forked from OSchip/llvm-project
420 lines
18 KiB
C
420 lines
18 KiB
C
//===-- fp_div_impl.inc - Floating point division -----------------*- C -*-===//
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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//
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//===----------------------------------------------------------------------===//
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//
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// This file implements soft-float division with the IEEE-754 default
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// rounding (to nearest, ties to even).
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//
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//===----------------------------------------------------------------------===//
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#include "fp_lib.h"
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// The __divXf3__ function implements Newton-Raphson floating point division.
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// It uses 3 iterations for float32, 4 for float64 and 5 for float128,
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// respectively. Due to number of significant bits being roughly doubled
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// every iteration, the two modes are supported: N full-width iterations (as
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// it is done for float32 by default) and (N-1) half-width iteration plus one
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// final full-width iteration. It is expected that half-width integer
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// operations (w.r.t rep_t size) can be performed faster for some hardware but
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// they require error estimations to be computed separately due to larger
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// computational errors caused by truncating intermediate results.
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// Half the bit-size of rep_t
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#define HW (typeWidth / 2)
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// rep_t-sized bitmask with lower half of bits set to ones
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#define loMask (REP_C(-1) >> HW)
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#if NUMBER_OF_FULL_ITERATIONS < 1
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#error At least one full iteration is required
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#endif
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static __inline fp_t __divXf3__(fp_t a, fp_t b) {
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const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
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const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
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const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
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rep_t aSignificand = toRep(a) & significandMask;
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rep_t bSignificand = toRep(b) & significandMask;
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int scale = 0;
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// Detect if a or b is zero, denormal, infinity, or NaN.
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if (aExponent - 1U >= maxExponent - 1U ||
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bExponent - 1U >= maxExponent - 1U) {
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const rep_t aAbs = toRep(a) & absMask;
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const rep_t bAbs = toRep(b) & absMask;
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// NaN / anything = qNaN
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if (aAbs > infRep)
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return fromRep(toRep(a) | quietBit);
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// anything / NaN = qNaN
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if (bAbs > infRep)
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return fromRep(toRep(b) | quietBit);
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if (aAbs == infRep) {
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// infinity / infinity = NaN
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if (bAbs == infRep)
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return fromRep(qnanRep);
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// infinity / anything else = +/- infinity
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else
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return fromRep(aAbs | quotientSign);
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}
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// anything else / infinity = +/- 0
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if (bAbs == infRep)
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return fromRep(quotientSign);
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if (!aAbs) {
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// zero / zero = NaN
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if (!bAbs)
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return fromRep(qnanRep);
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// zero / anything else = +/- zero
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else
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return fromRep(quotientSign);
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}
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// anything else / zero = +/- infinity
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if (!bAbs)
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return fromRep(infRep | quotientSign);
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// One or both of a or b is denormal. The other (if applicable) is a
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// normal number. Renormalize one or both of a and b, and set scale to
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// include the necessary exponent adjustment.
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if (aAbs < implicitBit)
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scale += normalize(&aSignificand);
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if (bAbs < implicitBit)
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scale -= normalize(&bSignificand);
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}
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// Set the implicit significand bit. If we fell through from the
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// denormal path it was already set by normalize( ), but setting it twice
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// won't hurt anything.
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aSignificand |= implicitBit;
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bSignificand |= implicitBit;
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int writtenExponent = (aExponent - bExponent + scale) + exponentBias;
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const rep_t b_UQ1 = bSignificand << (typeWidth - significandBits - 1);
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// Align the significand of b as a UQ1.(n-1) fixed-point number in the range
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// [1.0, 2.0) and get a UQ0.n approximate reciprocal using a small minimax
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// polynomial approximation: x0 = 3/4 + 1/sqrt(2) - b/2.
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// The max error for this approximation is achieved at endpoints, so
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// abs(x0(b) - 1/b) <= abs(x0(1) - 1/1) = 3/4 - 1/sqrt(2) = 0.04289...,
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// which is about 4.5 bits.
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// The initial approximation is between x0(1.0) = 0.9571... and x0(2.0) = 0.4571...
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// Then, refine the reciprocal estimate using a quadratically converging
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// Newton-Raphson iteration:
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// x_{n+1} = x_n * (2 - x_n * b)
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//
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// Let b be the original divisor considered "in infinite precision" and
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// obtained from IEEE754 representation of function argument (with the
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// implicit bit set). Corresponds to rep_t-sized b_UQ1 represented in
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// UQ1.(W-1).
