem_pio2) and
+// use reduced argument to get result via kernel_sin/kernel_cos
+//
+// HIGH-LEVEL CHANGES BETWEEN INTRINSICS AND FDLIBM:
+// 1) two_over_pi table fdlibm representation is int[], while intrinsic version
+// has these int values converted to double representation to load converted
+// double values directly (see stubRoutines_aarch4::_two_over_pi)
+// 2) Several loops are unrolled and vectorized: see comments in code after
+// labels: SKIP_F_LOAD, RECOMP_FOR1_CHECK, RECOMP_FOR2
+// 3) fdlibm npio2_hw table now has "prefix" with constants used in
+// calculation. These constants are loaded from npio2_hw table instead of
+// constructing it in code (see stubRoutines_aarch64.cpp)
+// 4) Polynomial coefficients for sin and cos are moved to table sin_coef
+// and cos_coef to use the same optimization as in 3). It allows to load most of
+// required constants via single instruction
+//
+//
+//
+///* __ieee754_rem_pio2(x,y)
+// *
+// * returns the remainder of x rem pi/2 in y[0]+y[1] (i.e. like x div pi/2)
+// * x is input argument, y[] is hi and low parts of reduced argument (x)
+// * uses __kernel_rem_pio2()
+// */
+// // use tables(see stubRoutines_aarch64.cpp): two_over_pi and modified npio2_hw
+//
+// BEGIN __ieee754_rem_pio2 PSEUDO CODE
+//
+//static int __ieee754_rem_pio2(double x, double *y) {
+// double z,w,t,r,fn;
+// double tx[3];
+// int e0,i,j,nx,n,ix,hx,i0;
+//
+// i0 = ((*(int*)&two24A)>>30)^1; /* high word index */
+// hx = *(i0+(int*)&x); /* high word of x */
+// ix = hx&0x7fffffff;
+// 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 */
+// // NOTE: y[0] = r-w; is moved from if/else below to be before "if"
+// y[0] = r-w;
+// if(n<32&&ix!=npio2_hw[n-1]) {
+// // y[0] = r-w; /* quick check no cancellation */ // NOTE: moved earlier
+// } else {
+// j = ix>>20;
+// // y[0] = r-w; // NOTE: moved earlier
+// 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;}
+// else return n;
+// }
+// /*
+// * all other (large) arguments
+// */
+// // NOTE: this check is removed, because it was checked in dsin/dcos
+// // 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);
+//
+// // NOTE: "for" loop below in unrolled. See comments in asm code
+// for(i=0;i<2;i++) {
+// tx[i] = (double)((int)(z));
+// z = (z-tx[i])*two24A;
+// }
+//
+// tx[2] = z;
+// nx = 3;
+//
+// // NOTE: while(tx[nx-1]==zeroA) nx--; is unrolled. See comments in asm code
+// 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;
+//}
+//
+// END __ieee754_rem_pio2 PSEUDO CODE
+//
+// Changes between fdlibm and intrinsic for __ieee754_rem_pio2:
+// 1. INF/NaN check for huge argument is removed in comparison with fdlibm
+// code, because this check is already done in dcos/dsin code
+// 2. Most constants are now loaded from table instead of direct initialization
+// 3. Two loops are unrolled
+// Assumptions:
+// 1. Assume |X| >= PI/4
+// 2. Assume rscratch1 = 0x3fe921fb00000000 (~ PI/4)
+// 3. Assume ix = r3
+// Input and output:
+// 1. Input: X = r0
+// 2. Return n in r2, y[0] == y0 == v4, y[1] == y1 == v5
+// NOTE: general purpose register names match local variable names in C code
+// NOTE: fpu registers are actively reused. See comments in code about their usage
+void MacroAssembler::generate__ieee754_rem_pio2(address npio2_hw,
+ address two_over_pi, address pio2) {
+ const long PIO2_1t = 0x3DD0B4611A626331UL;
+ const long PIO2_2 = 0x3DD0B4611A600000UL;
+ const long PIO2_2t = 0x3BA3198A2E037073UL;
+ Label X_IS_NEGATIVE, X_IS_MEDIUM_OR_LARGE, X_IS_POSITIVE_LONG_PI, LARGE_ELSE,
+ REDUCTION_DONE, X_IS_MEDIUM_BRANCH_DONE, X_IS_LARGE, NX_SET,
+ X_IS_NEGATIVE_LONG_PI;
+ Register X = r0, n = r2, ix = r3, jv = r4, tmp5 = r5, jx = r6,
+ tmp3 = r7, iqBase = r10, ih = r11, i = r17;
+ // initializing constants first
+ // rscratch1 = 0x3fe921fb00000000 (see assumptions)
+ movk(rscratch1, 0x3ff9, 48); // was 0x3fe921fb0..0 now it's 0x3ff921fb0..0
+ mov(rscratch2, 0x4002d97c); // 3*PI/4 high word
+ movk(rscratch1, 0x5440, 16); // now rscratch1 == PIO2_1
+ fmovd(v1, rscratch1); // v1 = PIO2_1
+ cmp(rscratch2, ix);
+ br(LE, X_IS_MEDIUM_OR_LARGE);
+
+ block_comment("if(ix<0x4002d97c) {... /* |x| ~< 3pi/4 */ "); {
+ cmp(X, zr);
+ br(LT, X_IS_NEGATIVE);
+
+ block_comment("if(hx>0) {"); {
+ fsubd(v2, v0, v1); // v2 = z = x - pio2_1
+ cmp(ix, rscratch1, LSR, 32);
+ mov(n, 1);
+ br(EQ, X_IS_POSITIVE_LONG_PI);
+
+ block_comment("case: hx > 0 && ix!=0x3ff921fb {"); { /* 33+53 bit pi is good enough */
+ mov(rscratch2, PIO2_1t);
+ fmovd(v27, rscratch2);
+ fsubd(v4, v2, v27); // v4 = y[0] = z - pio2_1t;
+ fsubd(v5, v2, v4);
+ fsubd(v5, v5, v27); // v5 = y[1] = (z-y[0])-pio2_1t
+ b(REDUCTION_DONE);
+ }
+
+ block_comment("case: hx > 0 &*& ix==0x3ff921fb {"); { /* near pi/2, use 33+33+53 bit pi */
+ bind(X_IS_POSITIVE_LONG_PI);
+ mov(rscratch1, PIO2_2);
+ mov(rscratch2, PIO2_2t);
+ fmovd(v27, rscratch1);
+ fmovd(v6, rscratch2);
+ fsubd(v2, v2, v27); // z-= pio2_2
+ fsubd(v4, v2, v6); // y[0] = z - pio2_2t
+ fsubd(v5, v2, v4);
+ fsubd(v5, v5, v6); // v5 = (z - y[0]) - pio2_2t
+ b(REDUCTION_DONE);
+ }
+ }
+
+ block_comment("case: hx <= 0)"); {
+ bind(X_IS_NEGATIVE);
+ faddd(v2, v0, v1); // v2 = z = x + pio2_1
+ cmp(ix, rscratch1, LSR, 32);
+ mov(n, -1);
+ br(EQ, X_IS_NEGATIVE_LONG_PI);
+
+ block_comment("case: hx <= 0 && ix!=0x3ff921fb) {"); { /* 33+53 bit pi is good enough */
+ mov(rscratch2, PIO2_1t);
+ fmovd(v27, rscratch2);
+ faddd(v4, v2, v27); // v4 = y[0] = z + pio2_1t;
+ fsubd(v5, v2, v4);
+ faddd(v5, v5, v27); // v5 = y[1] = (z-y[0]) + pio2_1t
+ b(REDUCTION_DONE);
+ }
+
+ block_comment("case: hx <= 0 && ix==0x3ff921fb"); { /* near pi/2, use 33+33+53 bit pi */
+ bind(X_IS_NEGATIVE_LONG_PI);
+ mov(rscratch1, PIO2_2);
+ mov(rscratch2, PIO2_2t);
+ fmovd(v27, rscratch1);
+ fmovd(v6, rscratch2);
+ faddd(v2, v2, v27); // z += pio2_2
+ faddd(v4, v2, v6); // y[0] = z + pio2_2t
+ fsubd(v5, v2, v4);
+ faddd(v5, v5, v6); // v5 = (z - y[0]) + pio2_2t
+ b(REDUCTION_DONE);
+ }
+ }
+ }
+ bind(X_IS_MEDIUM_OR_LARGE);
+ mov(rscratch1, 0x413921fb);
+ cmp(ix, rscratch1); // ix < = 0x413921fb ?
