Average Error: 1.9 → 1.8
Time: 1.1m
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\frac{\frac{e^{\mathsf{fma}\left(\log z, y, \log a \cdot \left(t - 1\right) - b\right)}}{\sqrt[3]{y}} \cdot x}{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\frac{\frac{e^{\mathsf{fma}\left(\log z, y, \log a \cdot \left(t - 1\right) - b\right)}}{\sqrt[3]{y}} \cdot x}{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}}
double f(double x, double y, double z, double t, double a, double b) {
        double r2998918 = x;
        double r2998919 = y;
        double r2998920 = z;
        double r2998921 = log(r2998920);
        double r2998922 = r2998919 * r2998921;
        double r2998923 = t;
        double r2998924 = 1.0;
        double r2998925 = r2998923 - r2998924;
        double r2998926 = a;
        double r2998927 = log(r2998926);
        double r2998928 = r2998925 * r2998927;
        double r2998929 = r2998922 + r2998928;
        double r2998930 = b;
        double r2998931 = r2998929 - r2998930;
        double r2998932 = exp(r2998931);
        double r2998933 = r2998918 * r2998932;
        double r2998934 = r2998933 / r2998919;
        return r2998934;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r2998935 = z;
        double r2998936 = log(r2998935);
        double r2998937 = y;
        double r2998938 = a;
        double r2998939 = log(r2998938);
        double r2998940 = t;
        double r2998941 = 1.0;
        double r2998942 = r2998940 - r2998941;
        double r2998943 = r2998939 * r2998942;
        double r2998944 = b;
        double r2998945 = r2998943 - r2998944;
        double r2998946 = fma(r2998936, r2998937, r2998945);
        double r2998947 = exp(r2998946);
        double r2998948 = cbrt(r2998937);
        double r2998949 = r2998947 / r2998948;
        double r2998950 = x;
        double r2998951 = r2998949 * r2998950;
        double r2998952 = cbrt(r2998948);
        double r2998953 = r2998952 * r2998952;
        double r2998954 = r2998952 * r2998953;
        double r2998955 = r2998951 / r2998954;
        double r2998956 = cbrt(r2998954);
        double r2998957 = r2998953 * r2998956;
        double r2998958 = r2998955 / r2998957;
        return r2998958;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 1.9

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.9

    \[\leadsto \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

  4. Applied associate-/r*1.9

    \[\leadsto \color{blue}{\frac{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt[3]{y}}}\]

  5. Simplified1.7

    \[\leadsto \frac{\color{blue}{\frac{x \cdot \frac{e^{\mathsf{fma}\left(\log z, y, \left(t - 1\right) \cdot \log a - b\right)}}{\sqrt[3]{y}}}{\sqrt[3]{y}}}}{\sqrt[3]{y}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt1.7

    \[\leadsto \frac{\frac{x \cdot \frac{e^{\mathsf{fma}\left(\log z, y, \left(t - 1\right) \cdot \log a - b\right)}}{\sqrt[3]{y}}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}}}{\sqrt[3]{y}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt1.7

    \[\leadsto \frac{\frac{x \cdot \frac{e^{\mathsf{fma}\left(\log z, y, \left(t - 1\right) \cdot \log a - b\right)}}{\sqrt[3]{y}}}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt1.8

    \[\leadsto \frac{\frac{x \cdot \frac{e^{\mathsf{fma}\left(\log z, y, \left(t - 1\right) \cdot \log a - b\right)}}{\sqrt[3]{y}}}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}}}\]

  12. Final simplification1.8

    \[\leadsto \frac{\frac{\frac{e^{\mathsf{fma}\left(\log z, y, \log a \cdot \left(t - 1\right) - b\right)}}{\sqrt[3]{y}} \cdot x}{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))