Fastest implementation of sine, cosine and square root in C++ (doesn't need to be much accurate)

Question

I am googling the question for past hour, but there are only points to Taylor Series or some sample code that is either too slow or does not compile at all. Well, most answe

Answer 1

So let me rephrase that, this idea comes from approximating the cosine & sine functions on an interval [-pi/4,+pi/4] with a bounded error using the Remez algorithm. Then using the range reduced float remainder and a LUT for the outputs cos & sine of the integer quotient, the approximation can be moved to any angular argument.

Its just unique and I thought it could be expanded on to make a more efficient algorithm in terms of a bounded error.

void sincos_fast(float x, float *pS, float *pC){
    float cosOff4LUT[] = { 0x1.000000p+00,  0x1.6A09E6p-01,  0x0.000000p+00, -0x1.6A09E6p-01, -0x1.000000p+00, -0x1.6A09E6p-01,  0x0.000000p+00,  0x1.6A09E6p-01 };

    int     m, ms, mc;
    float   xI, xR, xR2;
    float   c, s, cy, sy;

    // Cody & Waite's range reduction Algorithm, [-pi/4, pi/4]
    xI  = floorf(x * 0x1.45F306p+00 + 0.5);              // This is 4/pi.
    xR  = (x - xI * 0x1.920000p-01) - xI*0x1.FB5444p-13; // This is pi/4 in two parts per C&W.
    m   = (int) xI;
    xR2 = xR*xR;

    // Find cosine & sine index for angle offsets indices
    mc = (  m  ) & 0x7;     // two's complement permits upper modulus for negative numbers =P
    ms = (m + 6) & 0x7;     // phase correction for sine.

    // Find cosine & sine
    cy = cosOff4LUT[mc];     // Load angle offset neighborhood cosine value 
    sy = cosOff4LUT[ms];     // Load angle offset neighborhood sine value 

    c = 0xf.ff79fp-4 + xR2 * (-0x7.e58e9p-4);               // TOL = 1.2786e-4
    // c = 0xf.ffffdp-4 + xR2 * (-0x7.ffebep-4 + xR2 * 0xa.956a9p-8);  // TOL = 1.7882e-7

    s = xR * (0xf.ffbf7p-4 + xR2 * (-0x2.a41d0cp-4));   // TOL = 4.835251e-6
    // s = xR * (0xf.fffffp-4 + xR2 * (-0x2.aaa65cp-4 + xR2 * 0x2.1ea25p-8));  // TOL = 1.1841e-8

    *pC = c*cy - s*sy;      
    *pS = c*sy + s*cy;
}

float sqrt_fast(float x){
    union {float f; int i; } X, Y;
    float ScOff;
    uint8_t e;

    X.f = x;
    e = (X.i >> 23);           // f.SFPbits.e;

    if(x <= 0) return(0.0f);

    ScOff = ((e & 1) != 0) ? 1.0f : 0x1.6a09e6p0;  // NOTE: If exp=EVEN, b/c (exp-127) a (EVEN - ODD) := ODD; but a (ODD - ODD) := EVEN!!

    e = ((e + 127) >> 1);                            // NOTE: If exp=ODD,  b/c (exp-127) then flr((exp-127)/2)
    X.i = (X.i & ((1uL << 23) - 1)) | (0x7F << 23);  // Mask mantissa, force exponent to zero.
    Y.i = (((uint32_t) e) << 23);