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//
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// Let b_hw be an infinitely precise number obtained from the highest (HW-1)
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// bits of divisor significand (with the implicit bit set). Corresponds to
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// half_rep_t-sized b_UQ1_hw represented in UQ1.(HW-1) that is a **truncated**
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// version of b_UQ1.
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//
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// Let e_n := x_n - 1/b_hw
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// E_n := x_n - 1/b
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// abs(E_n) <= abs(e_n) + (1/b_hw - 1/b)
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// = abs(e_n) + (b - b_hw) / (b*b_hw)
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// <= abs(e_n) + 2 * 2^-HW
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// rep_t-sized iterations may be slower than the corresponding half-width
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// variant depending on the handware and whether single/double/quad precision
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// is selected.
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// NB: Using half-width iterations increases computation errors due to
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// rounding, so error estimations have to be computed taking the selected
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// mode into account!
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#if NUMBER_OF_HALF_ITERATIONS > 0
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// Starting with (n-1) half-width iterations
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const half_rep_t b_UQ1_hw = bSignificand >> (significandBits + 1 - HW);
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// C is (3/4 + 1/sqrt(2)) - 1 truncated to W0 fractional bits as UQ0.HW
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// with W0 being either 16 or 32 and W0 <= HW.
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// That is, C is the aforementioned 3/4 + 1/sqrt(2) constant (from which
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// b/2 is subtracted to obtain x0) wrapped to [0, 1) range.
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#if defined(SINGLE_PRECISION)
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// Use 16-bit initial estimation in case we are using half-width iterations
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// for float32 division. This is expected to be useful for some 16-bit
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// targets. Not used by default as it requires performing more work during
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// rounding and would hardly help on regular 32- or 64-bit targets.
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const half_rep_t C_hw = HALF_REP_C(0x7504);
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#else
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// HW is at least 32. Shifting into the highest bits if needed.
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const half_rep_t C_hw = HALF_REP_C(0x7504F333) << (HW - 32);
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#endif
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// b >= 1, thus an upper bound for 3/4 + 1/sqrt(2) - b/2 is about 0.9572,
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// so x0 fits to UQ0.HW without wrapping.
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half_rep_t x_UQ0_hw = C_hw - (b_UQ1_hw /* exact b_hw/2 as UQ0.HW */);
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// An e_0 error is comprised of errors due to
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// * x0 being an inherently imprecise first approximation of 1/b_hw
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// * C_hw being some (irrational) number **truncated** to W0 bits
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// Please note that e_0 is calculated against the infinitely precise
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// reciprocal of b_hw (that is, **truncated** version of b).
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//
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// e_0 <= 3/4 - 1/sqrt(2) + 2^-W0
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// By construction, 1 <= b < 2
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// f(x) = x * (2 - b*x) = 2*x - b*x^2
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// f'(x) = 2 * (1 - b*x)
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//
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// On the [0, 1] interval, f(0) = 0,
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// then it increses until f(1/b) = 1 / b, maximum on (0, 1),
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// then it decreses to f(1) = 2 - b
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//
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// Let g(x) = x - f(x) = b*x^2 - x.
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// On (0, 1/b), g(x) < 0 <=> f(x) > x
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// On (1/b, 1], g(x) > 0 <=> f(x) < x
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//
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// For half-width iterations, b_hw is used instead of b.
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REPEAT_N_TIMES(NUMBER_OF_HALF_ITERATIONS, {
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// corr_UQ1_hw can be **larger** than 2 - b_hw*x by at most 1*Ulp
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// of corr_UQ1_hw.
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// "0.0 - (...)" is equivalent to "2.0 - (...)" in UQ1.(HW-1).
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// On the other hand, corr_UQ1_hw should not overflow from 2.0 to 0.0 provided
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// no overflow occurred earlier: ((rep_t)x_UQ0_hw * b_UQ1_hw >> HW) is
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// expected to be strictly positive because b_UQ1_hw has its highest bit set
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// and x_UQ0_hw should be rather large (it converges to 1/2 < 1/b_hw <= 1).
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half_rep_t corr_UQ1_hw = 0 - ((rep_t)x_UQ0_hw * b_UQ1_hw >> HW);
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// Now, we should multiply UQ0.HW and UQ1.(HW-1) numbers, naturally
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// obtaining an UQ1.(HW-1) number and proving its highest bit could be
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// considered to be 0 to be able to represent it in UQ0.HW.
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// From the above analysis of f(x), if corr_UQ1_hw would be represented
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// without any intermediate loss of precision (that is, in twice_rep_t)
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// x_UQ0_hw could be at most [1.]000... if b_hw is exactly 1.0 and strictly
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// less otherwise. On the other hand, to obtain [1.]000..., one have to pass
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// 1/b_hw == 1.0 to f(x), so this cannot occur at all without overflow (due
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// to 1.0 being not representable as UQ0.HW).
