Average Error: 9.1 → 0.4
Time: 8.9s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \mathsf{fma}\left(\log \left(1 \cdot {y}^{\frac{1}{3}}\right), x, z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \mathsf{fma}\left(\log \left(1 \cdot {y}^{\frac{1}{3}}\right), x, z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r445634 = x;
        double r445635 = y;
        double r445636 = log(r445635);
        double r445637 = r445634 * r445636;
        double r445638 = z;
        double r445639 = 1.0;
        double r445640 = r445639 - r445635;
        double r445641 = log(r445640);
        double r445642 = r445638 * r445641;
        double r445643 = r445637 + r445642;
        double r445644 = t;
        double r445645 = r445643 - r445644;
        return r445645;
}
↓
double f(double x, double y, double z, double t) {
        double r445646 = x;
        double r445647 = y;
        double r445648 = cbrt(r445647);
        double r445649 = r445648 * r445648;
        double r445650 = log(r445649);
        double r445651 = r445646 * r445650;
        double r445652 = 1.0;
        double r445653 = 0.3333333333333333;
        double r445654 = pow(r445647, r445653);
        double r445655 = r445652 * r445654;
        double r445656 = log(r445655);
        double r445657 = z;
        double r445658 = 1.0;
        double r445659 = log(r445658);
        double r445660 = r445658 * r445647;
        double r445661 = 0.5;
        double r445662 = 2.0;
        double r445663 = pow(r445647, r445662);
        double r445664 = pow(r445658, r445662);
        double r445665 = r445663 / r445664;
        double r445666 = r445661 * r445665;
        double r445667 = r445660 + r445666;
        double r445668 = r445659 - r445667;
        double r445669 = r445657 * r445668;
        double r445670 = t;
        double r445671 = r445669 - r445670;
        double r445672 = fma(r445656, r445646, r445671);
        double r445673 = r445651 + r445672;
        return r445673;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.1
Target0.3
Herbie0.4
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.1

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

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]
  3. Using strategy rm

  4. Applied associate--l+0.4

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

  6. Applied add-cube-cbrt0.4

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

  7. Applied log-prod0.4

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

  8. Applied distribute-lft-in0.4

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

  9. Applied associate-+l+0.4

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

  10. Simplified0.4

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

  12. Applied *-un-lft-identity0.4

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

  13. Applied cbrt-prod0.4

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

  14. Simplified0.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))