Average Error: 0.1 → 0.6
Time: 37.9s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\left(\left(\sqrt[3]{\mathsf{fma}\left(\log y, x, \log t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, x, \log t\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\log y, x, \log t\right)} - z\right) - y\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\left(\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)} - z\right) - y
double f(double x, double y, double z, double t) {
        double r8019974 = x;
        double r8019975 = y;
        double r8019976 = log(r8019975);
        double r8019977 = r8019974 * r8019976;
        double r8019978 = r8019977 - r8019975;
        double r8019979 = z;
        double r8019980 = r8019978 - r8019979;
        double r8019981 = t;
        double r8019982 = log(r8019981);
        double r8019983 = r8019980 + r8019982;
        return r8019983;
}
↓
double f(double x, double y, double z, double t) {
        double r8019984 = y;
        double r8019985 = log(r8019984);
        double r8019986 = x;
        double r8019987 = t;
        double r8019988 = log(r8019987);
        double r8019989 = fma(r8019985, r8019986, r8019988);
        double r8019990 = cbrt(r8019989);
        double r8019991 = r8019990 * r8019990;
        double r8019992 = r8019991 * r8019990;
        double r8019993 = z;
        double r8019994 = r8019992 - r8019993;
        double r8019995 = r8019994 - r8019984;
        return r8019995;
}

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

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

  4. Applied add-cube-cbrt0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  (+ (- (- (* x (log y)) y) z) (log t)))