Average Error: 2.0 → 1.1
Time: 43.1s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\sqrt[3]{e^{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\frac{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right) - b}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\sqrt[3]{e^{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\frac{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right) - b}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r4533997 = x;
        double r4533998 = y;
        double r4533999 = z;
        double r4534000 = log(r4533999);
        double r4534001 = r4533998 * r4534000;
        double r4534002 = t;
        double r4534003 = 1.0;
        double r4534004 = r4534002 - r4534003;
        double r4534005 = a;
        double r4534006 = log(r4534005);
        double r4534007 = r4534004 * r4534006;
        double r4534008 = r4534001 + r4534007;
        double r4534009 = b;
        double r4534010 = r4534008 - r4534009;
        double r4534011 = exp(r4534010);
        double r4534012 = r4533997 * r4534011;
        double r4534013 = r4534012 / r4533998;
        return r4534013;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r4534014 = y;
        double r4534015 = z;
        double r4534016 = log(r4534015);
        double r4534017 = t;
        double r4534018 = 1.0;
        double r4534019 = r4534017 - r4534018;
        double r4534020 = a;
        double r4534021 = log(r4534020);
        double r4534022 = r4534019 * r4534021;
        double r4534023 = fma(r4534014, r4534016, r4534022);
        double r4534024 = b;
        double r4534025 = r4534023 - r4534024;
        double r4534026 = exp(r4534025);
        double r4534027 = cbrt(r4534026);
        double r4534028 = r4534027 * r4534027;
        double r4534029 = cbrt(r4534014);
        double r4534030 = r4534029 * r4534029;
        double r4534031 = r4534028 / r4534030;
        double r4534032 = x;
        double r4534033 = cbrt(r4534029);
        double r4534034 = r4534033 * r4534033;
        double r4534035 = r4534034 * r4534033;
        double r4534036 = r4534035 / r4534027;
        double r4534037 = r4534032 / r4534036;
        double r4534038 = r4534031 * r4534037;
        return r4534038;
}

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 2.0

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

  2. Simplified2.0

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

  4. Applied associate-/l*1.9

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

  6. Applied add-cube-cbrt1.9

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

  7. Applied add-cube-cbrt1.9

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

  8. Applied times-frac1.9

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

  9. Applied *-un-lft-identity1.9

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

  10. Applied times-frac1.1

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

  11. Simplified1.1

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

  13. Applied add-cube-cbrt1.1

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

  14. Final simplification1.1

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

Reproduce

herbie shell --seed 2019174 +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))