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/* Copyright (C) 2007, 2009 Free Software Foundation, Inc. This file is part of GCC. GCC is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. GCC 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 for more details. Under Section 7 of GPL version 3, you are granted additional permissions described in the GCC Runtime Library Exception, version 3.1, as published by the Free Software Foundation. You should have received a copy of the GNU General Public License and a copy of the GCC Runtime Library Exception along with this program; see the files COPYING3 and COPYING.RUNTIME respectively. If not, see <http://www.gnu.org/licenses/>. */ /***************************************************************************** * BID64 square root ***************************************************************************** * * Algorithm description: * * if(exponent_x is odd) * scale coefficient_x by 10, adjust exponent * - get lower estimate for number of digits in coefficient_x * - scale coefficient x to between 31 and 33 decimal digits * - in parallel, check for exact case and return if true * - get high part of result coefficient using double precision sqrt * - compute remainder and refine coefficient in one iteration (which * modifies it by at most 1) * - result exponent is easy to compute from the adjusted arg. exponent * ****************************************************************************/ #include "bid_internal.h" #include "bid_sqrt_macros.h" #ifdef UNCHANGED_BINARY_STATUS_FLAGS #include <fenv.h> #define FE_ALL_FLAGS FE_INVALID|FE_DIVBYZERO|FE_OVERFLOW|FE_UNDERFLOW|FE_INEXACT #endif extern double sqrt (double); #if DECIMAL_CALL_BY_REFERENCE void bid64_sqrt (UINT64 * pres, UINT64 * px _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { UINT64 x; #else UINT64 bid64_sqrt (UINT64 x _RND_MODE_PARAM _EXC_FLAGS_PARAM _EXC_MASKS_PARAM _EXC_INFO_PARAM) { #endif UINT128 CA, CT; UINT64 sign_x, coefficient_x; UINT64 Q, Q2, A10, C4, R, R2, QE, res; SINT64 D; int_double t_scale; int_float tempx; double da, dq, da_h, da_l, dqe; int exponent_x, exponent_q, bin_expon_cx; int digits_x; int scale; #ifdef UNCHANGED_BINARY_STATUS_FLAGS fexcept_t binaryflags = 0; #endif #if DECIMAL_CALL_BY_REFERENCE #if !DECIMAL_GLOBAL_ROUNDING _IDEC_round rnd_mode = *prnd_mode; #endif x = *px; #endif // unpack arguments, check for NaN or Infinity if (!unpack_BID64 (&sign_x, &exponent_x, &coefficient_x, x)) { // x is Inf. or NaN or 0 if ((x & INFINITY_MASK64) == INFINITY_MASK64) { res = coefficient_x; if ((coefficient_x & SSNAN_MASK64) == SINFINITY_MASK64) // -Infinity { res = NAN_MASK64; #ifdef