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