Average Error: 9.3 → 0.3
Time: 9.4s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\log \left(1 \cdot \left({\left({y}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{y}\right)}^{\frac{1}{3}}\right)\right) \cdot x + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\log \left(1 \cdot \left({\left({y}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot {\left(\sqrt[3]{y}\right)}^{\frac{1}{3}}\right)\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
double f(double x, double y, double z, double t) {
        double r518732 = x;
        double r518733 = y;
        double r518734 = log(r518733);
        double r518735 = r518732 * r518734;
        double r518736 = z;
        double r518737 = 1.0;
        double r518738 = r518737 - r518733;
        double r518739 = log(r518738);
        double r518740 = r518736 * r518739;
        double r518741 = r518735 + r518740;
        double r518742 = t;
        double r518743 = r518741 - r518742;
        return r518743;
}
↓
double f(double x, double y, double z, double t) {
        double r518744 = x;
        double r518745 = y;
        double r518746 = cbrt(r518745);
        double r518747 = r518746 * r518746;
        double r518748 = log(r518747);
        double r518749 = r518744 * r518748;
        double r518750 = 1.0;
        double r518751 = 0.3333333333333333;
        double r518752 = pow(r518745, r518751);
        double r518753 = 0.6666666666666666;
        double r518754 = pow(r518752, r518753);
        double r518755 = pow(r518746, r518751);
        double r518756 = r518754 * r518755;
        double r518757 = r518750 * r518756;
        double r518758 = log(r518757);
        double r518759 = r518758 * r518744;
        double r518760 = z;
        double r518761 = 1.0;
        double r518762 = log(r518761);
        double r518763 = r518761 * r518745;
        double r518764 = 0.5;
        double r518765 = 2.0;
        double r518766 = pow(r518745, r518765);
        double r518767 = pow(r518761, r518765);
        double r518768 = r518766 / r518767;
        double r518769 = r518764 * r518768;
        double r518770 = r518763 + r518769;
        double r518771 = r518762 - r518770;
        double r518772 = r518760 * r518771;
        double r518773 = r518759 + r518772;
        double r518774 = r518749 + r518773;
        double r518775 = t;
        double r518776 = r518774 - r518775;
        return r518776;
}

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

Original9.3
Target0.2
Herbie0.3
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.3

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

  2. Taylor expanded around 0 0.3

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

  4. Applied add-cube-cbrt0.3

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

  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-lft-in0.4

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

  7. Applied associate-+l+0.4

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

  8. Simplified0.4

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

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

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

  11. Applied cbrt-prod0.4

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

  12. Simplified0.4

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

  13. Simplified0.3

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

  15. Applied add-cube-cbrt0.3

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

  16. Simplified0.3

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

  17. Simplified0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2020100 
(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))