Average Error: 9.4 → 0.6
Time: 28.6s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(\frac{y}{1} \cdot \frac{y}{1}, \frac{1}{2}, 1 \cdot y\right), z, \left(\sqrt[3]{\log y} \cdot x\right) \cdot \sqrt[3]{\sqrt[3]{\left(\log y \cdot \left(\log y \cdot \log y\right)\right) \cdot \left(\log y \cdot \left(\log y \cdot \log y\right)\right)}} - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(\frac{y}{1} \cdot \frac{y}{1}, \frac{1}{2}, 1 \cdot y\right), z, \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)}} - t\right)
double f(double x, double y, double z, double t) {
        double r14239609 = x;
        double r14239610 = y;
        double r14239611 = log(r14239610);
        double r14239612 = r14239609 * r14239611;
        double r14239613 = z;
        double r14239614 = 1.0;
        double r14239615 = r14239614 - r14239610;
        double r14239616 = log(r14239615);
        double r14239617 = r14239613 * r14239616;
        double r14239618 = r14239612 + r14239617;
        double r14239619 = t;
        double r14239620 = r14239618 - r14239619;
        return r14239620;
}
↓
double f(double x, double y, double z, double t) {
        double r14239621 = 1.0;
        double r14239622 = log(r14239621);
        double r14239623 = y;
        double r14239624 = r14239623 / r14239621;
        double r14239625 = r14239624 * r14239624;
        double r14239626 = 0.5;
        double r14239627 = r14239621 * r14239623;
        double r14239628 = fma(r14239625, r14239626, r14239627);
        double r14239629 = r14239622 - r14239628;
        double r14239630 = z;
        double r14239631 = log(r14239623);
        double r14239632 = cbrt(r14239631);
        double r14239633 = x;
        double r14239634 = r14239632 * r14239633;
        double r14239635 = r14239631 * r14239631;
        double r14239636 = r14239631 * r14239635;
        double r14239637 = r14239636 * r14239636;
        double r14239638 = cbrt(r14239637);
        double r14239639 = cbrt(r14239638);
        double r14239640 = r14239634 * r14239639;
        double r14239641 = t;
        double r14239642 = r14239640 - r14239641;
        double r14239643 = fma(r14239629, r14239630, r14239642);
        return r14239643;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.4
Target0.3
Herbie0.6
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.4

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

  2. Simplified9.4

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  6. Applied add-cube-cbrt0.8

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

  7. Applied associate-*l*0.8

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

  9. Applied cbrt-unprod0.6

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

  11. Applied add-cbrt-cube0.6

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

  12. Applied add-cbrt-cube0.6

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

  13. Applied cbrt-unprod0.6

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

  14. Final simplification0.6

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

Reproduce

herbie shell --seed 2019170 +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))