+ br(GT, X_IS_LARGE);
+
+ block_comment("|x| ~<= 2^19*(pi/2), medium size"); {
+ lea(ih, ExternalAddress(npio2_hw));
+ ld1(v4, v5, v6, v7, T1D, ih);
+ fabsd(v31, v0); // v31 = t = |x|
+ add(ih, ih, 64);
+ fmaddd(v2, v31, v5, v4); // v2 = t * invpio2 + half (invpio2 = 53 bits of 2/pi, half = 0.5)
+ fcvtzdw(n, v2); // n = (int) v2
+ frintzd(v2, v2);
+ fmsubd(v3, v2, v6, v31); // v3 = r = t - fn * pio2_1
+ fmuld(v26, v2, v7); // v26 = w = fn * pio2_1t
+ fsubd(v4, v3, v26); // y[0] = r - w. Calculated before branch
+ cmp(n, 32);
+ br(GT, LARGE_ELSE);
+ subw(tmp5, n, 1); // tmp5 = n - 1
+ ldrw(jv, Address(ih, tmp5, Address::lsl(2)));
+ cmp(ix, jv);
+ br(NE, X_IS_MEDIUM_BRANCH_DONE);
+
+ block_comment("else block for if(n<32&&ix!=npio2_hw[n-1])"); {
+ bind(LARGE_ELSE);
+ fmovd(jx, v4);
+ lsr(tmp5, ix, 20); // j = ix >> 20
+ lsl(jx, jx, 1);
+ sub(tmp3, tmp5, jx, LSR, 32 + 20 + 1); // r7 = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff);
+
+ block_comment("if(i>16)"); {
+ cmp(tmp3, 16);
+ br(LE, X_IS_MEDIUM_BRANCH_DONE);
+ // i > 16. 2nd iteration needed
+ ldpd(v6, v7, Address(ih, -32));
+ fmovd(v28, v3); // t = r
+ fmuld(v29, v2, v6); // w = v29 = fn * pio2_2
+ fsubd(v3, v28, v29); // r = t - w
+ fsubd(v31, v28, v3); // v31 = (t - r)
+ fsubd(v31, v29, v31); // v31 = w - (t - r) = - ((t - r) - w)
+ fmaddd(v26, v2, v7, v31); // v26 = w = fn*pio2_2t - ((t - r) - w)
+ fsubd(v4, v3, v26); // y[0] = r - w
+ fmovd(jx, v4);
+ lsl(jx, jx, 1);
+ sub(tmp3, tmp5, jx, LSR, 32 + 20 + 1); // r7 = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff);
+
+ block_comment("if(i>49)"); {
+ cmp(tmp3, 49);
+ br(LE, X_IS_MEDIUM_BRANCH_DONE);
+ // 3rd iteration need, 151 bits acc
+ ldpd(v6, v7, Address(ih, -16));
+ fmovd(v28, v3); // save "r"
+ fmuld(v29, v2, v6); // v29 = fn * pio2_3
+ fsubd(v3, v28, v29); // r = r - w
+ fsubd(v31, v28, v3); // v31 = (t - r)
+ fsubd(v31, v29, v31); // v31 = w - (t - r) = - ((t - r) - w)
+ fmaddd(v26, v2, v7, v31); // v26 = w = fn*pio2_3t - ((t - r) - w)
+ fsubd(v4, v3, v26); // y[0] = r - w
+ }
+ }
+ }
+ block_comment("medium x tail"); {
+ bind(X_IS_MEDIUM_BRANCH_DONE);
+ fsubd(v5, v3, v4); // v5 = y[1] = (r - y[0])
+ fsubd(v5, v5, v26); // v5 = y[1] = (r - y[0]) - w
+ cmp(X, zr);
+ br(GT, REDUCTION_DONE);
+ fnegd(v4, v4);
+ negw(n, n);
+ fnegd(v5, v5);
+ b(REDUCTION_DONE);
+ }
+ }
+
+ block_comment("all other (large) arguments"); {
+ bind(X_IS_LARGE);
+ lsr(rscratch1, ix, 20); // ix >> 20
+ movz(tmp5, 0x4170, 48);
+ subw(rscratch1, rscratch1, 1046); // e0
+ fmovd(v10, tmp5); // init two24A value
+ subw(jv, ix, rscratch1, LSL, 20); // ix - (e0<<20)
+ lsl(jv, jv, 32);
+ subw(rscratch2, rscratch1, 3);
+ bfm(jv, X, 0, 31); // jv = z
+ movw(i, 24);
+ fmovd(v26, jv); // v26 = z
+
+ block_comment("unrolled for(i=0;i<2;i++) {tx[i] = (double)((int)(z));z = (z-tx[i])*two24A;}"); {
+ // tx[0,1,2] = v6,v7,v26
+ frintzd(v6, v26); // v6 = (double)((int)v26)
+ sdivw(jv, rscratch2, i); // jv = (e0 - 3)/24
+ fsubd(v26, v26, v6);
+ sub(sp, sp, 560);
+ fmuld(v26, v26, v10);
+ frintzd(v7, v26); // v7 = (double)((int)v26)
+ movw(jx, 2); // calculate jx as nx - 1, which is initially 2. Not a part of unrolled loop
+ fsubd(v26, v26, v7);
+ }
+
+ block_comment("nx calculation with unrolled while(tx[nx-1]==zeroA) nx--;"); {
+ fcmpd(v26, 0.0d); // if NE then jx == 2. else it's 1 or 0
+ add(iqBase, sp, 480); // base of iq[]
+ fmuld(v3, v26, v10);
+ br(NE, NX_SET);
+ fcmpd(v7, 0.0d); // v7 == 0 => jx = 0. Else jx = 1
+ csetw(jx, NE);
+ }
+ bind(NX_SET);
+ generate__kernel_rem_pio2(two_over_pi, pio2);
+ // now we have y[0] = v4, y[1] = v5 and n = r2
+ cmp(X, zr);
+ br(GE, REDUCTION_DONE);
+ fnegd(v4, v4);
+ fnegd(v5, v5);
+ negw(n, n);
+ }
+ bind(REDUCTION_DONE);
+}
+
+///*
+// * __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.
