MathF.Sin.cs

// This is a port of the amd/win-libm implementation provided in assembly here: https://github.com/amd/win-libm/blob/master/sinf.asm
// The original source is Copyright (c) 2002-2019 Advanced Micro Devices, Inc. and provided under the MIT License.
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this Software and associated documentaon files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
// 
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
// 
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE.

const double PiOverTwo = 1.5707963267948966;
const double PiOverTwoPartOne = 1.5707963267341256;
const double PiOverTwoPartOneTail = 6.077100506506192E-11;
const double PiOverTwoPartTwo = 6.077100506303966E-11;
const double PiOverTwoPartTwoTail = 2.0222662487959506E-21;
const double PiOverFour = 0.7853981633974483;
const double TwoOverPi = 0.6366197723675814;
const double TwoPowNegSeven = 0.0078125;
const double TwoPowNegThirteen = 0.0001220703125;

const double C0 = -1.0 / 2.0;       // 1 / 2!
const double C1 = +1.0 / 24.0;      // 1 / 4!
const double C2 = -1.0 / 720.0;     // 1 / 6!
const double C3 = +1.0 / 40320.0;   // 1 / 8!
const double C4 = -1.0 / 3628800.0; // 1 / 10!

const double S1 = -1.0 / 6.0;       // 1 / 3!
const double S2 = +1.0 / 120.0;     // 1 / 5!
const double S3 = -1.0 / 5040.0;    // 1 / 7!
const double S4 = +1.0 / 362880.0;  // 1 / 9!

private static readonly long[] PiBits = new long[]
{
    0,
    5215,
    13000023176,
    11362338026,
    67174558139,
    34819822259,
    10612056195,
    67816420731,
    57840157550,
    19558516809,
    50025467026,
    25186875954,
    18152700886
};

public static float Sin(float x)
{
    double result = x;

    if (float.IsFinite(x))
    {
        double ax = Math.Abs(x);

        if (ax <= PiOverFour)
        {
            if (ax >= TwoPowNegSeven)
            {
                result = SinTaylorSeriesFourIterations(x);
            }
            else if (ax >= TwoPowNegThirteen)
            {
                result = SinTaylorSeriesOneIteration(x);
            }
            else
            {
                result = x;
            }
        }
        else
        {
            int wasNegative = 0;

            if (float.IsNegative(x))
            {
                x = -x;
                wasNegative = 1;
            }

            int region;

            if (x < 16000000.0)
            {
                // Reduce x to be in the range of -(PI / 4) to (PI / 4), inclusive

                // This is done by subtracting multiples of (PI / 2). Double-precision
                // isn't quite accurate enough and introduces some error, but we account
                // for that using a tail value that helps account for this.

                long axExp = BitConverter.DoubleToInt64Bits(ax) >> 52;

                region = (int)(x * TwoOverPi + 0.5);
                double piOverTwoCount = region;

                double rHead = x - (piOverTwoCount * PiOverTwoPartOne);
                double rTail = (piOverTwoCount * PiOverTwoPartOneTail);

                double r = rHead - rTail;
                long rExp = (BitConverter.DoubleToInt64Bits(r) << 1) >> 53;

                if ((axExp - rExp) > 15)
                {
                    // The remainder is pretty small compared with x, which implies that x is
                    // near a multiple of (PI / 2). That is, x matches the multiple to at least
                    // 15 bits and so we perform an additional fixup to account for any error

                    r = rHead;

                    rTail = (piOverTwoCount * PiOverTwoPartTwo);
                    rHead = r - rTail;
                    rTail = (piOverTwoCount * PiOverTwoPartTwoTail) - ((r - rHead) - rTail);

                    r = rHead - rTail;
                }

                if (rExp >= 0x3F2)      // r >= 2^-13
                {
                    if ((region & 1) == 0)  // region 0 or 2
                    {
                        result = SinTaylorSeriesFourIterations(r);
                    }
                    else                    // region 1 or 3
                    {
                        result = CosTaylorSeriesFourIterations(r);
                    }
                }
                else if (rExp > 0x3DE)  // r > 1.1641532182693481E-10
                {
                    if ((region & 1) == 0)  // region 0 or 2
                    {
                        result = SinTaylorSeriesOneIteration(r);
                    }
                    else                    // region 1 or 3
                    {
                        result = CosTaylorSeriesOneIteration(r);

                    }
                }
                else
                {
                    if ((region & 1) == 0)  // region 0 or 2
                    {
                        result = r;
                    }
                    else                    // region 1 or 3
                    {
                        result = 1;
                    }
                }
            }
            else
            {
                double r = ReduceForLargeInput(x, out region);

                if ((region & 1) == 0)  // region 0 or 2
                {
                    result = SinTaylorSeriesFourIterations(r);
                }
                else                    // region 1 or 3
                {
                    result = CosTaylorSeriesFourIterations(r);
                }
            }

            region >>= 1;

            int tmp1 = region & wasNegative;

            region = ~region;
            wasNegative = ~wasNegative;

            int tmp2 = region & wasNegative;

            if (((tmp1 | tmp2) & 1) == 0)
            {
                // If the original region was 0/1 and arg is negative, then we negate the result.
                // -or-
                // If the original region was 2/3 and arg is positive, then we negate the result.

                result = -result;
            }
        }
    }

    return (float)result;

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    static double CosTaylorSeriesOneIteration(double x1)
    {
        // 1 - (x^2 / 2!)
        double x2 = x1 * x1;
        return 1.0 + (x2 * C1);
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    static double CosTaylorSeriesFourIterations(double x1)
    {
        // 1 - (x^2 / 2!) + (x^4 / 4!) - (x^6 / 6!) + (x^8 / 8!) - (x^10 / 10!)

