231 lines
9.2 KiB
Zig
231 lines
9.2 KiB
Zig
//! Ported from:
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//!
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//! https://github.com/llvm/llvm-project/commit/d674d96bc56c0f377879d01c9d8dfdaaa7859cdb/compiler-rt/lib/builtins/divdf3.c
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const std = @import("std");
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const builtin = @import("builtin");
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const arch = builtin.cpu.arch;
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const is_test = builtin.is_test;
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const common = @import("common.zig");
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const normalize = common.normalize;
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const wideMultiply = common.wideMultiply;
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pub const panic = common.panic;
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comptime {
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if (common.want_aeabi) {
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@export(__aeabi_ddiv, .{ .name = "__aeabi_ddiv", .linkage = common.linkage, .visibility = common.visibility });
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} else {
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@export(__divdf3, .{ .name = "__divdf3", .linkage = common.linkage, .visibility = common.visibility });
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}
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}
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pub fn __divdf3(a: f64, b: f64) callconv(.C) f64 {
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return div(a, b);
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}
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fn __aeabi_ddiv(a: f64, b: f64) callconv(.AAPCS) f64 {
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return div(a, b);
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}
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inline fn div(a: f64, b: f64) f64 {
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const Z = std.meta.Int(.unsigned, 64);
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const SignedZ = std.meta.Int(.signed, 64);
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const significandBits = std.math.floatMantissaBits(f64);
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const exponentBits = std.math.floatExponentBits(f64);
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const signBit = (@as(Z, 1) << (significandBits + exponentBits));
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const maxExponent = ((1 << exponentBits) - 1);
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const exponentBias = (maxExponent >> 1);
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const implicitBit = (@as(Z, 1) << significandBits);
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const quietBit = implicitBit >> 1;
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const significandMask = implicitBit - 1;
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const absMask = signBit - 1;
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const exponentMask = absMask ^ significandMask;
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const qnanRep = exponentMask | quietBit;
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const infRep = @as(Z, @bitCast(std.math.inf(f64)));
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const aExponent: u32 = @truncate((@as(Z, @bitCast(a)) >> significandBits) & maxExponent);
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const bExponent: u32 = @truncate((@as(Z, @bitCast(b)) >> significandBits) & maxExponent);
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const quotientSign: Z = (@as(Z, @bitCast(a)) ^ @as(Z, @bitCast(b))) & signBit;
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var aSignificand: Z = @as(Z, @bitCast(a)) & significandMask;
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var bSignificand: Z = @as(Z, @bitCast(b)) & significandMask;
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var scale: i32 = 0;
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// Detect if a or b is zero, denormal, infinity, or NaN.
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if (aExponent -% 1 >= maxExponent - 1 or bExponent -% 1 >= maxExponent - 1) {
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const aAbs: Z = @as(Z, @bitCast(a)) & absMask;
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const bAbs: Z = @as(Z, @bitCast(b)) & absMask;
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// NaN / anything = qNaN
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if (aAbs > infRep) return @bitCast(@as(Z, @bitCast(a)) | quietBit);
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// anything / NaN = qNaN
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if (bAbs > infRep) return @bitCast(@as(Z, @bitCast(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 @bitCast(qnanRep);
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}
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// infinity / anything else = +/- infinity
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else {
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return @bitCast(aAbs | quotientSign);
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}
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}
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// anything else / infinity = +/- 0
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if (bAbs == infRep) return @bitCast(quotientSign);
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if (aAbs == 0) {
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// zero / zero = NaN
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if (bAbs == 0) {
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return @bitCast(qnanRep);
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}
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// zero / anything else = +/- zero
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else {
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return @bitCast(quotientSign);
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}
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}
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// anything else / zero = +/- infinity
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if (bAbs == 0) return @bitCast(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) scale +%= normalize(f64, &aSignificand);
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if (bAbs < implicitBit) scale -%= normalize(f64, &bSignificand);
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}
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// Or in 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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var quotientExponent: i32 = @as(i32, @bitCast(aExponent -% bExponent)) +% scale;
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// Align the significand of b as a Q31 fixed-point number in the range
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// [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
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// polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This
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// is accurate to about 3.5 binary digits.
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const q31b: u32 = @truncate(bSignificand >> 21);
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var recip32 = @as(u32, 0x7504f333) -% q31b;
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// Now refine the reciprocal estimate using a Newton-Raphson iteration:
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//
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// x1 = x0 * (2 - x0 * b)
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//
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// This doubles the number of correct binary digits in the approximation
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// with each iteration, so after three iterations, we have about 28 binary
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// digits of accuracy.
