Average Error: 9.6 → 0.5
Time: 23.3s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \left(\sqrt[3]{\log y} \cdot x\right) \cdot \sqrt[3]{\sqrt[3]{\left(\log y \cdot \left(\log y \cdot \log y\right)\right) \cdot \left(\log y \cdot \left(\log y \cdot \log y\right)\right)}}\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \left(\sqrt[3]{\log y} \cdot x\right) \cdot \sqrt[3]{\sqrt[3]{\left(\log y \cdot \left(\log y \cdot \log y\right)\right) \cdot \left(\log y \cdot \left(\log y \cdot \log y\right)\right)}}\right) - t
double f(double x, double y, double z, double t) {
        double r15104092 = x;
        double r15104093 = y;
        double r15104094 = log(r15104093);
        double r15104095 = r15104092 * r15104094;
        double r15104096 = z;
        double r15104097 = 1.0;
        double r15104098 = r15104097 - r15104093;
        double r15104099 = log(r15104098);
        double r15104100 = r15104096 * r15104099;
        double r15104101 = r15104095 + r15104100;
        double r15104102 = t;
        double r15104103 = r15104101 - r15104102;
        return r15104103;
}
↓
double f(double x, double y, double z, double t) {
        double r15104104 = z;
        double r15104105 = 1.0;
        double r15104106 = log(r15104105);
        double r15104107 = y;
        double r15104108 = 0.5;
        double r15104109 = r15104105 / r15104107;
        double r15104110 = r15104109 * r15104109;
        double r15104111 = r15104108 / r15104110;
        double r15104112 = fma(r15104105, r15104107, r15104111);
        double r15104113 = r15104106 - r15104112;
        double r15104114 = log(r15104107);
        double r15104115 = cbrt(r15104114);
        double r15104116 = x;
        double r15104117 = r15104115 * r15104116;
        double r15104118 = r15104114 * r15104114;
        double r15104119 = r15104114 * r15104118;
        double r15104120 = r15104119 * r15104119;
        double r15104121 = cbrt(r15104120);
        double r15104122 = cbrt(r15104121);
        double r15104123 = r15104117 * r15104122;
        double r15104124 = fma(r15104104, r15104113, r15104123);
        double r15104125 = t;
        double r15104126 = r15104124 - r15104125;
        return r15104126;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.6
Target0.2
Herbie0.5
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.6

    \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

  2. Simplified9.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, \log \left(1 - y\right), \log y \cdot x\right) - t}\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \mathsf{fma}\left(z, \color{blue}{\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)}, \log y \cdot x\right) - t\]

  4. Simplified0.3

    \[\leadsto \mathsf{fma}\left(z, \color{blue}{\log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right)}, \log y \cdot x\right) - t\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.7

    \[\leadsto \mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \color{blue}{\left(\left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right) \cdot \sqrt[3]{\log y}\right)} \cdot x\right) - t\]

  7. Applied associate-*l*0.8

    \[\leadsto \mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \color{blue}{\left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right) \cdot \left(\sqrt[3]{\log y} \cdot x\right)}\right) - t\]
  8. Using strategy rm

  9. Applied cbrt-unprod0.5

    \[\leadsto \mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \color{blue}{\sqrt[3]{\log y \cdot \log y}} \cdot \left(\sqrt[3]{\log y} \cdot x\right)\right) - t\]
  10. Using strategy rm

  11. Applied add-cbrt-cube0.6

    \[\leadsto \mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \sqrt[3]{\log y \cdot \color{blue}{\sqrt[3]{\left(\log y \cdot \log y\right) \cdot \log y}}} \cdot \left(\sqrt[3]{\log y} \cdot x\right)\right) - t\]

  12. Applied add-cbrt-cube0.6

    \[\leadsto \mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \sqrt[3]{\color{blue}{\sqrt[3]{\left(\log y \cdot \log y\right) \cdot \log y}} \cdot \sqrt[3]{\left(\log y \cdot \log y\right) \cdot \log y}} \cdot \left(\sqrt[3]{\log y} \cdot x\right)\right) - t\]

  13. Applied cbrt-unprod0.5

    \[\leadsto \mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \sqrt[3]{\color{blue}{\sqrt[3]{\left(\left(\log y \cdot \log y\right) \cdot \log y\right) \cdot \left(\left(\log y \cdot \log y\right) \cdot \log y\right)}}} \cdot \left(\sqrt[3]{\log y} \cdot x\right)\right) - t\]

  14. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \left(\sqrt[3]{\log y} \cdot x\right) \cdot \sqrt[3]{\sqrt[3]{\left(\log y \cdot \left(\log y \cdot \log y\right)\right) \cdot \left(\log y \cdot \left(\log y \cdot \log y\right)\right)}}\right) - t\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))