Average Error: 1.8 → 1.1
Time: 40.8s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}} \cdot \left(x \cdot \left(\frac{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}}\right)\right)\]
\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(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}} \cdot \left(x \cdot \left(\frac{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}}\right)\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r3102658 = x;
        double r3102659 = y;
        double r3102660 = z;
        double r3102661 = log(r3102660);
        double r3102662 = r3102659 * r3102661;
        double r3102663 = t;
        double r3102664 = 1.0;
        double r3102665 = r3102663 - r3102664;
        double r3102666 = a;
        double r3102667 = log(r3102666);
        double r3102668 = r3102665 * r3102667;
        double r3102669 = r3102662 + r3102668;
        double r3102670 = b;
        double r3102671 = r3102669 - r3102670;
        double r3102672 = exp(r3102671);
        double r3102673 = r3102658 * r3102672;
        double r3102674 = r3102673 / r3102659;
        return r3102674;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r3102675 = t;
        double r3102676 = 1.0;
        double r3102677 = r3102675 - r3102676;
        double r3102678 = a;
        double r3102679 = log(r3102678);
        double r3102680 = y;
        double r3102681 = z;
        double r3102682 = log(r3102681);
        double r3102683 = r3102680 * r3102682;
        double r3102684 = b;
        double r3102685 = r3102683 - r3102684;
        double r3102686 = fma(r3102677, r3102679, r3102685);
        double r3102687 = exp(r3102686);
        double r3102688 = cbrt(r3102687);
        double r3102689 = cbrt(r3102680);
        double r3102690 = r3102688 / r3102689;
        double r3102691 = x;
        double r3102692 = r3102690 * r3102690;
        double r3102693 = r3102691 * r3102692;
        double r3102694 = r3102690 * r3102693;
        return r3102694;
}

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 1.8

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

  3. Applied *-un-lft-identity1.8

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

  4. Applied times-frac2.2

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

  5. Simplified2.2

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

  6. Simplified2.2

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

  8. Applied add-cube-cbrt2.2

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

  9. Applied add-cube-cbrt2.2

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

  10. Applied times-frac2.2

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

  11. Applied associate-*r*1.1

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

  12. Simplified1.1

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

  13. Final simplification1.1

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

Reproduce

herbie shell --seed 2019170 +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))