Average Error: 0.2 → 0.2
Time: 35.0s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]
\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(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r3534126 = x;
        double r3534127 = y;
        double r3534128 = r3534126 + r3534127;
        double r3534129 = log(r3534128);
        double r3534130 = z;
        double r3534131 = log(r3534130);
        double r3534132 = r3534129 + r3534131;
        double r3534133 = t;
        double r3534134 = r3534132 - r3534133;
        double r3534135 = a;
        double r3534136 = 0.5;
        double r3534137 = r3534135 - r3534136;
        double r3534138 = log(r3534133);
        double r3534139 = r3534137 * r3534138;
        double r3534140 = r3534134 + r3534139;
        return r3534140;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r3534141 = t;
        double r3534142 = log(r3534141);
        double r3534143 = a;
        double r3534144 = 0.5;
        double r3534145 = r3534143 - r3534144;
        double r3534146 = 2.0;
        double r3534147 = z;
        double r3534148 = cbrt(r3534147);
        double r3534149 = log(r3534148);
        double r3534150 = y;
        double r3534151 = x;
        double r3534152 = r3534150 + r3534151;
        double r3534153 = log(r3534152);
        double r3534154 = fma(r3534146, r3534149, r3534153);
        double r3534155 = r3534154 + r3534149;
        double r3534156 = r3534155 - r3534141;
        double r3534157 = fma(r3534142, r3534145, r3534156);
        return r3534157;
}

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 0.2

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

  2. Simplified0.2

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

  4. Applied add-cube-cbrt0.2

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(x + y\right) + \log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right) - t\right)\]

  5. Applied log-prod0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(x + y\right) + \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)}\right) - t\right)\]

  6. Applied associate-+r+0.2

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \color{blue}{\left(\left(\log \left(x + y\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(\sqrt[3]{z}\right)\right)} - t\right)\]

  7. Simplified0.2

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]

  8. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]

Reproduce

herbie shell --seed 2019174 +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))))