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var correction32: u32 = undefined;
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correction32 = @truncate(~(@as(u64, recip32) *% q31b >> 32) +% 1);
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recip32 = @truncate(@as(u64, recip32) *% correction32 >> 31);
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correction32 = @truncate(~(@as(u64, recip32) *% q31b >> 32) +% 1);
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recip32 = @truncate(@as(u64, recip32) *% correction32 >> 31);
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correction32 = @truncate(~(@as(u64, recip32) *% q31b >> 32) +% 1);
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recip32 = @truncate(@as(u64, recip32) *% correction32 >> 31);
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// recip32 might have overflowed to exactly zero in the preceding
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// computation if the high word of b is exactly 1.0. This would sabotage
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// the full-width final stage of the computation that follows, so we adjust
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// recip32 downward by one bit.
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recip32 -%= 1;
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// We need to perform one more iteration to get us to 56 binary digits;
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// The last iteration needs to happen with extra precision.
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const q63blo: u32 = @truncate(bSignificand << 11);
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var correction: u64 = undefined;
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var reciprocal: u64 = undefined;
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correction = ~(@as(u64, recip32) *% q31b +% (@as(u64, recip32) *% q63blo >> 32)) +% 1;
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const cHi: u32 = @truncate(correction >> 32);
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const cLo: u32 = @truncate(correction);
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reciprocal = @as(u64, recip32) *% cHi +% (@as(u64, recip32) *% cLo >> 32);
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// We already adjusted the 32-bit estimate, now we need to adjust the final
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// 64-bit reciprocal estimate downward to ensure that it is strictly smaller
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// than the infinitely precise exact reciprocal. Because the computation
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// of the Newton-Raphson step is truncating at every step, this adjustment
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// is small; most of the work is already done.
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reciprocal -%= 2;
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// The numerical reciprocal is accurate to within 2^-56, lies in the
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// interval [0.5, 1.0), and is strictly smaller than the true reciprocal
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// of b. Multiplying a by this reciprocal thus gives a numerical q = a/b
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// in Q53 with the following properties:
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//
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// 1. q < a/b
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// 2. q is in the interval [0.5, 2.0)
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// 3. the error in q is bounded away from 2^-53 (actually, we have a
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// couple of bits to spare, but this is all we need).
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// We need a 64 x 64 multiply high to compute q, which isn't a basic
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// operation in C, so we need to be a little bit fussy.
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var quotient: Z = undefined;
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var quotientLo: Z = undefined;
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wideMultiply(Z, aSignificand << 2, reciprocal, "ient, "ientLo);
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// Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
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// In either case, we are going to compute a residual of the form
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//
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// r = a - q*b
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//
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// We know from the construction of q that r satisfies:
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//
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// 0 <= r < ulp(q)*b
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//
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// if r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we
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// already have the correct result. The exact halfway case cannot occur.
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// We also take this time to right shift quotient if it falls in the [1,2)
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// range and adjust the exponent accordingly.
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var residual: Z = undefined;
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if (quotient < (implicitBit << 1)) {
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residual = (aSignificand << 53) -% quotient *% bSignificand;
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quotientExponent -%= 1;
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} else {
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quotient >>= 1;
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residual = (aSignificand << 52) -% quotient *% bSignificand;
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}
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const writtenExponent = quotientExponent +% exponentBias;
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if (writtenExponent >= maxExponent) {
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// If we have overflowed the exponent, return infinity.
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return @bitCast(infRep | quotientSign);
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} else if (writtenExponent < 1) {
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if (writtenExponent == 0) {
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// Check whether the rounded result is normal.
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const round = @intFromBool((residual << 1) > bSignificand);
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// Clear the implicit bit.
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var absResult = quotient & significandMask;
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// Round.
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absResult += round;
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if ((absResult & ~significandMask) != 0) {
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// The rounded result is normal; return it.
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return @bitCast(absResult | quotientSign);
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}
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}
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// Flush denormals to zero. In the future, it would be nice to add
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// code to round them correctly.
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return @bitCast(quotientSign);
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} else {
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const round = @intFromBool((residual << 1) > bSignificand);
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// Clear the implicit bit
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var absResult = quotient & significandMask;
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// Insert the exponent
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absResult |= @as(Z, @bitCast(@as(SignedZ, writtenExponent))) << significandBits;
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// Round
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absResult +%= round;
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// Insert the sign and return
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return @bitCast(absResult | quotientSign);
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}
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}
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test {
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_ = @import("divdf3_test.zig");
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}
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