        double x2 = x1 * x1;
        double x4 = x2 * x2;

        return 1.0 + (x2 * C0) + (x4 * ((C1 + (x2 * C2)) + (x4 * (C3 + (x2 * C4)))));
    }

    static unsafe double ReduceForLargeInput(double x, out int region)
    {
        Debug.Assert(!double.IsNegative(x));

        // This method simulates multi-precision floating-point
        // arithmetic and is accurate for all 1 <= x < infinity

        const int BitsPerIteration = 36;
        long ux = BitConverter.DoubleToInt64Bits(x);

        int xExp = (int)(((ux & 0x7FF0000000000000) >> 52) - 1023);
        ux = ((ux & 0x000FFFFFFFFFFFFF) | 0x0010000000000000) >> 29;

        // Now ux is the mantissa bit pattern of x as a long integer
        long mask = 1;
        mask = (mask << BitsPerIteration) - 1;

        // Set first and last to the positions of the first and last chunks of (2 / PI) that we need
        int first = xExp / BitsPerIteration;
        int resultExp = xExp - (first * BitsPerIteration);

        // 120 is the theoretical maximum number of bits (actually
        // 115 for IEEE single precision) that we need to extract
        // from the middle of (2 / PI) to compute the reduced argument
        // accurately enough for our purposes

        int last = first + (120 / BitsPerIteration);

        // Unroll the loop. This is only correct because we know that bitsper is fixed as 36.

        long* result = stackalloc long[10];
        long u, carry;

        result[4] = 0;
        u = PiBits[last] * ux;

        result[3] = u & mask;
        carry = u >> BitsPerIteration;
        u = PiBits[last - 1] * ux + carry;

        result[2] = u & mask;
        carry = u >> BitsPerIteration;
        u = PiBits[last - 2] * ux + carry;

        result[1] = u & mask;
        carry = u >> BitsPerIteration;
        u = PiBits[first] * ux + carry;

        result[0] = u & mask;

        // Reconstruct the result
        int ltb = (int)((((result[0] << BitsPerIteration) | result[1]) >> (BitsPerIteration - 1 - resultExp)) & 7);

        long mantissa;
        long nextBits;

        // determ says whether the fractional part is >= 0.5
        bool determ = (ltb & 1) != 0;

        int i = 1;

        if (determ)
        {
            // The mantissa is >= 0.5. We want to subtract it from 1.0 by negating all the bits
            region = ((ltb >> 1) + 1) & 3;

            mantissa = 1;
            mantissa = ~(result[1]) & ((mantissa << (BitsPerIteration - resultExp)) - 1);

            while (mantissa < 0x0000000000010000)
            {
                i++;
                mantissa = (mantissa << BitsPerIteration) | (~(result[i]) & mask);
            }

            nextBits = (~(result[i + 1]) & mask);
        }
        else
        {
            region = (ltb >> 1);

            mantissa = 1;
            mantissa = result[1] & ((mantissa << (BitsPerIteration - resultExp)) - 1);

            while (mantissa < 0x0000000000010000)
            {
                i++;
                mantissa = (mantissa << BitsPerIteration) | result[i];
            }

            nextBits = result[i + 1];
        }

        // Normalize the mantissa.
        // The shift value 6 here, determined by trial and error, seems to give optimal speed.

        int bc = 0;

        while (mantissa < 0x0000400000000000)
        {
            bc += 6;
            mantissa <<= 6;
        }

        while (mantissa < 0x0010000000000000)
        {
            bc++;
            mantissa <<= 1;
        }

        mantissa |= nextBits >> (BitsPerIteration - bc);

        int rExp = 52 + resultExp - bc - i * BitsPerIteration;

        // Put the result exponent rexp onto the mantissa pattern
        u = (rExp + 1023L) << 52;
        ux = (mantissa & 0x000FFFFFFFFFFFFF) | u;

        if (determ)
        {
            // If we negated the mantissa we negate x too
            ux |= unchecked((long)(0x8000000000000000));
        }

        x = BitConverter.Int64BitsToDouble(ux);

        // x is a double precision version of the fractional part of
        // (x * (2 / PI)). Multiply x by (PI / 2) in double precision
        // to get the reduced result.

        return x * PiOverTwo;
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    static double SinTaylorSeriesOneIteration(double x1)
    {
        // x - (x^3 / 3!)

        double x2 = x1 * x1;
        double x3 = x2 * x1;

        return x1 + (x3 * S1);
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    static double SinTaylorSeriesFourIterations(double x1)
    {
        // x - (x^3 / 3!) + (x^5 / 5!) - (x^7 / 7!) + (x^9 / 9!)

        double x2 = x1 * x1;
        double x3 = x2 * x1;
        double x4 = x2 * x2;

        return x1 + ((S1 + (x2 * S2) + (x4 * (S3 + (x2 * S4)))) * x3);
    }
}

Perf.md

Test Env	Method	Mean	Error	StdDev	Median	Min	Max	Gen 0	Gen 1	Gen 2	Allocated
x64 Windows Native	Sin	22.82 us	0.015 us	0.014 us	22.82 us	22.80 us	22.85 us	-	-	-	-
x86 Windows Native	Sin	60.48 us	0.087 us	0.068 us	60.48 us	60.38 us	60.60 us	-	-	-	-
x64 Windows Managed	Sin	21.26 us	0.305 us	0.285 us	21.27 us	20.85 us	21.65 us	-	-	-	-
x86 Windows Managed	Sin	38.10 us	0.390 us	0.346 us	38.06 us	37.66 us	38.71 us	-	-	-	-

@tannergooding should we add this benchmarks to the performance repo?

@adamsitnik, these numbers are from the perf repo benchmark. The only code above is the managed implementation

@tannergooding thanks for the explanation! BTW these numbers are very impressive!