Average Error: 9.4 → 0.5
Time: 22.0s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x + \left(x \cdot \log \left({\left({\left(\frac{1}{y}\right)}^{\left(\sqrt[3]{\frac{-1}{3}} \cdot \sqrt[3]{\frac{-1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{-1}{3}}\right)}\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x + \left(x \cdot \log \left({\left({\left(\frac{1}{y}\right)}^{\left(\sqrt[3]{\frac{-1}{3}} \cdot \sqrt[3]{\frac{-1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{-1}{3}}\right)}\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r278737 = x;
        double r278738 = y;
        double r278739 = log(r278738);
        double r278740 = r278737 * r278739;
        double r278741 = z;
        double r278742 = 1.0;
        double r278743 = r278742 - r278738;
        double r278744 = log(r278743);
        double r278745 = r278741 * r278744;
        double r278746 = r278740 + r278745;
        double r278747 = t;
        double r278748 = r278746 - r278747;
        return r278748;
}
↓
double f(double x, double y, double z, double t) {
        double r278749 = y;
        double r278750 = cbrt(r278749);
        double r278751 = r278750 * r278750;
        double r278752 = log(r278751);
        double r278753 = x;
        double r278754 = r278752 * r278753;
        double r278755 = 1.0;
        double r278756 = r278755 / r278749;
        double r278757 = -0.3333333333333333;
        double r278758 = cbrt(r278757);
        double r278759 = r278758 * r278758;
        double r278760 = pow(r278756, r278759);
        double r278761 = pow(r278760, r278758);
        double r278762 = log(r278761);
        double r278763 = r278753 * r278762;
        double r278764 = z;
        double r278765 = 1.0;
        double r278766 = log(r278765);
        double r278767 = r278765 * r278749;
        double r278768 = 0.5;
        double r278769 = 2.0;
        double r278770 = pow(r278749, r278769);
        double r278771 = pow(r278765, r278769);
        double r278772 = r278770 / r278771;
        double r278773 = r278768 * r278772;
        double r278774 = r278767 + r278773;
        double r278775 = r278766 - r278774;
        double r278776 = r278764 * r278775;
        double r278777 = r278763 + r278776;
        double r278778 = r278754 + r278777;
        double r278779 = t;
        double r278780 = r278778 - r278779;
        return r278780;
}

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
Herbie0.5
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.4

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

  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. Using strategy rm

  4. Applied add-cube-cbrt0.4

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

  5. Applied log-prod0.4

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

  6. Applied distribute-rgt-in0.4

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

  7. Applied associate-+l+0.4

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

  8. Simplified0.4

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

  9. Taylor expanded around inf 0.4

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

  11. Applied add-cube-cbrt0.5

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

  12. Applied pow-unpow0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019198 
(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))