Average Error: 9.6 → 0.4
Time: 19.9s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[x \cdot \left(\log \left({y}^{\frac{1}{3}} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
x \cdot \left(\log \left({y}^{\frac{1}{3}} \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)
double f(double x, double y, double z, double t) {
        double r320748 = x;
        double r320749 = y;
        double r320750 = log(r320749);
        double r320751 = r320748 * r320750;
        double r320752 = z;
        double r320753 = 1.0;
        double r320754 = r320753 - r320749;
        double r320755 = log(r320754);
        double r320756 = r320752 * r320755;
        double r320757 = r320751 + r320756;
        double r320758 = t;
        double r320759 = r320757 - r320758;
        return r320759;
}
↓
double f(double x, double y, double z, double t) {
        double r320760 = x;
        double r320761 = y;
        double r320762 = 0.3333333333333333;
        double r320763 = pow(r320761, r320762);
        double r320764 = cbrt(r320761);
        double r320765 = r320763 * r320764;
        double r320766 = log(r320765);
        double r320767 = log(r320764);
        double r320768 = r320766 + r320767;
        double r320769 = r320760 * r320768;
        double r320770 = z;
        double r320771 = 1.0;
        double r320772 = log(r320771);
        double r320773 = r320771 * r320761;
        double r320774 = 0.5;
        double r320775 = 2.0;
        double r320776 = pow(r320761, r320775);
        double r320777 = pow(r320771, r320775);
        double r320778 = r320776 / r320777;
        double r320779 = r320774 * r320778;
        double r320780 = r320773 + r320779;
        double r320781 = r320772 - r320780;
        double r320782 = r320770 * r320781;
        double r320783 = t;
        double r320784 = r320782 - r320783;
        double r320785 = r320769 + r320784;
        return r320785;
}

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

Original9.6
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 9.6

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

  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-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 pow1/30.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019291 
(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) (* (/ 0.333333333333333315 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

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