SET_STATUS_FLAGS __set_status_flags (pfpsf, INVALID_EXCEPTION); #endif } #ifdef SET_STATUS_FLAGS if ((x & SNAN_MASK64) == SNAN_MASK64) // sNaN __set_status_flags (pfpsf, INVALID_EXCEPTION); #endif BID_RETURN (res & QUIET_MASK64); } // x is 0 exponent_x = (exponent_x + DECIMAL_EXPONENT_BIAS) >> 1; res = sign_x | (((UINT64) exponent_x) << 53); BID_RETURN (res); } // x<0? if (sign_x && coefficient_x) { res = NAN_MASK64; #ifdef SET_STATUS_FLAGS __set_status_flags (pfpsf, INVALID_EXCEPTION); #endif BID_RETURN (res); } #ifdef UNCHANGED_BINARY_STATUS_FLAGS (void) fegetexceptflag (&binaryflags, FE_ALL_FLAGS); #endif //--- get number of bits in the coefficient of x --- tempx.d = (float) coefficient_x; bin_expon_cx = ((tempx.i >> 23) & 0xff) - 0x7f; digits_x = estimate_decimal_digits[bin_expon_cx]; // add test for range if (coefficient_x >= power10_index_binexp[bin_expon_cx]) digits_x++; A10 = coefficient_x; if (exponent_x & 1) { A10 = (A10 << 2) + A10; A10 += A10; } dqe = sqrt ((double) A10); QE = (UINT32) dqe; if (QE * QE == A10) { res = very_fast_get_BID64 (0, (exponent_x + DECIMAL_EXPONENT_BIAS) >> 1, QE); #ifdef UNCHANGED_BINARY_STATUS_FLAGS (void) fesetexceptflag (&binaryflags, FE_ALL_FLAGS); #endif BID_RETURN (res); } // if exponent is odd, scale coefficient by 10 scale = 31 - digits_x; exponent_q = exponent_x - scale; scale += (exponent_q & 1); // exp. bias is even CT = power10_table_128[scale]; __mul_64x128_short (CA, coefficient_x, CT); // 2^64 t_scale.i = 0x43f0000000000000ull; // convert CA to DP da_h = CA.w[1]; da_l = CA.w[0]; da = da_h * t_scale.d + da_l; dq = sqrt (da); Q = (UINT64) dq; // get sign(sqrt(CA)-Q) R = CA.w[0] - Q * Q; R = ((SINT64) R) >> 63; D = R + R + 1; exponent_q = (exponent_q + DECIMAL_EXPONENT_BIAS) >> 1; #ifdef SET_STATUS_FLAGS __set_status_flags (pfpsf, INEXACT_EXCEPTION); #endif #ifndef IEEE_ROUND_NEAREST #ifndef IEEE_ROUND_NEAREST_TIES_AWAY if (!((rnd_mode) & 3)) { #endif #endif // midpoint to check Q2 = Q + Q + D; C4 = CA.w[0] << 2; // get sign(-sqrt(CA)+Midpoint) R2 = Q2 * Q2 - C4; R2 = ((SINT64) R2) >> 63; // adjust Q if R!=R2 Q += (D & (R ^ R2)); #ifndef IEEE_ROUND_NEAREST #ifndef IEEE_ROUND_NEAREST_TIES_AWAY } else { C4 = CA.w[0]; Q += D; if ((SINT64) (Q * Q - C4) > 0) Q--; if (rnd_mode == ROUNDING_UP) Q++; } #endif #endif res = fast_get_BID64 (0, exponent_q, Q); #ifdef UNCHANGED_BINARY_STATUS_FLAGS (void) fesetexceptflag (&binaryflags, FE_ALL_FLAGS); #endif BID_RETURN (res); } TYPE0_FUNCTION_ARG1 (UINT64, bid64q_sqrt, x) UINT256 M256, C4, C8; UINT128 CX, CX2, A10, S2, T128, CS, CSM, CS2, C256, CS1, mul_factor2_long = { {0x0ull, 0x0ull} }, QH, Tmp, TP128, Qh, Ql; UINT64 