+// *
+// * NOTE: 2/pi int representation is converted to double
+// * // (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[] ouput 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 interger indicating the precision:
+// * 0 24 bits (single)
+// * 1 53 bits (double)
+// * 2 64 bits (extended)
+// * 3 113 bits (quad)
+// *
+// * NOTE: ipio2[] array below is converted to double representation
+// * //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)).
+// *
+// * 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.
+// *
+// */
+//
+// Use PIo2 table(see stubRoutines_aarch64.cpp)
+//
+// BEGIN __kernel_rem_pio2 PSEUDO CODE
+//
+//static int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, /* NOTE: converted to double */ const double *ipio2 // const int *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];
+//
+// /* initialize jk*/
+// // jk = init_jk[prec]; // NOTE: prec==2 for double. jk is always 4.
+// jp = jk; // NOTE: always 4
+//
+// /* determine jx,jv,q0, note that 3>q0 */
+// jx = nx-1;
+// jv = (e0-3)/24; if(jv<0) jv=0;
+// q0 = e0-24*(jv+1);
+//
+// /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
+// j = jv-jx; m = jx+jk;
+//
+// // NOTE: split into two for-loops: one with zeroB and one with ipio2[j]. It
+// // allows the use of wider loads/stores
+// for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : /* NOTE: converted to double */ ipio2[j]; //(double) ipio2[j];
+//
+// // NOTE: unrolled and vectorized "for". See comments in asm code
+// /* compute q[0],q[1],...q[jk] */
+// for (i=0;i<=jk;i++) {
+// for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
+// }
+//
+// jz = jk;
+//recompute:
+// /* distill q[] into iq[] reversingly */
+// for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
+// fw = (double)((int)(twon24* z));
+// iq[i] = (int)(z-two24B*fw);
+// z = q[j-1]+fw;
+// }
+//
+// /* 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);
+// }
+// else if(q0==0) ih = iq[jz-1]>>23;
+// else if(z>=0.5) ih=2;
+//
+// if(ih>0) { /* q > 0.5 */
+// n += 1; carry = 0;
+// for(i=0;i<jz ;i++) { /* compute 1-q */
+// j = iq[i];
+// if(carry==0) {
+// if(j!=0) {
+// carry = 1; iq[i] = 0x1000000- j;
+// }
+// } else iq[i] = 0xffffff - j;
+// }
+// if(q0>0) { /* rare case: chance is 1 in 12 */
+// switch(q0) {
+// case 1:
+// iq[jz-1] &= 0x7fffff; break;
+// case 2:
+// iq[jz-1] &= 0x3fffff; break;
+// }
+// }
+// if(ih==2) {
+// z = one - z;
+// if(carry!=0) z -= scalbnA(one,q0);
+// }
+// }
+//
+// /* check if recomputation is needed */
+// if(z==zeroB) {
+// j = 0;
+// for (i=jz-1;i>=jk;i--) j |= iq[i];
+// if(j==0) { /* need recomputation */
+// for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
+//
+// for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
+// f[jx+i] = /* NOTE: converted to double */ ipio2[jv+i]; //(double) ipio2[jv+i];
+// for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
+// q[i] = fw;
+// }
+// jz += k;
+// goto recompute;
+// }
+// }
+//
+// /* chop off zero terms */
+// if(z==0.0) {
+// jz -= 1; q0 -= 24;
+// while(iq[jz]==0) { jz--; q0-=24;}
+// } else { /* break z into 24-bit if necessary */
+// z = scalbnA(z,-q0);
+// if(z>=two24B) {
+// fw = (double)((int)(twon24*z));
+// iq[jz] = (int)(z-two24B*fw);
+// jz += 1; q0 += 24;
+// iq[jz] = (int) fw;
+// } else iq[jz] = (int) z ;
+// }
+//
+// /* convert integer "bit" chunk to floating-point value */
+// fw = scalbnA(one,q0);
+// for(i=jz;i>=0;i--) {
+// q[i] = fw*(double)iq[i]; fw*=twon24;
+// }
+//
+// /* compute PIo2[0,...,jp]*q[jz,...,0] */
+// for(i=jz;i>=0;i--) {
+// for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
+// fq[jz-i] = fw;
+// }
+//
+// // NOTE: switch below is eliminated, because prec is always 2 for doubles
+// /* compress fq[] into y[] */
+// //switch(prec) {
+// //case 0:
+// // fw = 0.0;
+// // for (i=jz;i>=0;i--) fw += fq[i];
+// // y[0] = (ih==0)? fw: -fw;
+// // break;
+// //case 1:
+// //case 2:
+// fw = 0.0;
+// for (i=jz;i>=0;i--) fw += fq[i];
+// y[0] = (ih==0)? fw: -fw;
+// fw = fq[0]-fw;
+// for (i=1;i<=jz;i++) fw += fq[i];
+// y[1] = (ih==0)? fw: -fw;
+// // break;
+// //case 3: /* painful */
+// // for (i=jz;i>0;i--) {
+// // fw = fq[i-1]+fq[i];
+// // fq[i] += fq[i-1]-fw;
+// // fq[i-1] = fw;
+// // }
+// // for (i=jz;i>1;i--) {
+// // fw = fq[i-1]+fq[i];
+// // fq[i] += fq[i-1]-fw;
+// // fq[i-1] = fw;
+// // }
+// // for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
+// // if(ih==0) {
+// // y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
+// // } else {
+// // y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
+// // }
+// //}
+// return n&7;
+//}
+//
+// END __kernel_rem_pio2 PSEUDO CODE
+//
+// Changes between fdlibm and intrinsic:
+// 1. One loop is unrolled and vectorized (see comments in code)
+// 2. One loop is split into 2 loops (see comments in code)
+// 3. Non-double code is removed(last switch). Sevaral variables became
