213 lines
8.6 KiB
Zig
213 lines
8.6 KiB
Zig
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 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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@export(__divxf3, .{ .name = "__divxf3", .linkage = common.linkage });
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}
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pub fn __divxf3(a: f80, b: f80) callconv(.C) f80 {
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const T = f80;
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const Z = std.meta.Int(.unsigned, @bitSizeOf(T));
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const significandBits = std.math.floatMantissaBits(T);
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const fractionalBits = std.math.floatFractionalBits(T);
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const exponentBits = std.math.floatExponentBits(T);
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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 integerBit = (@as(Z, 1) << fractionalBits);
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const quietBit = integerBit >> 1;
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const significandMask = (@as(Z, 1) << significandBits) - 1;
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const absMask = signBit - 1;
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const qnanRep = @bitCast(Z, std.math.nan(T)) | quietBit;
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const infRep = @bitCast(Z, std.math.inf(T));
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const aExponent = @truncate(u32, (@bitCast(Z, a) >> significandBits) & maxExponent);
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const bExponent = @truncate(u32, (@bitCast(Z, b) >> significandBits) & maxExponent);
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const quotientSign: Z = (@bitCast(Z, a) ^ @bitCast(Z, b)) & signBit;
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var aSignificand: Z = @bitCast(Z, a) & significandMask;
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var bSignificand: Z = @bitCast(Z, 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 = @bitCast(Z, a) & absMask;
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const bAbs: Z = @bitCast(Z, b) & absMask;
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// NaN / anything = qNaN
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if (aAbs > infRep) return @bitCast(T, @bitCast(Z, a) | quietBit);
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// anything / NaN = qNaN
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if (bAbs > infRep) return @bitCast(T, @bitCast(Z, 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(T, qnanRep);
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}
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// infinity / anything else = +/- infinity
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else {
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return @bitCast(T, 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(T, 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(T, qnanRep);
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}
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// zero / anything else = +/- zero
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else {
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return @bitCast(T, 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(T, 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 < integerBit) scale +%= normalize(T, &aSignificand);
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if (bAbs < integerBit) scale -%= normalize(T, &bSignificand);
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}
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var quotientExponent: i32 = @bitCast(i32, aExponent -% bExponent) +% scale;
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// Align the significand of b as a Q63 fixed-point number in the range
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// [1, 2.0) and get a Q64 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 q63b = @intCast(u64, bSignificand);
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var recip64 = @as(u64, 0x7504f333F9DE6484) -% q63b;
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// 0x7504f333F9DE6484 / 2^64 + 1 = 3/4 + 1/sqrt(2)
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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.
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var correction64: u64 = undefined;
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correction64 = @truncate(u64, ~(@as(u128, recip64) *% q63b >> 64) +% 1);
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recip64 = @truncate(u64, @as(u128, recip64) *% correction64 >> 63);
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correction64 = @truncate(u64, ~(@as(u128, recip64) *% q63b >> 64) +% 1);
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recip64 = @truncate(u64, @as(u128, recip64) *% correction64 >> 63);
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correction64 = @truncate(u64, ~(@as(u128, recip64) *% q63b >> 64) +% 1);
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recip64 = @truncate(u64, @as(u128, recip64) *% correction64 >> 63);
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correction64 = @truncate(u64, ~(@as(u128, recip64) *% q63b >> 64) +% 1);
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recip64 = @truncate(u64, @as(u128, recip64) *% correction64 >> 63);
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correction64 = @truncate(u64, ~(@as(u128, recip64) *% q63b >> 64) +% 1);
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recip64 = @truncate(u64, @as(u128, recip64) *% correction64 >> 63);
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// The reciprocal may have overflowed to zero if the upper half of b is
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// exactly 1.0. This would sabatoge the full-width final stage of the
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// computation that follows, so we adjust the reciprocal down by one bit.
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recip64 -%= 1;
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// We need to perform one more iteration to get us to 112 binary digits;
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// The last iteration needs to happen with extra precision.
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// NOTE: This operation is equivalent to __multi3, which is not implemented
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// in some architechures
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var reciprocal: u128 = undefined;
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var correction: u128 = undefined;
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var dummy: u128 = undefined;
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wideMultiply(u128, recip64, q63b, &dummy, &correction);
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correction = -%correction;
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const cHi = @truncate(u64, correction >> 64);
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const cLo = @truncate(u64, correction);
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var r64cH: u128 = undefined;
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var r64cL: u128 = undefined;
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wideMultiply(u128, recip64, cHi, &dummy, &r64cH);
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wideMultiply(u128, recip64, cLo, &dummy, &r64cL);
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reciprocal = r64cH + (r64cL >> 64);
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// Adjust the final 128-bit reciprocal estimate downward to ensure that it
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// is strictly smaller than the infinitely precise exact reciprocal. Because
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// the computation of the Newton-Raphson step is truncating at every step,
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// this adjustment 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^-112, 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 Q127 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^-63 (actually, we have
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// many bits to spare, but this is all we need).
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// We need a 128 x 128 multiply high to compute q.
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var quotient128: u128 = undefined;
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var quotientLo: u128 = undefined;
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wideMultiply(u128, aSignificand << 2, reciprocal, "ient128, "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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// Right shift the quotient if it falls in the [1,2) range and adjust the
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// exponent accordingly.
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var quotient: u64 = if (quotient128 < (integerBit << 1)) b: {
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quotientExponent -= 1;
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break :b @intCast(u64, quotient128);
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} else @intCast(u64, quotient128 >> 1);
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// 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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var residual: u64 = -%(quotient *% q63b);
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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(T, 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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if (residual > (bSignificand >> 1)) { // round
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if (quotient == (integerBit - 1)) // If the rounded result is normal, return it
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return @bitCast(T, @bitCast(Z, std.math.floatMin(T)) | 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(T, quotientSign);
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} else {
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const round = @boolToInt(residual > (bSignificand >> 1));
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// Insert the exponent
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var absResult = quotient | (@intCast(Z, 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(T, absResult | quotientSign | integerBit);
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}
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}
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test {
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if (builtin.zig_backend == .stage2_llvm and builtin.os.tag == .windows) return error.SkipZigTest; // https://github.com/ziglang/zig/issues/12603
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_ = @import("divxf3_test.zig");
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}
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