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// The fact corr_UQ1_hw was virtually round up (due to result of
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// multiplication being **first** truncated, then negated - to improve
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// error estimations) can increase x_UQ0_hw by up to 2*Ulp of x_UQ0_hw.
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x_UQ0_hw = (rep_t)x_UQ0_hw * corr_UQ1_hw >> (HW - 1);
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// Now, either no overflow occurred or x_UQ0_hw is 0 or 1 in its half_rep_t
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// representation. In the latter case, x_UQ0_hw will be either 0 or 1 after
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// any number of iterations, so just subtract 2 from the reciprocal
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// approximation after last iteration.
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// In infinite precision, with 0 <= eps1, eps2 <= U = 2^-HW:
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// corr_UQ1_hw = 2 - (1/b_hw + e_n) * b_hw + 2*eps1
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// = 1 - e_n * b_hw + 2*eps1
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// x_UQ0_hw = (1/b_hw + e_n) * (1 - e_n*b_hw + 2*eps1) - eps2
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// = 1/b_hw - e_n + 2*eps1/b_hw + e_n - e_n^2*b_hw + 2*e_n*eps1 - eps2
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// = 1/b_hw + 2*eps1/b_hw - e_n^2*b_hw + 2*e_n*eps1 - eps2
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// e_{n+1} = -e_n^2*b_hw + 2*eps1/b_hw + 2*e_n*eps1 - eps2
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// = 2*e_n*eps1 - (e_n^2*b_hw + eps2) + 2*eps1/b_hw
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// \------ >0 -------/ \-- >0 ---/
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// abs(e_{n+1}) <= 2*abs(e_n)*U + max(2*e_n^2 + U, 2 * U)
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})
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// For initial half-width iterations, U = 2^-HW
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// Let abs(e_n) <= u_n * U,
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// then abs(e_{n+1}) <= 2 * u_n * U^2 + max(2 * u_n^2 * U^2 + U, 2 * U)
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// u_{n+1} <= 2 * u_n * U + max(2 * u_n^2 * U + 1, 2)
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// Account for possible overflow (see above). For an overflow to occur for the
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// first time, for "ideal" corr_UQ1_hw (that is, without intermediate
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// truncation), the result of x_UQ0_hw * corr_UQ1_hw should be either maximum
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// value representable in UQ0.HW or less by 1. This means that 1/b_hw have to
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// be not below that value (see g(x) above), so it is safe to decrement just
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// once after the final iteration. On the other hand, an effective value of
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// divisor changes after this point (from b_hw to b), so adjust here.
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x_UQ0_hw -= 1U;
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rep_t x_UQ0 = (rep_t)x_UQ0_hw << HW;
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x_UQ0 -= 1U;
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#else
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// C is (3/4 + 1/sqrt(2)) - 1 truncated to 32 fractional bits as UQ0.n
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const rep_t C = REP_C(0x7504F333) << (typeWidth - 32);
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rep_t x_UQ0 = C - b_UQ1;
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// E_0 <= 3/4 - 1/sqrt(2) + 2 * 2^-32
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#endif
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// Error estimations for full-precision iterations are calculated just
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// as above, but with U := 2^-W and taking extra decrementing into account.
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// We need at least one such iteration.
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#ifdef USE_NATIVE_FULL_ITERATIONS
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REPEAT_N_TIMES(NUMBER_OF_FULL_ITERATIONS, {
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rep_t corr_UQ1 = 0 - ((twice_rep_t)x_UQ0 * b_UQ1 >> typeWidth);
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x_UQ0 = (twice_rep_t)x_UQ0 * corr_UQ1 >> (typeWidth - 1);
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})
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#else
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#if NUMBER_OF_FULL_ITERATIONS != 1
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#error Only a single emulated full iteration is supported
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#endif
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#if !(NUMBER_OF_HALF_ITERATIONS > 0)
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// Cannot normally reach here: only one full-width iteration is requested and
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// the total number of iterations should be at least 3 even for float32.
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#error Check NUMBER_OF_HALF_ITERATIONS, NUMBER_OF_FULL_ITERATIONS and USE_NATIVE_FULL_ITERATIONS.
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#endif
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// Simulating operations on a twice_rep_t to perform a single final full-width
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// iteration. Using ad-hoc multiplication implementations to take advantage
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// of particular structure of operands.
