Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Average Error: 9.2 → 0.4

Time: 23.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r338767 = x;
        double r338768 = y;
        double r338769 = log(r338768);
        double r338770 = r338767 * r338769;
        double r338771 = z;
        double r338772 = 1.0;
        double r338773 = r338772 - r338768;
        double r338774 = log(r338773);
        double r338775 = r338771 * r338774;
        double r338776 = r338770 + r338775;
        double r338777 = t;
        double r338778 = r338776 - r338777;
        return r338778;
}

↓

double f(double x, double y, double z, double t) {
        double r338779 = 2.0;
        double r338780 = y;
        double r338781 = cbrt(r338780);
        double r338782 = log(r338781);
        double r338783 = r338779 * r338782;
        double r338784 = x;
        double r338785 = r338783 * r338784;
        double r338786 = 1.0;
        double r338787 = r338780 / r338786;
        double r338788 = r338786 / r338780;
        double r338789 = r338787 / r338788;
        double r338790 = -0.5;
        double r338791 = r338789 * r338790;
        double r338792 = log(r338786);
        double r338793 = r338786 * r338780;
        double r338794 = r338792 - r338793;
        double r338795 = r338791 + r338794;
        double r338796 = z;
        double r338797 = r338795 * r338796;
        double r338798 = r338785 + r338797;
        double r338799 = cbrt(r338781);
        double r338800 = log(r338799);
        double r338801 = r338784 * r338800;
        double r338802 = r338779 * r338784;
        double r338803 = r338800 * r338802;
        double r338804 = r338801 + r338803;
        double r338805 = t;
        double r338806 = r338804 - r338805;
        double r338807 = r338798 + r338806;
        return r338807;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.2
Target	0.3
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.2

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

2. Simplified 9.2

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

3. Taylor expanded around 0 0.3

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

4. Simplified 0.3

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

6. Applied add-cube-cbrt 0.3

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

7. Applied log-prod 0.4

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

8. Applied distribute-lft-in 0.4

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

9. Applied associate--l+ 0.4

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

10. Applied associate-+r+ 0.4

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

11. Simplified 0.4

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

13. Applied add-cube-cbrt 0.4

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

14. Applied log-prod 0.4

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

15. Applied distribute-lft-in 0.4

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

16. Simplified 0.4

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

17. Simplified 0.4

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

18. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019194 
(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))