sign_x, Carry, B10, res, mul_factor, mul_factor2 = 0x0ull, CS0; SINT64 D; int_float fx, f64; int exponent_x, bin_expon_cx, done = 0; int digits, scale, exponent_q = 0, exact = 1, amount, extra_digits; #ifdef UNCHANGED_BINARY_STATUS_FLAGS fexcept_t binaryflags = 0; #endif // unpack arguments, check for NaN or Infinity if (!unpack_BID128_value (&sign_x, &exponent_x, &CX, x)) { res = CX.w[1]; // NaN ? if ((x.w[1] & 0x7c00000000000000ull) == 0x7c00000000000000ull) { #ifdef SET_STATUS_FLAGS if ((x.w[1] & 0x7e00000000000000ull) == 0x7e00000000000000ull) // sNaN __set_status_flags (pfpsf, INVALID_EXCEPTION); #endif Tmp.w[1] = (CX.w[1] & 0x00003fffffffffffull); Tmp.w[0] = CX.w[0]; TP128 = reciprocals10_128[18]; __mul_128x128_full (Qh, Ql, Tmp, TP128); amount = recip_scale[18]; __shr_128 (Tmp, Qh, amount); res = (CX.w[1] & 0xfc00000000000000ull) | Tmp.w[0]; BID_RETURN (res); } // x is Infinity? if ((x.w[1] & 0x7800000000000000ull) == 0x7800000000000000ull) { if (sign_x) { // -Inf, return NaN res = 0x7c00000000000000ull; #ifdef SET_STATUS_FLAGS __set_status_flags (pfpsf, INVALID_EXCEPTION); #endif } BID_RETURN (res); } // x is 0 otherwise exponent_x = ((exponent_x - DECIMAL_EXPONENT_BIAS_128) >> 1) + DECIMAL_EXPONENT_BIAS; if (exponent_x < 0) exponent_x = 0; if (exponent_x > DECIMAL_MAX_EXPON_64) exponent_x = DECIMAL_MAX_EXPON_64; //res= sign_x | (((UINT64)exponent_x)<<53); res = get_BID64 (sign_x, exponent_x, 0, rnd_mode, pfpsf); BID_RETURN (res); } if (sign_x) { res = 0x7c00000000000000ull; #ifdef SET_STATUS_FLAGS __set_status_flags (pfpsf, INVALID_EXCEPTION); #endif BID_RETURN (res); } #ifdef UNCHANGED_BINARY_STATUS_FLAGS (void) fegetexceptflag (&binaryflags, FE_ALL_FLAGS); #endif // 2^64 f64.i = 0x5f800000; // fx ~ CX fx.d = (float) CX.w[1] * f64.d + (float) CX.w[0]; bin_expon_cx = ((fx.i >> 23) & 0xff) - 0x7f; digits = estimate_decimal_digits[bin_expon_cx]; A10 = CX; if (exponent_x & 1) { A10.w[1] = (CX.w[1] << 3) | (CX.w[0] >> 61); A10.w[0] = CX.w[0] << 3; CX2.w[1] = (CX.w[1] << 1) | (CX.w[0] >> 63); CX2.w[0] = CX.w[0] << 1; __add_128_128 (A10, A10, CX2); } C256.w[1] = A10.w[1]; C256.w[0] = A10.w[0]; CS.w[0] = short_sqrt128 (A10); CS.w[1] = 0; mul_factor = 0; // check for exact result if (CS.w[0] < 10000000000000000ull) { if (CS.w[0] * CS.w[0] == A10.w[0]) { __sqr64_fast (S2, CS.w[0]); if (S2.w[1] == A10.w[1]) // && S2.w[0]==A10.w[0]) { res = get_BID64 (0, ((exponent_x - DECIMAL_EXPONENT_BIAS_128) >> 1) + DECIMAL_EXPONENT_BIAS, CS.w[0], rnd_mode, pfpsf); #ifdef UNCHANGED_BINARY_STATUS_FLAGS (void) fesetexceptflag (&binaryflags, FE_ALL_FLAGS); #endif BID_RETURN (res); } } if (CS.w[0] >= 1000000000000000ull) { done = 