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rep_t blo = b_UQ1 & loMask;
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// x_UQ0 = x_UQ0_hw * 2^HW - 1
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// x_UQ0 * b_UQ1 = (x_UQ0_hw * 2^HW) * (b_UQ1_hw * 2^HW + blo) - b_UQ1
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//
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// <--- higher half ---><--- lower half --->
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// [x_UQ0_hw * b_UQ1_hw]
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// + [ x_UQ0_hw * blo ]
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// - [ b_UQ1 ]
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// = [ result ][.... discarded ...]
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rep_t corr_UQ1 = 0U - ( (rep_t)x_UQ0_hw * b_UQ1_hw
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+ ((rep_t)x_UQ0_hw * blo >> HW)
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- REP_C(1)); // account for *possible* carry
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rep_t lo_corr = corr_UQ1 & loMask;
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rep_t hi_corr = corr_UQ1 >> HW;
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// x_UQ0 * corr_UQ1 = (x_UQ0_hw * 2^HW) * (hi_corr * 2^HW + lo_corr) - corr_UQ1
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x_UQ0 = ((rep_t)x_UQ0_hw * hi_corr << 1)
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+ ((rep_t)x_UQ0_hw * lo_corr >> (HW - 1))
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- REP_C(2); // 1 to account for the highest bit of corr_UQ1 can be 1
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// 1 to account for possible carry
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// Just like the case of half-width iterations but with possibility
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// of overflowing by one extra Ulp of x_UQ0.
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x_UQ0 -= 1U;
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// ... and then traditional fixup by 2 should work
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// On error estimation:
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// abs(E_{N-1}) <= (u_{N-1} + 2 /* due to conversion e_n -> E_n */) * 2^-HW
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// + (2^-HW + 2^-W))
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// abs(E_{N-1}) <= (u_{N-1} + 3.01) * 2^-HW
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// Then like for the half-width iterations:
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// With 0 <= eps1, eps2 < 2^-W
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// E_N = 4 * E_{N-1} * eps1 - (E_{N-1}^2 * b + 4 * eps2) + 4 * eps1 / b
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// abs(E_N) <= 2^-W * [ 4 * abs(E_{N-1}) + max(2 * abs(E_{N-1})^2 * 2^W + 4, 8)) ]
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// abs(E_N) <= 2^-W * [ 4 * (u_{N-1} + 3.01) * 2^-HW + max(4 + 2 * (u_{N-1} + 3.01)^2, 8) ]
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#endif
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// Finally, account for possible overflow, as explained above.
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x_UQ0 -= 2U;
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// u_n for different precisions (with N-1 half-width iterations):
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// W0 is the precision of C
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// u_0 = (3/4 - 1/sqrt(2) + 2^-W0) * 2^HW
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// Estimated with bc:
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// define half1(un) { return 2.0 * (un + un^2) / 2.0^hw + 1.0; }
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// define half2(un) { return 2.0 * un / 2.0^hw + 2.0; }
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// define full1(un) { return 4.0 * (un + 3.01) / 2.0^hw + 2.0 * (un + 3.01)^2 + 4.0; }
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// define full2(un) { return 4.0 * (un + 3.01) / 2.0^hw + 8.0; }
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// | f32 (0 + 3) | f32 (2 + 1) | f64 (3 + 1) | f128 (4 + 1)
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// u_0 | < 184224974 | < 2812.1 | < 184224974 | < 791240234244348797
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// u_1 | < 15804007 | < 242.7 | < 15804007 | < 67877681371350440
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// u_2 | < 116308 | < 2.81 | < 116308 | < 499533100252317
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// u_3 | < 7.31 | | < 7.31 | < 27054456580
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// u_4 | | | | < 80.4
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// Final (U_N) | same as u_3 | < 72 | < 218 | < 13920
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// Add 2 to U_N due to final decrement.
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#if defined(SINGLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 2 && NUMBER_OF_FULL_ITERATIONS == 1
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#define RECIPROCAL_PRECISION REP_C(74)
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#elif defined(SINGLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 0 && NUMBER_OF_FULL_ITERATIONS == 3
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#define RECIPROCAL_PRECISION REP_C(10)
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#elif defined(DOUBLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 3 && NUMBER_OF_FULL_ITERATIONS == 1
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#define RECIPROCAL_PRECISION REP_C(220)
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#elif defined(QUAD_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 4 && NUMBER_OF_FULL_ITERATIONS == 1
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#define RECIPROCAL_PRECISION REP_C(13922)
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#else
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#error Invalid number of iterations
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#endif
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// Suppose 1/b - P * 2^-W < x < 1/b + P * 2^-W
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x_UQ0 -= RECIPROCAL_PRECISION;
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// Now 1/b - (2*P) * 2^-W < x < 1/b
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// FIXME Is x_UQ0 still >= 0.5?
