Average Error: 2.0 → 2.0
Time: 41.9s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{1}{\left(\sqrt[3]{\frac{y}{x \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)\right)}} \cdot \sqrt[3]{\frac{y}{x \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)\right)}}\right) \cdot \sqrt[3]{\frac{y}{x \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)\right)}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}
↓
\frac{1}{\left(\sqrt[3]{\frac{y}{x \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)\right)}} \cdot \sqrt[3]{\frac{y}{x \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)\right)}}\right) \cdot \sqrt[3]{\frac{y}{x \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)\right)}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r1237065 = x;
        double r1237066 = y;
        double r1237067 = z;
        double r1237068 = log(r1237067);
        double r1237069 = r1237066 * r1237068;
        double r1237070 = t;
        double r1237071 = 1.0;
        double r1237072 = r1237070 - r1237071;
        double r1237073 = a;
        double r1237074 = log(r1237073);
        double r1237075 = r1237072 * r1237074;
        double r1237076 = r1237069 + r1237075;
        double r1237077 = b;
        double r1237078 = r1237076 - r1237077;
        double r1237079 = exp(r1237078);
        double r1237080 = r1237065 * r1237079;
        double r1237081 = r1237080 / r1237066;
        return r1237081;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r1237082 = 1.0;
        double r1237083 = y;
        double r1237084 = x;
        double r1237085 = z;
        double r1237086 = log(r1237085);
        double r1237087 = a;
        double r1237088 = log(r1237087);
        double r1237089 = t;
        double r1237090 = 1.0;
        double r1237091 = r1237089 - r1237090;
        double r1237092 = r1237088 * r1237091;
        double r1237093 = fma(r1237083, r1237086, r1237092);
        double r1237094 = b;
        double r1237095 = r1237093 - r1237094;
        double r1237096 = exp(r1237095);
        double r1237097 = cbrt(r1237096);
        double r1237098 = r1237097 * r1237097;
        double r1237099 = r1237097 * r1237098;
        double r1237100 = cbrt(r1237099);
        double r1237101 = r1237100 * r1237097;
        double r1237102 = r1237097 * r1237101;
        double r1237103 = r1237084 * r1237102;
        double r1237104 = r1237083 / r1237103;
        double r1237105 = cbrt(r1237104);
        double r1237106 = r1237105 * r1237105;
        double r1237107 = r1237106 * r1237105;
        double r1237108 = r1237082 / r1237107;
        return r1237108;
}

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

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
  2. Using strategy rm

  3. Applied clear-num2.0

    \[\leadsto \color{blue}{\frac{1}{\frac{y}{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}\]

  4. Simplified2.0

    \[\leadsto \frac{1}{\color{blue}{\frac{y}{x \cdot e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt1.9

    \[\leadsto \frac{1}{\frac{y}{x \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right)}}}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt2.0

    \[\leadsto \frac{1}{\frac{y}{x \cdot \left(\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right)}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt2.0

    \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{\frac{y}{x \cdot \left(\left(\sqrt[3]{\left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right)}} \cdot \sqrt[3]{\frac{y}{x \cdot \left(\left(\sqrt[3]{\left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right)}}\right) \cdot \sqrt[3]{\frac{y}{x \cdot \left(\left(\sqrt[3]{\left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\left(t - 1.0\right) \cdot \log a\right)\right) - b}}\right)}}}}\]

  11. Final simplification2.0

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{y}{x \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)\right)}} \cdot \sqrt[3]{\frac{y}{x \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)\right)}}\right) \cdot \sqrt[3]{\frac{y}{x \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \left(\log z\right), \left(\log a \cdot \left(t - 1.0\right)\right)\right) - b}}\right)\right)}}}\]

Reproduce

herbie shell --seed 2019128 +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))