Average Error: 2.1 → 1.4
Time: 34.3s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x \cdot \frac{{a}^{\left(-1\right)}}{e^{\left(\sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)} \cdot \sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)}\right) \cdot \sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)}}}}{y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{x \cdot \frac{{a}^{\left(-1\right)}}{e^{\left(\sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)} \cdot \sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)}\right) \cdot \sqrt[3]{b - \mathsf{fma}\left(y, \log z, t \cdot \log a\right)}}}}{y}
double f(double x, double y, double z, double t, double a, double b) {
        double r53781 = x;
        double r53782 = y;
        double r53783 = z;
        double r53784 = log(r53783);
        double r53785 = r53782 * r53784;
        double r53786 = t;
        double r53787 = 1.0;
        double r53788 = r53786 - r53787;
        double r53789 = a;
        double r53790 = log(r53789);
        double r53791 = r53788 * r53790;
        double r53792 = r53785 + r53791;
        double r53793 = b;
        double r53794 = r53792 - r53793;
        double r53795 = exp(r53794);
        double r53796 = r53781 * r53795;
        double r53797 = r53796 / r53782;
        return r53797;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r53798 = x;
        double r53799 = a;
        double r53800 = 1.0;
        double r53801 = -r53800;
        double r53802 = pow(r53799, r53801);
        double r53803 = b;
        double r53804 = y;
        double r53805 = z;
        double r53806 = log(r53805);
        double r53807 = t;
        double r53808 = log(r53799);
        double r53809 = r53807 * r53808;
        double r53810 = fma(r53804, r53806, r53809);
        double r53811 = r53803 - r53810;
        double r53812 = cbrt(r53811);
        double r53813 = r53812 * r53812;
        double r53814 = r53813 * r53812;
        double r53815 = exp(r53814);
        double r53816 = r53802 / r53815;
        double r53817 = r53798 * r53816;
        double r53818 = r53817 / r53804;
        return r53818;
}

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 2.1

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

  2. Taylor expanded around inf 2.1

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

  3. Simplified1.4

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

  5. Applied add-cube-cbrt1.4

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

  6. Simplified1.4

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

  7. Simplified1.4

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

  8. Final simplification1.4

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

Reproduce

herbie shell --seed 2019194 +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))