1; exponent_q = exponent_x; C256.w[1] = A10.w[1]; C256.w[0] = A10.w[0]; } #ifdef SET_STATUS_FLAGS __set_status_flags (pfpsf, INEXACT_EXCEPTION); #endif exact = 0; } else { B10 = 0x3333333333333334ull; __mul_64x64_to_128_full (CS2, CS.w[0], B10); CS0 = CS2.w[1] >> 1; if (CS.w[0] != ((CS0 << 3) + (CS0 << 1))) { #ifdef SET_STATUS_FLAGS __set_status_flags (pfpsf, INEXACT_EXCEPTION); #endif exact = 0; } done = 1; CS.w[0] = CS0; exponent_q = exponent_x + 2; mul_factor = 10; mul_factor2 = 100; if (CS.w[0] >= 10000000000000000ull) { __mul_64x64_to_128_full (CS2, CS.w[0], B10); CS0 = CS2.w[1] >> 1; if (CS.w[0] != ((CS0 << 3) + (CS0 << 1))) { #ifdef SET_STATUS_FLAGS __set_status_flags (pfpsf, INEXACT_EXCEPTION); #endif exact = 0; } exponent_q += 2; CS.w[0] = CS0; mul_factor = 100; mul_factor2 = 10000; } if (exact) { CS0 = CS.w[0] * mul_factor; __sqr64_fast (CS1, CS0) if ((CS1.w[0] != A10.w[0]) || (CS1.w[1] != A10.w[1])) { #ifdef SET_STATUS_FLAGS __set_status_flags (pfpsf, INEXACT_EXCEPTION); #endif exact = 0; } } } if (!done) { // get number of digits in CX D = CX.w[1] - power10_index_binexp_128[bin_expon_cx].w[1]; if (D > 0 || (!D && CX.w[0] >= power10_index_binexp_128[bin_expon_cx].w[0])) digits++; // if exponent is odd, scale coefficient by 10 scale = 31 - digits; exponent_q = exponent_x - scale; scale += (exponent_q & 1); // exp. bias is even T128 = power10_table_128[scale]; __mul_128x128_low (C256, CX, T128); CS.w[0] = short_sqrt128 (C256); } //printf("CS=%016I64x\n",CS.w[0]); exponent_q = ((exponent_q - DECIMAL_EXPONENT_BIAS_128) >> 1) + DECIMAL_EXPONENT_BIAS; if ((exponent_q < 0) && (exponent_q + MAX_FORMAT_DIGITS >= 0)) { extra_digits = -exponent_q; exponent_q = 0; // get coeff*(2^M[extra_digits])/10^extra_digits __mul_64x64_to_128 (QH, CS.w[0], reciprocals10_64[extra_digits]); // now get P/10^extra_digits: shift Q_high right by M[extra_digits]-128 amount = short_recip_scale[extra_digits]; CS0 = QH.w[1] >> amount; #ifdef SET_STATUS_FLAGS if (exact) { if (CS.w[0] != CS0 * power10_table_128[extra_digits].w[0]) exact = 0; } if (!exact) __set_status_flags (pfpsf, UNDERFLOW_EXCEPTION | INEXACT_EXCEPTION); #endif CS.w[0] = CS0; if (!mul_factor) mul_factor = 1; mul_factor *= power10_table_128[extra_digits].w[0]; __mul_64x64_to_128 (mul_factor2_long, mul_factor, mul_factor); if (mul_factor2_long.w[1]) mul_factor2 = 0; else mul_factor2 = mul_factor2_long.w[1]; } // 4*C256 C4.w[1] = (C256.w[1] << 2) | (C256.w[0] >> 62); C4.w[0] = C256.w[0] << 2; #ifndef IEEE_ROUND_NEAREST #ifndef IEEE_ROUND_NEAREST_TIES_AWAY if (!