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rep_t quotient_UQ1, dummy;
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wideMultiply(x_UQ0, aSignificand << 1, "ient_UQ1, &dummy);
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// Now, a/b - 4*P * 2^-W < q < a/b for q=<quotient_UQ1:dummy> in UQ1.(SB+1+W).
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// quotient_UQ1 is in [0.5, 2.0) as UQ1.(SB+1),
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// adjust it to be in [1.0, 2.0) as UQ1.SB.
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rep_t residualLo;
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if (quotient_UQ1 < (implicitBit << 1)) {
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// Highest bit is 0, so just reinterpret quotient_UQ1 as UQ1.SB,
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// effectively doubling its value as well as its error estimation.
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residualLo = (aSignificand << (significandBits + 1)) - quotient_UQ1 * bSignificand;
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writtenExponent -= 1;
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aSignificand <<= 1;
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} else {
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// Highest bit is 1 (the UQ1.(SB+1) value is in [1, 2)), convert it
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// to UQ1.SB by right shifting by 1. Least significant bit is omitted.
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quotient_UQ1 >>= 1;
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residualLo = (aSignificand << significandBits) - quotient_UQ1 * bSignificand;
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}
|
|
// NB: residualLo is calculated above for the normal result case.
|
|
// It is re-computed on denormal path that is expected to be not so
|
|
// performance-sensitive.
|
|
|
|
// Now, q cannot be greater than a/b and can differ by at most 8*P * 2^-W + 2^-SB
|
|
// Each NextAfter() increments the floating point value by at least 2^-SB
|
|
// (more, if exponent was incremented).
|
|
// Different cases (<---> is of 2^-SB length, * = a/b that is shown as a midpoint):
|
|
// q
|
|
// | | * | | | | |
|
|
// <---> 2^t
|
|
// | | | | | * | |
|
|
// q
|
|
// To require at most one NextAfter(), an error should be less than 1.5 * 2^-SB.
|
|
// (8*P) * 2^-W + 2^-SB < 1.5 * 2^-SB
|
|
// (8*P) * 2^-W < 0.5 * 2^-SB
|
|
// P < 2^(W-4-SB)
|
|
// Generally, for at most R NextAfter() to be enough,
|
|
// P < (2*R - 1) * 2^(W-4-SB)
|
|
// For f32 (0+3): 10 < 32 (OK)
|
|
// For f32 (2+1): 32 < 74 < 32 * 3, so two NextAfter() are required
|
|
// For f64: 220 < 256 (OK)
|
|
// For f128: 4096 * 3 < 13922 < 4096 * 5 (three NextAfter() are required)
|
|
|
|
// If we have overflowed the exponent, return infinity
|
|
if (writtenExponent >= maxExponent)
|
|
return fromRep(infRep | quotientSign);
|
|
|
|
// Now, quotient_UQ1_SB <= the correctly-rounded result
|
|
// and may need taking NextAfter() up to 3 times (see error estimates above)
|
|
// r = a - b * q
|
|
rep_t absResult;
|
|
if (writtenExponent > 0) {
|
|
// Clear the implicit bit
|
|
absResult = quotient_UQ1 & significandMask;
|
|
// Insert the exponent
|
|
absResult |= (rep_t)writtenExponent << significandBits;
|
|
residualLo <<= 1;
|
|
} else {
|
|
// Prevent shift amount from being negative
|
|
if (significandBits + writtenExponent < 0)
|
|
return fromRep(quotientSign);
|
|
|
|
absResult = quotient_UQ1 >> (-writtenExponent + 1);
|
|
|
|
// multiplied by two to prevent shift amount to be negative
|
|
residualLo = (aSignificand << (significandBits + writtenExponent)) - (absResult * bSignificand << 1);
|
|
}
|
|
|
|
// Round
|
|
residualLo += absResult & 1; // tie to even
|
|
// The above line conditionally turns the below LT comparison into LTE
|
|
absResult += residualLo > bSignificand;
|
|
#if defined(QUAD_PRECISION) || (defined(SINGLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS > 0)
|
|
// Do not round Infinity to NaN
|
|
absResult += absResult < infRep && residualLo > (2 + 1) * bSignificand;
|
|
#endif
|
|
#if defined(QUAD_PRECISION)
|
|
absResult += absResult < infRep && residualLo > (4 + 1) * bSignificand;
|
|
#endif
|
|
return fromRep(absResult | quotientSign);
|
|
}
|