zig

divtf3.zig (9925B) - Raw

---

 const std = @import("std");
 const builtin = @import("builtin");
 
 const common = @import("common.zig");
 const normalize = common.normalize;
 const wideMultiply = common.wideMultiply;
 
 pub const panic = common.panic;
 
 comptime {
     if (common.want_ppc_abi) {
         @export(&__divtf3, .{ .name = "__divkf3", .linkage = common.linkage, .visibility = common.visibility });
     } else if (common.want_sparc_abi) {
         @export(&_Qp_div, .{ .name = "_Qp_div", .linkage = common.linkage, .visibility = common.visibility });
     }
     @export(&__divtf3, .{ .name = "__divtf3", .linkage = common.linkage, .visibility = common.visibility });
 }
 
 pub fn __divtf3(a: f128, b: f128) callconv(.c) f128 {
     return div(a, b);
 }
 
 fn _Qp_div(c: *f128, a: *const f128, b: *const f128) callconv(.c) void {
     c.* = div(a.*, b.*);
 }
 
 inline fn div(a: f128, b: f128) f128 {
     const Z = std.meta.Int(.unsigned, 128);
 
     const significandBits = std.math.floatMantissaBits(f128);
     const exponentBits = std.math.floatExponentBits(f128);
 
     const signBit = (@as(Z, 1) << (significandBits + exponentBits));
     const maxExponent = ((1 << exponentBits) - 1);
     const exponentBias = (maxExponent >> 1);
 
     const implicitBit = (@as(Z, 1) << significandBits);
     const quietBit = implicitBit >> 1;
     const significandMask = implicitBit - 1;
 
     const absMask = signBit - 1;
     const exponentMask = absMask ^ significandMask;
     const qnanRep = exponentMask | quietBit;
     const infRep: Z = @bitCast(std.math.inf(f128));
 
     const aExponent: u32 = @truncate((@as(Z, @bitCast(a)) >> significandBits) & maxExponent);
     const bExponent: u32 = @truncate((@as(Z, @bitCast(b)) >> significandBits) & maxExponent);
     const quotientSign: Z = (@as(Z, @bitCast(a)) ^ @as(Z, @bitCast(b))) & signBit;
 
     var aSignificand: Z = @as(Z, @bitCast(a)) & significandMask;
     var bSignificand: Z = @as(Z, @bitCast(b)) & significandMask;
     var scale: i32 = 0;
 
     // Detect if a or b is zero, denormal, infinity, or NaN.
     if (aExponent -% 1 >= maxExponent - 1 or bExponent -% 1 >= maxExponent - 1) {
         const aAbs: Z = @as(Z, @bitCast(a)) & absMask;
         const bAbs: Z = @as(Z, @bitCast(b)) & absMask;
 
         // NaN / anything = qNaN
         if (aAbs > infRep) return @bitCast(@as(Z, @bitCast(a)) | quietBit);
         // anything / NaN = qNaN
         if (bAbs > infRep) return @bitCast(@as(Z, @bitCast(b)) | quietBit);
 
         if (aAbs == infRep) {
             // infinity / infinity = NaN
             if (bAbs == infRep) {
                 return @bitCast(qnanRep);
             }
             // infinity / anything else = +/- infinity
             else {
                 return @bitCast(aAbs | quotientSign);
             }
         }
 
         // anything else / infinity = +/- 0
         if (bAbs == infRep) return @bitCast(quotientSign);
 
         if (aAbs == 0) {
             // zero / zero = NaN
             if (bAbs == 0) {
                 return @bitCast(qnanRep);
             }
             // zero / anything else = +/- zero
             else {
                 return @bitCast(quotientSign);
             }
         }
         // anything else / zero = +/- infinity
         if (bAbs == 0) return @bitCast(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(f128, &aSignificand);
         if (bAbs < implicitBit) scale -%= normalize(f128, &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;
     var quotientExponent: i32 = @as(i32, @bitCast(aExponent -% bExponent)) +% scale;
 
     // Align the significand of b as a Q63 fixed-point number in the range
     // [1, 2.0) and get a Q64 approximate reciprocal using a small minimax
     // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2.  This
     // is accurate to about 3.5 binary digits.
     const q63b: u64 = @truncate(bSignificand >> 49);
     var recip64 = @as(u64, 0x7504f333F9DE6484) -% q63b;
     // 0x7504f333F9DE6484 / 2^64 + 1 = 3/4 + 1/sqrt(2)
 
     // Now refine the reciprocal estimate using a Newton-Raphson iteration:
     //
     //     x1 = x0 * (2 - x0 * b)
     //
     // This doubles the number of correct binary digits in the approximation
     // with each iteration.
     var correction64: u64 = undefined;
     correction64 = @truncate(~(@as(u128, recip64) *% q63b >> 64) +% 1);
     recip64 = @truncate(@as(u128, recip64) *% correction64 >> 63);
     correction64 = @truncate(~(@as(u128, recip64) *% q63b >> 64) +% 1);
     recip64 = @truncate(@as(u128, recip64) *% correction64 >> 63);
     correction64 = @truncate(~(@as(u128, recip64) *% q63b >> 64) +% 1);
     recip64 = @truncate(@as(u128, recip64) *% correction64 >> 63);
     correction64 = @truncate(~(@as(u128, recip64) *% q63b >> 64) +% 1);
     recip64 = @truncate(@as(u128, recip64) *% correction64 >> 63);
     correction64 = @truncate(~(@as(u128, recip64) *% q63b >> 64) +% 1);
     recip64 = @truncate(@as(u128, recip64) *% correction64 >> 63);
 