+// constants because of that (see comments in code)
+// 4. Use of jx, which is nx-1 instead of nx
+// Assumptions:
+// 1. Assume |X| >= PI/4
+// Input and output:
+// 1. Input: X = r0, jx == nx - 1 == r6, e0 == rscratch1
+// 2. Return n in r2, y[0] == y0 == v4, y[1] == y1 == v5
+// NOTE: general purpose register names match local variable names in C code
+// NOTE: fpu registers are actively reused. See comments in code about their usage
+void MacroAssembler::generate__kernel_rem_pio2(address two_over_pi, address pio2) {
+ Label Q_DONE, JX_IS_0, JX_IS_2, COMP_INNER_LOOP, RECOMP_FOR2, Q0_ZERO_CMP_LT,
+ RECOMP_CHECK_DONE_NOT_ZERO, Q0_ZERO_CMP_DONE, COMP_FOR, Q0_ZERO_CMP_EQ,
+ INIT_F_ZERO, RECOMPUTE, IH_FOR_INCREMENT, IH_FOR_STORE, RECOMP_CHECK_DONE,
+ Z_IS_LESS_THAN_TWO24B, Z_IS_ZERO, FW_Y1_NO_NEGATION,
+ RECOMP_FW_UPDATED, Z_ZERO_CHECK_DONE, FW_FOR1, IH_AFTER_SWITCH, IH_HANDLED,
+ CONVERTION_FOR, FW_Y0_NO_NEGATION, FW_FOR1_DONE, FW_FOR2, FW_FOR2_DONE,
+ IH_FOR, SKIP_F_LOAD, RECOMP_FOR1, RECOMP_FIRST_FOR, INIT_F_COPY,
+ RECOMP_FOR1_CHECK;
+ Register tmp2 = r1, n = r2, jv = r4, tmp5 = r5, jx = r6,
+ tmp3 = r7, iqBase = r10, ih = r11, tmp4 = r12, tmp1 = r13,
+ jz = r14, j = r15, twoOverPiBase = r16, i = r17, qBase = r18;
+ // jp = jk == init_jk[prec] = init_jk[2] == {2,3,4,6}[2] == 4
+ // jx = nx - 1
+ lea(twoOverPiBase, ExternalAddress(two_over_pi));
+ cmpw(jv, zr);
+ addw(tmp4, jx, 4); // tmp4 = m = jx + jk = jx + 4. jx is in {0,1,2} so m is in [4,5,6]
+ cselw(jv, jv, zr, GE);
+ fmovd(v26, 0.0d);
+ addw(tmp5, jv, 1); // jv+1
+ subsw(j, jv, jx);
+ add(qBase, sp, 320); // base of q[]
+ msubw(rscratch1, i, tmp5, rscratch1); // q0 = e0-24*(jv+1)
+ // use double f[20], fq[20], q[20], iq[20] on stack, which is
+ // (20 + 20 + 20) x 8 + 20 x 4 = 560 bytes. From lower to upper addresses it
+ // will contain f[20], fq[20], q[20], iq[20]
+ // now initialize f[20] indexes 0..m (inclusive)
+ // for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : /* NOTE: converted to double */ ipio2[j]; // (double) ipio2[j];
+ mov(tmp5, sp);
+
+ block_comment("for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : /* NOTE: converted to double */ ipio2[j]; // (double) ipio2[j];"); {
+ eorw(i, i, i);
+ br(GE, INIT_F_COPY);
+ bind(INIT_F_ZERO);
+ stpq(v26, v26, Address(post(tmp5, 32)));
+ addw(i, i, 4);
+ addsw(j, j, 4);
+ br(LT, INIT_F_ZERO);
+ subw(i, i, j);
+ movw(j, zr);
+ bind(INIT_F_COPY);
+ add(tmp1, twoOverPiBase, j, LSL, 3); // ipio2[j] start address
+ ld1(v18, v19, v20, v21, T16B, tmp1);
+ add(tmp5, sp, i, ext::uxtx, 3);
+ st1(v18, v19, v20, v21, T16B, tmp5);
+ }
+ // v18..v21 can actually contain f[0..7]
+ cbz(i, SKIP_F_LOAD); // i == 0 => f[i] == f[0] => already loaded
+ ld1(v18, v19, v20, v21, T2D, Address(sp)); // load f[0..7]
+ bind(SKIP_F_LOAD);
+ // calculate 2^q0 and 2^-q0, which we'll need further.
+ // q0 is exponent. So, calculate biased exponent(q0+1023)
+ negw(tmp4, rscratch1);
+ addw(tmp5, rscratch1, 1023);
+ addw(tmp4, tmp4, 1023);
+ // Unroll following for(s) depending on jx in [0,1,2]
+ // for (i=0;i<=jk;i++) {
+ // for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
+ // }
+ // Unrolling for jx == 0 case:
+ // q[0] = x[0] * f[0]
+ // q[1] = x[0] * f[1]
+ // q[2] = x[0] * f[2]
+ // q[3] = x[0] * f[3]
+ // q[4] = x[0] * f[4]
+ //
+ // Vectorization for unrolled jx == 0 case:
+ // {q[0], q[1]} = {f[0], f[1]} * x[0]
+ // {q[2], q[3]} = {f[2], f[3]} * x[0]
+ // q[4] = f[4] * x[0]
+ //
+ // Unrolling for jx == 1 case:
+ // q[0] = x[0] * f[1] + x[1] * f[0]
+ // q[1] = x[0] * f[2] + x[1] * f[1]
+ // q[2] = x[0] * f[3] + x[1] * f[2]
+ // q[3] = x[0] * f[4] + x[1] * f[3]
+ // q[4] = x[0] * f[5] + x[1] * f[4]
+ //
+ // Vectorization for unrolled jx == 1 case:
+ // {q[0], q[1]} = {f[0], f[1]} * x[1]
+ // {q[2], q[3]} = {f[2], f[3]} * x[1]
+ // q[4] = f[4] * x[1]
+ // {q[0], q[1]} += {f[1], f[2]} * x[0]
+ // {q[2], q[3]} += {f[3], f[4]} * x[0]
+ // q[4] += f[5] * x[0]
+ //
+ // Unrolling for jx == 2 case:
+ // q[0] = x[0] * f[2] + x[1] * f[1] + x[2] * f[0]
+ // q[1] = x[0] * f[3] + x[1] * f[2] + x[2] * f[1]
+ // q[2] = x[0] * f[4] + x[1] * f[3] + x[2] * f[2]
+ // q[3] = x[0] * f[5] + x[1] * f[4] + x[2] * f[3]
+ // q[4] = x[0] * f[6] + x[1] * f[5] + x[2] * f[4]
+ //
+ // Vectorization for unrolled jx == 2 case:
+ // {q[0], q[1]} = {f[0], f[1]} * x[2]
+ // {q[2], q[3]} = {f[2], f[3]} * x[2]
+ // q[4] = f[4] * x[2]
+ // {q[0], q[1]} += {f[1], f[2]} * x[1]
+ // {q[2], q[3]} += {f[3], f[4]} * x[1]
+ // q[4] += f[5] * x[1]
+ // {q[0], q[1]} += {f[2], f[3]} * x[0]
+ // {q[2], q[3]} += {f[4], f[5]} * x[0]
+ // q[4] += f[6] * x[0]
+ block_comment("unrolled and vectorized computation of q[0]..q[jk]"); {
+ cmpw(jx, 1);
+ lsl(tmp5, tmp5, 52); // now it's 2^q0 double value
+ lsl(tmp4, tmp4, 52); // now it's 2^-q0 double value
+ br(LT, JX_IS_0);
+ add(i, sp, 8);
+ ldpq(v26, v27, i); // load f[1..4]
+ br(GT, JX_IS_2);
+ // jx == 1
+ fmulxvs(v28, T2D, v18, v7); // f[0,1] * x[1]
