Average Error: 0.1 → 0.1
Time: 6.3s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\mathsf{fma}\left(2, \log \left(\sqrt[3]{t}\right), \left(\log y \cdot x - y\right) - z\right) + \log \left(\sqrt[3]{t}\right)\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\mathsf{fma}\left(2, \log \left(\sqrt[3]{t}\right), \left(\log y \cdot x - y\right) - z\right) + \log \left(\sqrt[3]{t}\right)
double f(double x, double y, double z, double t) {
        double r97872 = x;
        double r97873 = y;
        double r97874 = log(r97873);
        double r97875 = r97872 * r97874;
        double r97876 = r97875 - r97873;
        double r97877 = z;
        double r97878 = r97876 - r97877;
        double r97879 = t;
        double r97880 = log(r97879);
        double r97881 = r97878 + r97880;
        return r97881;
}
↓
double f(double x, double y, double z, double t) {
        double r97882 = 2.0;
        double r97883 = t;
        double r97884 = cbrt(r97883);
        double r97885 = log(r97884);
        double r97886 = y;
        double r97887 = log(r97886);
        double r97888 = x;
        double r97889 = r97887 * r97888;
        double r97890 = r97889 - r97886;
        double r97891 = z;
        double r97892 = r97890 - r97891;
        double r97893 = fma(r97882, r97885, r97892);
        double r97894 = r97893 + r97885;
        return r97894;
}

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. Using strategy rm

  3. Applied add-cube-cbrt0.1

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

  4. Applied log-prod0.1

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

  5. Applied associate-+r+0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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)))