Average Error: 0.3 → 0.3
Time: 36.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\log z + \left(-t\right)\right) + \log \left(y + x\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log t, a - 0.5, \left(\log z + \left(-t\right)\right) + \log \left(y + x\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r2831143 = x;
        double r2831144 = y;
        double r2831145 = r2831143 + r2831144;
        double r2831146 = log(r2831145);
        double r2831147 = z;
        double r2831148 = log(r2831147);
        double r2831149 = r2831146 + r2831148;
        double r2831150 = t;
        double r2831151 = r2831149 - r2831150;
        double r2831152 = a;
        double r2831153 = 0.5;
        double r2831154 = r2831152 - r2831153;
        double r2831155 = log(r2831150);
        double r2831156 = r2831154 * r2831155;
        double r2831157 = r2831151 + r2831156;
        return r2831157;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r2831158 = t;
        double r2831159 = log(r2831158);
        double r2831160 = a;
        double r2831161 = 0.5;
        double r2831162 = r2831160 - r2831161;
        double r2831163 = z;
        double r2831164 = log(r2831163);
        double r2831165 = -r2831158;
        double r2831166 = r2831164 + r2831165;
        double r2831167 = y;
        double r2831168 = x;
        double r2831169 = r2831167 + r2831168;
        double r2831170 = log(r2831169);
        double r2831171 = r2831166 + r2831170;
        double r2831172 = fma(r2831159, r2831162, r2831171);
        return r2831172;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Initial program 0.3

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  2. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, a - 0.5, \log \left(y + x\right) + \left(\log z - t\right)\right)}\]

  3. Taylor expanded around inf 0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \log \left(y + x\right) + \color{blue}{\left(-\left(t + \log \left(\frac{1}{z}\right)\right)\right)}\right)\]

  4. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \log \left(y + x\right) + \color{blue}{\left(\left(-t\right) + \log z\right)}\right)\]

  5. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\log z + \left(-t\right)\right) + \log \left(y + x\right)\right)\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))