Average Error: 9.4 → 0.4
Time: 42.5s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(e^{\log y \cdot \frac{1}{3}}\right)\right) + z \cdot \left(\left(\log 1 - 1 \cdot y\right) - \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(e^{\log y \cdot \frac{1}{3}}\right)\right) + z \cdot \left(\left(\log 1 - 1 \cdot y\right) - \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right)\right) - t
double f(double x, double y, double z, double t) {
        double r84239706 = x;
        double r84239707 = y;
        double r84239708 = log(r84239707);
        double r84239709 = r84239706 * r84239708;
        double r84239710 = z;
        double r84239711 = 1.0;
        double r84239712 = r84239711 - r84239707;
        double r84239713 = log(r84239712);
        double r84239714 = r84239710 * r84239713;
        double r84239715 = r84239709 + r84239714;
        double r84239716 = t;
        double r84239717 = r84239715 - r84239716;
        return r84239717;
}
↓
double f(double x, double y, double z, double t) {
        double r84239718 = x;
        double r84239719 = y;
        double r84239720 = cbrt(r84239719);
        double r84239721 = r84239720 * r84239720;
        double r84239722 = log(r84239721);
        double r84239723 = r84239718 * r84239722;
        double r84239724 = log(r84239719);
        double r84239725 = 0.3333333333333333;
        double r84239726 = r84239724 * r84239725;
        double r84239727 = exp(r84239726);
        double r84239728 = log(r84239727);
        double r84239729 = r84239718 * r84239728;
        double r84239730 = r84239723 + r84239729;
        double r84239731 = z;
        double r84239732 = 1.0;
        double r84239733 = log(r84239732);
        double r84239734 = r84239732 * r84239719;
        double r84239735 = r84239733 - r84239734;
        double r84239736 = 0.5;
        double r84239737 = r84239732 / r84239719;
        double r84239738 = r84239737 * r84239737;
        double r84239739 = r84239736 / r84239738;
        double r84239740 = r84239735 - r84239739;
        double r84239741 = r84239731 * r84239740;
        double r84239742 = r84239730 + r84239741;
        double r84239743 = t;
        double r84239744 = r84239742 - r84239743;
        return r84239744;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 9.4

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

  2. 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. Simplified0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Taylor expanded around inf 0.4

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

  10. Applied add-exp-log0.4

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

  11. Applied rec-exp0.4

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

  12. Applied pow-exp0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019173 
(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))