Average Error: 9.9 → 0.3
Time: 19.4s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) + \left(x \cdot \left(\log \left({y}^{\left(\frac{1}{3}\right)}\right) + 2 \cdot \log \left(\sqrt[3]{y}\right)\right) - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) + \left(x \cdot \left(\log \left({y}^{\left(\frac{1}{3}\right)}\right) + 2 \cdot \log \left(\sqrt[3]{y}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r351779 = x;
        double r351780 = y;
        double r351781 = log(r351780);
        double r351782 = r351779 * r351781;
        double r351783 = z;
        double r351784 = 1.0;
        double r351785 = r351784 - r351780;
        double r351786 = log(r351785);
        double r351787 = r351783 * r351786;
        double r351788 = r351782 + r351787;
        double r351789 = t;
        double r351790 = r351788 - r351789;
        return r351790;
}
↓
double f(double x, double y, double z, double t) {
        double r351791 = z;
        double r351792 = 1.0;
        double r351793 = log(r351792);
        double r351794 = y;
        double r351795 = r351792 * r351794;
        double r351796 = 1.0;
        double r351797 = 2.0;
        double r351798 = r351796 / r351797;
        double r351799 = pow(r351794, r351797);
        double r351800 = pow(r351792, r351797);
        double r351801 = r351799 / r351800;
        double r351802 = r351798 * r351801;
        double r351803 = r351795 + r351802;
        double r351804 = r351793 - r351803;
        double r351805 = r351791 * r351804;
        double r351806 = x;
        double r351807 = 3.0;
        double r351808 = r351796 / r351807;
        double r351809 = pow(r351794, r351808);
        double r351810 = log(r351809);
        double r351811 = cbrt(r351794);
        double r351812 = log(r351811);
        double r351813 = r351797 * r351812;
        double r351814 = r351810 + r351813;
        double r351815 = r351806 * r351814;
        double r351816 = t;
        double r351817 = r351815 - r351816;
        double r351818 = r351805 + r351817;
        return r351818;
}

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

Original9.9
Target0.2
Herbie0.3
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.9

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

  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. Simplified0.3

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  9. Taylor expanded around 0 0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 
(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))