+ fmulxvs(v29, T2D, v19, v7); // f[2,3] * x[1]
+ fmuld(v30, v20, v7); // f[4] * x[1]
+ fmlavs(v28, T2D, v26, v6, 0);
+ fmlavs(v29, T2D, v27, v6, 0);
+ fmlavs(v30, T2D, v6, v20, 1); // v30 += f[5] * x[0]
+ b(Q_DONE);
+ bind(JX_IS_2);
+ fmulxvs(v28, T2D, v18, v3); // f[0,1] * x[2]
+ fmulxvs(v29, T2D, v19, v3); // f[2,3] * x[2]
+ fmuld(v30, v20, v3); // f[4] * x[2]
+ fmlavs(v28, T2D, v26, v7, 0);
+ fmlavs(v29, T2D, v27, v7, 0);
+ fmlavs(v30, T2D, v7, v20, 1); // v30 += f[5] * x[1]
+ fmlavs(v28, T2D, v19, v6, 0);
+ fmlavs(v29, T2D, v20, v6, 0);
+ fmlavs(v30, T2D, v6, v21, 0); // v30 += f[6] * x[0]
+ b(Q_DONE);
+ bind(JX_IS_0);
+ fmulxvs(v28, T2D, v18, v6); // f[0,1] * x[0]
+ fmulxvs(v29, T2D, v19, v6); // f[2,3] * x[0]
+ fmuld(v30, v20, v6); // f[4] * x[0]
+ bind(Q_DONE);
+ st1(v28, v29, v30, T2D, Address(qBase)); // save calculated q[0]...q[jk]
+ }
+ movz(i, 0x3E70, 48);
+ movw(jz, 4);
+ fmovd(v17, i); // v17 = twon24
+ fmovd(v30, tmp5); // 2^q0
+ fmovd(v21, 0.125d);
+ fmovd(v20, 8.0d);
+ fmovd(v22, tmp4); // 2^-q0
+
+ block_comment("recompute loop"); {
+ bind(RECOMPUTE);
+ // for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
+ // fw = (double)((int)(twon24* z));
+ // iq[i] = (int)(z-two24A*fw);
+ // z = q[j-1]+fw;
+ // }
+ block_comment("distill q[] into iq[] reversingly"); {
+ eorw(i, i, i);
+ movw(j, jz);
+ add(tmp2, qBase, jz, LSL, 3); // q[jz] address
+ ldrd(v18, post(tmp2, -8)); // z = q[j] and moving address to q[j-1]
+ bind(RECOMP_FIRST_FOR);
+ ldrd(v27, post(tmp2, -8));
+ fmuld(v29, v17, v18); // twon24*z
+ frintzd(v29, v29); // (double)(int)
+ fmsubd(v28, v10, v29, v18); // v28 = z-two24A*fw
+ fcvtzdw(tmp1, v28); // (int)(z-two24A*fw)
+ strw(tmp1, Address(iqBase, i, Address::lsl(2)));
+ faddd(v18, v27, v29);
+ add(i, i, 1);
+ subs(j, j, 1);
+ br(GT, RECOMP_FIRST_FOR);
+ }
+ // compute n
+ fmuld(v18, v18, v30);
+ fmuld(v2, v18, v21);
+ frintmd(v2, v2); // v2 = floor(v2) == rounding towards -inf
+ fmsubd(v18, v2, v20, v18); // z -= 8.0*floor(z*0.125);
+ movw(ih, 2);
+ frintzd(v2, v18); // v2 = (double)((int)z)
+ fcvtzdw(n, v18); // n = (int) z;
+ fsubd(v18, v18, v2); // z -= (double)n;
+
+ block_comment("q0-dependent initialization"); {
+ cmpw(rscratch1, 0); // if (q0 > 0)
+ br(LT, Q0_ZERO_CMP_LT);
+ subw(j, jz, 1); // j = jz - 1
+ ldrw(tmp2, Address(iqBase, j, Address::lsl(2))); // tmp2 = iq[jz-1]
+ br(EQ, Q0_ZERO_CMP_EQ);
+ movw(tmp4, 24);
+ subw(tmp4, tmp4, rscratch1); // == 24 - q0
+ lsrvw(i, tmp2, tmp4); // i = iq[jz-1] >> (24-q0)
+ lslvw(tmp5, i, tmp4);
+ subw(tmp2, tmp2, tmp5); // iq[jz-1] -= i<<(24-q0);
+ strw(tmp2, Address(iqBase, j, Address::lsl(2))); // store iq[jz-1]
+ subw(rscratch2, tmp4, 1); // == 23 - q0
+ addw(n, n, i); // n+=i
+ lsrvw(ih, tmp2, rscratch2); // ih = iq[jz-1] >> (23-q0)
+ b(Q0_ZERO_CMP_DONE);
+ bind(Q0_ZERO_CMP_EQ);
+ lsr(ih, tmp2, 23); // ih = iq[z-1] >> 23
+ b(Q0_ZERO_CMP_DONE);
+ bind(Q0_ZERO_CMP_LT);
+ fmovd(v4, 0.5d);
+ fcmpd(v18, v4);
+ cselw(ih, zr, ih, LT); // if (z<0.5) ih = 0
+ }
+ bind(Q0_ZERO_CMP_DONE);
+ cmpw(ih, zr);
+ br(LE, IH_HANDLED);
+
+ block_comment("if(ih>) {"); {
+ // use rscratch2 as carry
+
+ block_comment("for(i=0;i<jz ;i++) {...}"); {
+ addw(n, n, 1);
+ eorw(i, i, i);
+ eorw(rscratch2, rscratch2, rscratch2);
+ bind(IH_FOR);
+ ldrw(j, Address(iqBase, i, Address::lsl(2))); // j = iq[i]
+ movw(tmp3, 0x1000000);
+ subw(tmp3, tmp3, rscratch2);
+ cbnzw(rscratch2, IH_FOR_STORE);
+ cbzw(j, IH_FOR_INCREMENT);
+ movw(rscratch2, 1);
+ bind(IH_FOR_STORE);
+ subw(tmp3, tmp3, j);
+ strw(tmp3, Address(iqBase, i, Address::lsl(2))); // iq[i] = 0xffffff - j
+ bind(IH_FOR_INCREMENT);
+ addw(i, i, 1);
+ cmpw(i, jz);
+ br(LT, IH_FOR);
+ }
+
+ block_comment("if(q0>0) {"); {
+ cmpw(rscratch1, zr);
+ br(LE, IH_AFTER_SWITCH);
+ // tmp3 still has iq[jz-1] value. no need to reload
+ // now, zero high tmp3 bits (rscratch1 number of bits)
+ movw(j, -1);
+ subw(i, jz, 1); // set i to jz-1
+ lsrv(j, j, rscratch1);
+ andw(tmp3, tmp3, j, LSR, 8); // we have 24-bit-based constants
+ strw(tmp3, Address(iqBase, i, Address::lsl(2))); // save iq[jz-1]
+ }
+ bind(IH_AFTER_SWITCH);
+ cmpw(ih, 2);
+ br(NE, IH_HANDLED);
+
+ block_comment("if(ih==2) {"); {
+ fmovd(v25, 1.0d);
+ fsubd(v18, v25, v18); // z = one - z;
+ cbzw(rscratch2, IH_HANDLED);
+ fsubd(v18, v18, v30); // z -= scalbnA(one,q0);
+ }
+ }
+ bind(IH_HANDLED);
+ // check if recomputation is needed
+ fcmpd(v18, 0.0d);
+ br(NE, RECOMP_CHECK_DONE_NOT_ZERO);
+
+ block_comment("if(z==zeroB) {"); {
+
+ block_comment("for (i=jz-1;i>=jk;i--) j |= iq[i];"); {
+ subw(i, jz, 1);
+ eorw(j, j, j);
+ b(RECOMP_FOR1_CHECK);
+ bind(RECOMP_FOR1);
+ ldrw(tmp1, Address(iqBase, i, Address::lsl(2)));
+ orrw(j, j, tmp1);
+ subw(i, i, 1);
+ bind(RECOMP_FOR1_CHECK);
+ cmpw(i, 4);
+ br(GE, RECOMP_FOR1);
+ }
+ cbnzw(j, RECOMP_CHECK_DONE);
+
+ block_comment("if(j==0) {"); {
+ // for(k=1;iq[jk-k]==0;k++); // let's unroll it. jk == 4. So, read