     // The reciprocal may have overflowed to zero if the upper half of b is
     // exactly 1.0.  This would sabatoge the full-width final stage of the
     // computation that follows, so we adjust the reciprocal down by one bit.
     recip64 -%= 1;
 
     // We need to perform one more iteration to get us to 112 binary digits;
     // The last iteration needs to happen with extra precision.
     const q127blo: u64 = @truncate(bSignificand << 15);
     var correction: u128 = undefined;
     var reciprocal: u128 = undefined;
 
     // NOTE: This operation is equivalent to __multi3, which is not implemented
     //       in some architecture
     var r64q63: u128 = undefined;
     var r64q127: u128 = undefined;
     var r64cH: u128 = undefined;
     var r64cL: u128 = undefined;
     var dummy: u128 = undefined;
     wideMultiply(u128, recip64, q63b, &dummy, &r64q63);
     wideMultiply(u128, recip64, q127blo, &dummy, &r64q127);
 
     correction = -%(r64q63 + (r64q127 >> 64));
 
     const cHi: u64 = @truncate(correction >> 64);
     const cLo: u64 = @truncate(correction);
 
     wideMultiply(u128, recip64, cHi, &dummy, &r64cH);
     wideMultiply(u128, recip64, cLo, &dummy, &r64cL);
 
     reciprocal = r64cH + (r64cL >> 64);
 
     // Adjust the final 128-bit reciprocal estimate downward to ensure that it
     // is strictly smaller than the infinitely precise exact reciprocal. Because
     // the computation of the Newton-Raphson step is truncating at every step,
     // this adjustment is small; most of the work is already done.
     reciprocal -%= 2;
 
     // The numerical reciprocal is accurate to within 2^-112, lies in the
     // interval [0.5, 1.0), and is strictly smaller than the true reciprocal
     // of b.  Multiplying a by this reciprocal thus gives a numerical q = a/b
     // in Q127 with the following properties:
     //
     //    1. q < a/b
     //    2. q is in the interval [0.5, 2.0)
     //    3. The error in q is bounded away from 2^-113 (actually, we have a
     //       couple of bits to spare, but this is all we need).
 
     // We need a 128 x 128 multiply high to compute q.
     var quotient: u128 = undefined;
     var quotientLo: u128 = undefined;
     wideMultiply(u128, aSignificand << 2, reciprocal, &quotient, &quotientLo);
 
     // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
     // In either case, we are going to compute a residual of the form
     //
     //     r = a - q*b
     //
     // We know from the construction of q that r satisfies:
     //
     //     0 <= r < ulp(q)*b
     //
     // If r is greater than 1/2 ulp(q)*b, then q rounds up.  Otherwise, we
     // already have the correct result.  The exact halfway case cannot occur.
     // We also take this time to right shift quotient if it falls in the [1,2)
     // range and adjust the exponent accordingly.
     var residual: u128 = undefined;
     var qb: u128 = undefined;
 
     if (quotient < (implicitBit << 1)) {
         wideMultiply(u128, quotient, bSignificand, &dummy, &qb);
         residual = (aSignificand << 113) -% qb;
         quotientExponent -%= 1;
     } else {
         quotient >>= 1;
         wideMultiply(u128, quotient, bSignificand, &dummy, &qb);
         residual = (aSignificand << 112) -% qb;
     }
 
     const writtenExponent = quotientExponent +% exponentBias;
 
     if (writtenExponent >= maxExponent) {
         // If we have overflowed the exponent, return infinity.
         return @bitCast(infRep | quotientSign);
     } else if (writtenExponent < 1) {
         if (writtenExponent == 0) {
             // Check whether the rounded result is normal.
             const round = @intFromBool((residual << 1) > bSignificand);
             // Clear the implicit bit.
             var absResult = quotient & significandMask;
             // Round.
             absResult += round;
             if ((absResult & ~significandMask) > 0) {
                 // The rounded result is normal; return it.
                 return @bitCast(absResult | quotientSign);
             }
         }
         // Flush denormals to zero.  In the future, it would be nice to add
         // code to round them correctly.
         return @bitCast(quotientSign);
     } else {
         const round = @intFromBool((residual << 1) >= bSignificand);
         // Clear the implicit bit
         var absResult = quotient & significandMask;
         // Insert the exponent
         absResult |= @as(Z, @intCast(writtenExponent)) << significandBits;
         // Round
         absResult +%= round;
         // Insert the sign and return
         return @bitCast(absResult | quotientSign);
     }
 }
 
 test {
     _ = @import("divtf3_test.zig");
 }