    // Error grows with square root of the exponent. Unfortunately no work around like inverse square root... :(
    // Y.f *= ScOff * (0x9.5f61ap-4 + X.f*(0x6.a09e68p-4));        // Error = +-1.78e-2 * 2^(flr(log2(x)/2))
    // Y.f *= ScOff * (0x7.2181d8p-4 + X.f*(0xa.05406p-4 + X.f*(-0x1.23a14cp-4)));      // Error = +-7.64e-5 * 2^(flr(log2(x)/2))
    // Y.f *= ScOff * (0x5.f10e7p-4 + X.f*(0xc.8f2p-4 +X.f*(-0x2.e41a4cp-4 + X.f*(0x6.441e6p-8))));     // Error =  8.21e-5 * 2^(flr(log2(x)/2))
    // Y.f *= ScOff * (0x5.32eb88p-4 + X.f*(0xe.abbf5p-4 + X.f*(-0x5.18ee2p-4 + X.f*(0x1.655efp-4 + X.f*(-0x2.b11518p-8)))));   // Error = +-9.92e-6 * 2^(flr(log2(x)/2))
    // Y.f *= ScOff * (0x4.adde5p-4 + X.f*(0x1.08448cp0 + X.f*(-0x7.ae1248p-4 + X.f*(0x3.2cf7a8p-4 + X.f*(-0xc.5c1e2p-8 + X.f*(0x1.4b6dp-8))))));   // Error = +-1.38e-6 * 2^(flr(log2(x)/2))
    // Y.f *= ScOff * (0x4.4a17fp-4 + X.f*(0x1.22d44p0 + X.f*(-0xa.972e8p-4 + X.f*(0x5.dd53fp-4 + X.f*(-0x2.273c08p-4 + X.f*(0x7.466cb8p-8 + X.f*(-0xa.ac00ep-12)))))));    // Error = +-2.9e-7 * 2^(flr(log2(x)/2))
    Y.f *= ScOff * (0x3.fbb3e8p-4 + X.f*(0x1.3b2a3cp0 + X.f*(-0xd.cbb39p-4 + X.f*(0x9.9444ep-4 + X.f*(-0x4.b5ea38p-4 + X.f*(0x1.802f9ep-4 + X.f*(-0x4.6f0adp-8 + X.f*(0x5.c24a28p-12 ))))))));   // Error = +-2.7e-6 * 2^(flr(log2(x)/2))

    return(Y.f);
}

The longer expressions are longer, slower, but more precise. Polynomials are written per Horner's rule.

Answer 2

An approximation for the sine function that preserves the derivatives at multiples of 90 degrees is given by this formula. The derivation is similar to Bhaskara I's sine approximation formula, but the constraints are to set the values and derivatives at 0, 90, and 180 degrees to that of the sine function. You can use this if you need the function to be smooth everywhere.

#define PI 3.141592653589793

double fast_sin(double x) {
    x /= 2 * PI;
    x -= (int) x;

    if (x <= 0.5) {
        double t = 2 * x * (2 * x - 1);
        return (PI * t) / ((PI - 4) * t - 1);
    }
    else {
        double t = 2 * (1 - x) * (1 - 2 * x);
        return -(PI * t) / ((PI - 4) * t - 1);
    }
}

double fast_cos(double x) {
    return fast_sin(x + 0.5 * PI);
}

As for its speed, it at least outperforms the std::sin() function by an average of 0.3 microseconds per call. And the maximum absolute error is 0.0051.

Answer 3

Based on the idea of http://forum.devmaster.net/t/fast-and-accurate-sine-cosine/9648 and some manual rewriting to improve the performance in a micro benchmark I ended up with the following cosine implementation which is used in a HPC physics simulation that is bottlenecked by repeated cos calls on a large number space. It's accurate enough and much faster than a lookup table, most notably no division is required.

template<typename T>
inline T cos(T x) noexcept
{
    constexpr T tp = 1./(2.*M_PI);
    x *= tp;
    x -= T(.25) + std::floor(x + T(.25));
    x *= T(16.) * (std::abs(x) - T(.5));
    #if EXTRA_PRECISION
    x += T(.225) * x * (std::abs(x) - T(1.));
    #endif
    return x;
}

The Intel compiler at least is also smart enough in vectorizing this function when used in a loop.

If EXTRA_PRECISION is defined, the maximum error is about 0.00109 for the range -π to π, assuming T is double as it's usually defined in most C++ implementations. Otherwise, the maximum error is about 0.056 for the same range.