+ // iq[3], iq[2], iq[1], iq[0] until non-zero value
+ ldp(tmp1, tmp3, iqBase); // iq[0..3]
+ movw(j, 2);
+ cmp(tmp3, zr);
+ csel(tmp1, tmp1, tmp3, EQ); // set register for further consideration
+ cselw(j, j, zr, EQ); // set initial k. Use j as k
+ cmp(zr, tmp1, LSR, 32);
+ addw(i, jz, 1);
+ csincw(j, j, j, NE);
+
+ block_comment("for(i=jz+1;i<=jz+k;i++) {...}"); {
+ addw(jz, i, j); // i = jz+1, j = k-1. j+i = jz+k (which is a new jz)
+ bind(RECOMP_FOR2);
+ addw(tmp1, jv, i);
+ ldrd(v29, Address(twoOverPiBase, tmp1, Address::lsl(3)));
+ addw(tmp2, jx, i);
+ strd(v29, Address(sp, tmp2, Address::lsl(3)));
+ // f[jx+i] = /* NOTE: converted to double */ ipio2[jv+i]; //(double) ipio2[jv+i];
+ // since jx = 0, 1 or 2 we can unroll it:
+ // for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
+ // f[jx+i-j] == (for first iteration) f[jx+i], which is already v29
+ add(tmp2, sp, tmp2, ext::uxtx, 3); // address of f[jx+i]
+ ldpd(v4, v5, Address(tmp2, -16)); // load f[jx+i-2] and f[jx+i-1]
+ fmuld(v26, v6, v29); // initial fw
+ cbzw(jx, RECOMP_FW_UPDATED);
+ fmaddd(v26, v7, v5, v26);
+ cmpw(jx, 1);
+ br(EQ, RECOMP_FW_UPDATED);
+ fmaddd(v26, v3, v4, v26);
+ bind(RECOMP_FW_UPDATED);
+ strd(v26, Address(qBase, i, Address::lsl(3))); // q[i] = fw;
+ addw(i, i, 1);
+ cmpw(i, jz); // jz here is "old jz" + k
+ br(LE, RECOMP_FOR2);
+ }
+ b(RECOMPUTE);
+ }
+ }
+ }
+ bind(RECOMP_CHECK_DONE);
+ // chop off zero terms
+ fcmpd(v18, 0.0d);
+ br(EQ, Z_IS_ZERO);
+
+ block_comment("else block of if(z==0.0) {"); {
+ bind(RECOMP_CHECK_DONE_NOT_ZERO);
+ fmuld(v18, v18, v22);
+ fcmpd(v18, v10); // v10 is stil two24A
+ br(LT, Z_IS_LESS_THAN_TWO24B);
+ fmuld(v1, v18, v17); // twon24*z
+ frintzd(v1, v1); // v1 = (double)(int)(v1)
+ fmaddd(v2, v10, v1, v18);
+ fcvtzdw(tmp3, v1); // (int)fw
+ fcvtzdw(tmp2, v2); // double to int
+ strw(tmp2, Address(iqBase, jz, Address::lsl(2)));
+ addw(rscratch1, rscratch1, 24);
+ addw(jz, jz, 1);
+ strw(tmp3, Address(iqBase, jz, Address::lsl(2))); // iq[jz] = (int) fw
+ b(Z_ZERO_CHECK_DONE);
+ bind(Z_IS_LESS_THAN_TWO24B);
+ fcvtzdw(tmp3, v18); // (int)z
+ strw(tmp3, Address(iqBase, jz, Address::lsl(2))); // iq[jz] = (int) z
+ b(Z_ZERO_CHECK_DONE);
+ }
+
+ block_comment("if(z==0.0) {"); {
+ bind(Z_IS_ZERO);
+ subw(jz, jz, 1);
+ ldrw(tmp1, Address(iqBase, jz, Address::lsl(2)));
+ subw(rscratch1, rscratch1, 24);
+ cbz(tmp1, Z_IS_ZERO);
+ }
+ bind(Z_ZERO_CHECK_DONE);
+ // convert integer "bit" chunk to floating-point value
+ // v17 = twon24
+ // update v30, which was scalbnA(1.0, <old q0>);
+ addw(tmp2, rscratch1, 1023); // biased exponent
+ lsl(tmp2, tmp2, 52); // put at correct position
+ mov(i, jz);
+ fmovd(v30, tmp2);
+
+ block_comment("for(i=jz;i>=0;i--) {q[i] = fw*(double)iq[i]; fw*=twon24;}"); {
+ bind(CONVERTION_FOR);
+ ldrw(tmp1, Address(iqBase, i, Address::lsl(2)));
+ scvtfwd(v31, tmp1);
+ fmuld(v31, v31, v30);
+ strd(v31, Address(qBase, i, Address::lsl(3)));
+ fmuld(v30, v30, v17);
+ subsw(i, i, 1);
+ br(GE, CONVERTION_FOR);
+ }
+ add(rscratch2, sp, 160); // base for fq
+ // reusing twoOverPiBase
+ lea(twoOverPiBase, ExternalAddress(pio2));
+
+ block_comment("compute PIo2[0,...,jp]*q[jz,...,0]. for(i=jz;i>=0;i--) {...}"); {
+ movw(i, jz);
+ movw(tmp2, zr); // tmp2 will keep jz - i == 0 at start
+ bind(COMP_FOR);
+ // for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
+ fmovd(v30, 0.0d);
+ add(tmp5, qBase, i, LSL, 3); // address of q[i+k] for k==0
+ movw(tmp3, 4);
+ movw(tmp4, zr); // used as k
+ cmpw(tmp2, 4);
+ add(tmp1, qBase, i, LSL, 3); // used as q[i] address
+ cselw(tmp3, tmp2, tmp3, LE); // min(jz - i, jp)
+
+ block_comment("for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];"); {
+ bind(COMP_INNER_LOOP);
+ ldrd(v18, Address(tmp1, tmp4, Address::lsl(3))); // q[i+k]
+ ldrd(v19, Address(twoOverPiBase, tmp4, Address::lsl(3))); // PIo2[k]
+ fmaddd(v30, v18, v19, v30); // fw += PIo2[k]*q[i+k];
+ addw(tmp4, tmp4, 1); // k++
+ cmpw(tmp4, tmp3);
+ br(LE, COMP_INNER_LOOP);
+ }
+ strd(v30, Address(rscratch2, tmp2, Address::lsl(3))); // fq[jz-i]
+ add(tmp2, tmp2, 1);
+ subsw(i, i, 1);
+ br(GE, COMP_FOR);
+ }
+
+ block_comment("switch(prec) {...}. case 2:"); {
+ // compress fq into y[]
+ // remember prec == 2
+
+ block_comment("for (i=jz;i>=0;i--) fw += fq[i];"); {
+ fmovd(v4, 0.0d);
+ mov(i, jz);
+ bind(FW_FOR1);
+ ldrd(v1, Address(rscratch2, i, Address::lsl(3)));
+ subsw(i, i, 1);
+ faddd(v4, v4, v1);
+ br(GE, FW_FOR1);
+ }
+ bind(FW_FOR1_DONE);
+ // v1 contains fq[0]. so, keep it so far
+ fsubd(v5, v1, v4); // fw = fq[0] - fw
+ cbzw(ih, FW_Y0_NO_NEGATION);
+ fnegd(v4, v4);
+ bind(FW_Y0_NO_NEGATION);
+
+ block_comment("for (i=1;i<=jz;i++) fw += fq[i];"); {
+ movw(i, 1);
+ cmpw(jz, 1);
+ br(LT, FW_FOR2_DONE);
+ bind(FW_FOR2);
+ ldrd(v1, Address(rscratch2, i, Address::lsl(3)));
+ addw(i, i, 1);
+ cmp(i, jz);
+ faddd(v5, v5, v1);
+ br(LE, FW_FOR2);
+ }
+ bind(FW_FOR2_DONE);
+ cbz(ih, FW_Y1_NO_NEGATION);
+ fnegd(v5, v5);
+ bind(FW_Y1_NO_NEGATION);
