Average Error: 0.1 → 0.6
Time: 11.1s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left(\log y, x, \log t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, x, \log t\right)}, \sqrt[3]{\mathsf{fma}\left(\log y, x, \log t\right)}, -\left(y + z\right)\right)\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left(\log y, x, \log t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, x, \log t\right)}, \sqrt[3]{\mathsf{fma}\left(\log y, x, \log t\right)}, -\left(y + z\right)\right)
double f(double x, double y, double z, double t) {
        double r117907 = x;
        double r117908 = y;
        double r117909 = log(r117908);
        double r117910 = r117907 * r117909;
        double r117911 = r117910 - r117908;
        double r117912 = z;
        double r117913 = r117911 - r117912;
        double r117914 = t;
        double r117915 = log(r117914);
        double r117916 = r117913 + r117915;
        return r117916;
}
↓
double f(double x, double y, double z, double t) {
        double r117917 = y;
        double r117918 = log(r117917);
        double r117919 = x;
        double r117920 = t;
        double r117921 = log(r117920);
        double r117922 = fma(r117918, r117919, r117921);
        double r117923 = cbrt(r117922);
        double r117924 = r117923 * r117923;
        double r117925 = z;
        double r117926 = r117917 + r117925;
        double r117927 = -r117926;
        double r117928 = fma(r117924, r117923, r117927);
        return r117928;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  4. Applied add-cube-cbrt0.6

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

  5. Applied fma-neg0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))