((rnd_mode) & 3)) { #endif #endif // compare to midpoints CSM.w[0] = (CS.w[0] + CS.w[0]) | 1; //printf("C256=%016I64x %016I64x, CSM=%016I64x %016I64x %016I64x\n",C4.w[1],C4.w[0],CSM.w[1],CSM.w[0], CS.w[0]); if (mul_factor) CSM.w[0] *= mul_factor; // CSM^2 __mul_64x64_to_128 (M256, CSM.w[0], CSM.w[0]); //__mul_128x128_to_256(M256, CSM, CSM); if (C4.w[1] > M256.w[1] || (C4.w[1] == M256.w[1] && C4.w[0] > M256.w[0])) { // round up CS.w[0]++; } else { C8.w[0] = CS.w[0] << 3; C8.w[1] = 0; if (mul_factor) { if (mul_factor2) { __mul_64x64_to_128 (C8, C8.w[0], mul_factor2); } else { __mul_64x128_low (C8, C8.w[0], mul_factor2_long); } } // M256 - 8*CSM __sub_borrow_out (M256.w[0], Carry, M256.w[0], C8.w[0]); M256.w[1] = M256.w[1] - C8.w[1] - Carry; // if CSM' > C256, round up if (M256.w[1] > C4.w[1] || (M256.w[1] == C4.w[1] && M256.w[0] > C4.w[0])) { // round down if (CS.w[0]) CS.w[0]--; } } #ifndef IEEE_ROUND_NEAREST #ifndef IEEE_ROUND_NEAREST_TIES_AWAY } else { CS.w[0]++; CSM.w[0] = CS.w[0]; C8.w[0] = CSM.w[0] << 1; if (mul_factor) CSM.w[0] *= mul_factor; __mul_64x64_to_128 (M256, CSM.w[0], CSM.w[0]); C8.w[1] = 0; if (mul_factor) { if (mul_factor2) { __mul_64x64_to_128 (C8, C8.w[0], mul_factor2); } else { __mul_64x128_low (C8, C8.w[0], mul_factor2_long); } } //printf("C256=%016I64x %016I64x, CSM=%016I64x %016I64x %016I64x\n",C256.w[1],C256.w[0],M256.w[1],M256.w[0], CS.w[0]); if (M256.w[1] > C256.w[1] || (M256.w[1] == C256.w[1] && M256.w[0] > C256.w[0])) { __sub_borrow_out (M256.w[0], Carry, M256.w[0], C8.w[0]); M256.w[1] = M256.w[1] - Carry - C8.w[1]; M256.w[0]++; if (!M256.w[0]) { M256.w[1]++; } if ((M256.w[1] > C256.w[1] || (M256.w[1] == C256.w[1] && M256.w[0] > C256.w[0])) && (CS.w[0] > 1)) { CS.w[0]--; if (CS.w[0] > 1) { __sub_borrow_out (M256.w[0], Carry, M256.w[0], C8.w[0]); M256.w[1] = M256.w[1] - Carry - C8.w[1]; M256.w[0]++; if (!M256.w[0]) { M256.w[1]++; } if (M256.w[1] > C256.w[1] || (M256.w[1] == C256.w[1] && M256.w[0] > C256.w[0])) CS.w[0]--; } } } else { /*__add_carry_out(M256.w[0], Carry, M256.w[0], C8.w[0]); M256.w[1] = M256.w[1] + Carry + C8.w[1]; M256.w[0]++; if(!M256.w[0]) { M256.w[1]++; } CS.w[0]++; if(M256.w[1]<C256.w[1] || (M256.w[1]==C256.w[1] && M256.w[0]<=C256.w[0])) { CS.w[0]++; }*/ CS.w[0]++; } //printf("C256=%016I64x %016I64x, CSM=%016I64x %016I64x %016I64x %d\n",C4.w[1],C4.w[0],M256.w[1],M256.w[0], CS.w[0], exact); // RU? if (((rnd_mode) != ROUNDING_UP) || exact) { if (CS.w[0]) CS.w[0]--; } } #endif #endif //printf("C256=%016I64x %016I64x, CSM=%016I64x %016I64x %016I64x %d\n",C4.w[1],C4.w[0],M256.w[1],M256.w[0], CS.w[0], exact); res = get_BID64 (0, exponent_q, CS.w[0], rnd_mode, pfpsf); #ifdef UNCHANGED_BINARY_STATUS_FLAGS (void) fesetexceptflag (&binaryflags, FE_ALL_FLAGS); #endif BID_RETURN (res); }