+ add(sp, sp, 560);
+ }
+}
+
+///* __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))
+// */
+//static const double
+//S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */
+//S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */
+//S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */
+//S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */
+//S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */
+//S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */
+//
+// NOTE: S1..S6 were moved into a table: StubRoutines::aarch64::_dsin_coef
+//
+// BEGIN __kernel_sin PSEUDO CODE
+//
+//static double __kernel_sin(double x, double y, bool iy)
+//{
+// double z,r,v;
+//
+// // NOTE: not needed. moved to dsin/dcos
+// //int ix;
+// //ix = high(x)&0x7fffffff; /* high word of x */
+//
+// // NOTE: moved to dsin/dcos
+// //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);
+// else return x-((z*(half*y-v*r)-y)-v*S1);
+//}
+//
+// END __kernel_sin PSEUDO CODE
+//
+// Changes between fdlibm and intrinsic:
+// 1. Removed |x| < 2**-27 check, because if was done earlier in dsin/dcos
+// 2. Constants are now loaded from table dsin_coef
+// 3. C code parameter "int iy" was modified to "bool iyIsOne", because
+// iy is always 0 or 1. Also, iyIsOne branch was moved into
+// generation phase instead of taking it during code execution
+// Input ans output:
+// 1. Input for generated function: X argument = x
+// 2. Input for generator: x = register to read argument from, iyIsOne
+// = flag to use low argument low part or not, dsin_coef = coefficients
+// table address
+// 3. Return sin(x) value in v0
+void MacroAssembler::generate_kernel_sin(FloatRegister x, bool iyIsOne,
+ address dsin_coef) {
+ FloatRegister y = v5, z = v6, v = v7, r = v16, S1 = v17, S2 = v18,
+ S3 = v19, S4 = v20, S5 = v21, S6 = v22, half = v23;
+ lea(rscratch2, ExternalAddress(dsin_coef));
+ ldpd(S5, S6, Address(rscratch2, 32));
+ fmuld(z, x, x); // z = x*x;
+ ld1(S1, S2, S3, S4, T1D, Address(rscratch2));
+ fmuld(v, z, x); // v = z*x;
+
+ block_comment("calculate r = S2+z*(S3+z*(S4+z*(S5+z*S6)))"); {
+ fmaddd(r, z, S6, S5);
+ // initialize "half" in current block to utilize 2nd FPU. However, it's
+ // not a part of this block
+ fmovd(half, 0.5);
+ fmaddd(r, z, r, S4);
+ fmaddd(r, z, r, S3);
+ fmaddd(r, z, r, S2);
+ }
+
+ if (!iyIsOne) {
+ // return x+v*(S1+z*r);
+ fmaddd(S1, z, r, S1);
+ fmaddd(v0, v, S1, x);
+ } else {
+ // return x-((z*(half*y-v*r)-y)-v*S1);
+ fmuld(S6, half, y); // half*y
+ fmsubd(S6, v, r, S6); // half*y-v*r
+ fmsubd(S6, z, S6, y); // y - z*(half*y-v*r) = - (z*(half*y-v*r)-y)
+ fmaddd(S6, v, S1, S6); // - (z*(half*y-v*r)-y) + v*S1 == -((z*(half*y-v*r)-y)-v*S1)
+ faddd(v0, x, S6);
+ }
+}
+
+///*
+// * __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.
+// */
+//
+//static const double
+//C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
+//C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
+//C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
+//C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
+//C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
+//C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
+//
+// NOTE: C1..C6 were moved into a table: StubRoutines::aarch64::_dcos_coef
+//
+// BEGIN __kernel_cos PSEUDO CODE
+//
+//static double __kernel_cos(double x, double y)
+//{
+// double a,h,z,r,qx=0;
+//
+// // NOTE: ix is already initialized in dsin/dcos. Reuse value from register
+// //int ix;
+// //ix = high(x)&0x7fffffff; /* ix = |x|'s high word*/
+//
+// // NOTE: moved to dsin/dcos
+// //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));
+// }
+//}
+//
+// END __kernel_cos PSEUDO CODE
+//
+// Changes between fdlibm and intrinsic:
+// 1. Removed |x| < 2**-27 check, because if was done earlier in dsin/dcos
+// 2. Constants are now loaded from table dcos_coef
+// Input and output:
+// 1. Input for generated function: X argument = x
+// 2. Input for generator: x = register to read argument from, dcos_coef
+// = coefficients table address
+// 2. Return cos(x) value in v0
+void MacroAssembler::generate_kernel_cos(FloatRegister x, address dcos_coef) {
+ Register ix = r3;
+ FloatRegister qx = v1, h = v2, a = v3, y = v5, z = v6, r = v7, C1 = v18,
+ C2 = v19, C3 = v20, C4 = v21, C5 = v22, C6 = v23, one = v25, half = v26;
+ Label IX_IS_LARGE, SET_QX_CONST, DONE, QX_SET;
+ lea(rscratch2, ExternalAddress(dcos_coef));
+ ldpd(C5, C6, Address(rscratch2, 32)); // load C5, C6
+ fmuld(z, x, x); // z=x^2
+ ld1(C1, C2, C3, C4, T1D, Address(rscratch2)); // load C1..C3\4
+ block_comment("calculate r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))))"); {
+ fmaddd(r, z, C6, C5);
+ fmovd(half, 0.5d);
+ fmaddd(r, z, r, C4);
+ fmuld(y, x, y);
+ fmaddd(r, z, r, C3);
+ mov(rscratch1, 0x3FD33333);
+ fmaddd(r, z, r, C2);
+ fmuld(x, z, z); // x = z^2
+ fmaddd(r, z, r, C1); // r = C1+z(C2+z(C4+z(C5+z*C6)))
+ }
+ // need to multiply r by z to have "final" r value
+ fmovd(one, 1.0d);
+ cmp(ix, rscratch1);
+ br(GT, IX_IS_LARGE);
+ block_comment("if(ix < 0x3FD33333) return one - (0.5*z - (z*r - x*y))"); {
+ // return 1.0 - (0.5*z - (z*r - x*y)) = 1.0 - (0.5*z + (x*y - z*r))
+ fmsubd(v0, x, r, y);
+ fmaddd(v0, half, z, v0);
+ fsubd(v0, one, v0);
+ b(DONE);
+ }
+ block_comment("if(ix >= 0x3FD33333)"); {
+ bind(IX_IS_LARGE);
+ movz(rscratch2, 0x3FE9, 16);
+ cmp(ix, rscratch2);
+ br(GT, SET_QX_CONST);
+ block_comment("set_high(&qx, ix-0x00200000); set_low(&qx, 0);"); {
+ subw(rscratch2, ix, 0x00200000);
+ lsl(rscratch2, rscratch2, 32);
+ fmovd(qx, rscratch2);
+ }
+ b(QX_SET);
+ bind(SET_QX_CONST);
+ block_comment("if(ix > 0x3fe90000) qx = 0.28125;"); {
+ fmovd(qx, 0.28125d);
+ }
+ bind(QX_SET);
+ fnmsub(C6, x, r, y); // z*r - xy
+ fnmsub(h, half, z, qx); // h = 0.5*z - qx
+ fsubd(a, one, qx); // a = 1-qx
+ fsubd(C6, h, C6); // = h - (z*r - x*y)
+ fsubd(v0, a, C6);
+ }
+ bind(DONE);
+}
+
+// generate_dsin_dcos creates stub for dsin and dcos
+// Generation is done via single call because dsin and dcos code is almost the
+// same(see C code below). These functions work as follows:
+// 1) handle corner cases: |x| ~< pi/4, x is NaN or INF, |x| < 2**-27
+// 2) perform argument reduction if required
+// 3) call kernel_sin or kernel_cos which approximate sin/cos via polynomial
+//
+// BEGIN dsin/dcos PSEUDO CODE
+//
+//dsin_dcos(jdouble x, bool isCos) {
+// double y[2],z=0.0;
+// int n, ix;
+//
+// /* High word of x. */
+// ix = high(x);
+//
+// /* |x| ~< pi/4 */
+// ix &= 0x7fffffff;
+// if(ix <= 0x3fe921fb) return isCos ? __kernel_cos : __kernel_sin(x,z,0);
+//
+// /* sin/cos(Inf or NaN) is NaN */
+// else if (ix>=0x7ff00000) return x-x;
+// else if (ix<0x3e400000) { /* if ix < 2**27 */
+// if(((int)x)==0) return isCos ? one : x; /* generate inexact */
+// }
+// /* argument reduction needed */
+// else {
+// n = __ieee754_rem_pio2(x,y);
+// switch(n&3) {
+// case 0: return isCos ? __kernel_cos(y[0],y[1]) : __kernel_sin(y[0],y[1], true);
+// case 1: return isCos ? -__kernel_sin(y[0],y[1],true) : __kernel_cos(y[0],y[1]);
+// case 2: return isCos ? -__kernel_cos(y[0],y[1]) : -__kernel_sin(y[0],y[1], true);
+// default:
+// return isCos ? __kernel_sin(y[0],y[1],1) : -__kernel_cos(y[0],y[1]);
+// }
+// }
+//}
+// END dsin/dcos PSEUDO CODE
+//
+// Changes between fdlibm and intrinsic:
+// 1. Moved ix < 2**27 from kernel_sin/kernel_cos into dsin/dcos
+// 2. Final switch use equivalent bit checks(tbz/tbnz)
+// Input ans output:
+// 1. Input for generated function: X = r0
+// 2. Input for generator: isCos = generate sin or cos, npio2_hw = address
+// of npio2_hw table, two_over_pi = address of two_over_pi table,
+// pio2 = address if pio2 table, dsin_coef = address if dsin_coef table,
+// dcos_coef = address of dcos_coef table
+// 3. Return result in v0
+// NOTE: general purpose register names match local variable names in C code
+void MacroAssembler::generate_dsin_dcos(bool isCos, address npio2_hw,
+ address two_over_pi, address pio2, address dsin_coef, address dcos_coef) {
+ const int POSITIVE_INFINITY_OR_NAN_PREFIX = 0x7FF0;
+
+ Label DONE, ARG_REDUCTION, TINY_X, RETURN_SIN, EARLY_CASE;
+ Register X = r0, absX = r1, n = r2, ix = r3;
+ FloatRegister y0 = v4, y1 = v5;
+ block_comment("check |x| ~< pi/4, NaN, Inf and |x| < 2**-27 cases"); {
+ fmovd(X, v0);
+ mov(rscratch2, 0x3e400000);
+ mov(rscratch1, 0x3fe921fb00000000); // pi/4. shifted to reuse later
+ ubfm(absX, X, 0, 62); // absX
+ movz(r10, POSITIVE_INFINITY_OR_NAN_PREFIX, 48);
+ cmp(rscratch2, absX, LSR, 32);
+ lsr(ix, absX, 32); // set ix
+ br(GT, TINY_X); // handle tiny x (|x| < 2^-27)
+ cmp(ix, rscratch1, LSR, 32);
+ br(LE, EARLY_CASE); // if(ix <= 0x3fe921fb) return
+ cmp(absX, r10);
+ br(LT, ARG_REDUCTION);
+ // X is NaN or INF(i.e. 0x7FF* or 0xFFF*). Return NaN (mantissa != 0).
+ // Set last bit unconditionally to make it NaN
+ orr(r10, r10, 1);
+ fmovd(v0, r10);
+ ret(lr);
+ }
+ block_comment("kernel_sin/kernel_cos: if(ix<0x3e400000) {<fast return>}"); {
+ bind(TINY_X);
+ if (isCos) {
+ fmovd(v0, 1.0d);
+ }
+ ret(lr);
+ }
+ bind(ARG_REDUCTION); /* argument reduction needed */
+ block_comment("n = __ieee754_rem_pio2(x,y);"); {
+ generate__ieee754_rem_pio2(npio2_hw, two_over_pi, pio2);
+ }
+ block_comment("switch(n&3) {case ... }"); {
+ if (isCos) {
+ eorw(absX, n, n, LSR, 1);
+ tbnz(n, 0, RETURN_SIN);
+ } else {
+ tbz(n, 0, RETURN_SIN);
+ }
+ generate_kernel_cos(y0, dcos_coef);
+ if (isCos) {
+ tbz(absX, 0, DONE);
+ } else {
+ tbz(n, 1, DONE);
+ }
+ fnegd(v0, v0);
+ ret(lr);
+ bind(RETURN_SIN);
+ generate_kernel_sin(y0, true, dsin_coef);
+ if (isCos) {
+ tbz(absX, 0, DONE);
+ } else {
+ tbz(n, 1, DONE);
+ }
+ fnegd(v0, v0);
+ ret(lr);
+ }
+ bind(EARLY_CASE);
+ eor(y1, T8B, y1, y1);
+ if (isCos) {
+ generate_kernel_cos(v0, dcos_coef);
+ } else {
+ generate_kernel_sin(v0, false, dsin_coef);
+ }
+ bind(DONE);
+ ret(lr);
+}