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

Average Error: 9.2 → 0.4

Time: 19.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r403837 = x;
        double r403838 = y;
        double r403839 = log(r403838);
        double r403840 = r403837 * r403839;
        double r403841 = z;
        double r403842 = 1.0;
        double r403843 = r403842 - r403838;
        double r403844 = log(r403843);
        double r403845 = r403841 * r403844;
        double r403846 = r403840 + r403845;
        double r403847 = t;
        double r403848 = r403846 - r403847;
        return r403848;
}

↓

double f(double x, double y, double z, double t) {
        double r403849 = x;
        double r403850 = y;
        double r403851 = cbrt(r403850);
        double r403852 = r403851 * r403851;
        double r403853 = cbrt(r403852);
        double r403854 = log(r403853);
        double r403855 = 2.0;
        double r403856 = r403854 * r403855;
        double r403857 = cbrt(r403851);
        double r403858 = log(r403857);
        double r403859 = r403858 * r403855;
        double r403860 = log(r403851);
        double r403861 = r403859 + r403860;
        double r403862 = r403856 + r403861;
        double r403863 = r403849 * r403862;
        double r403864 = z;
        double r403865 = 1.0;
        double r403866 = log(r403865);
        double r403867 = r403865 * r403850;
        double r403868 = 0.5;
        double r403869 = pow(r403850, r403855);
        double r403870 = pow(r403865, r403855);
        double r403871 = r403869 / r403870;
        double r403872 = r403868 * r403871;
        double r403873 = r403867 + r403872;
        double r403874 = r403866 - r403873;
        double r403875 = r403864 * r403874;
        double r403876 = t;
        double r403877 = r403875 - r403876;
        double r403878 = r403863 + r403877;
        return r403878;
}

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. 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-cbrt 0.4

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

5. Applied log-prod 0.4

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

6. Applied distribute-lft-in 0.4

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

7. Simplified 0.4

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

9. Applied add-cube-cbrt 0.4

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

10. Applied cbrt-prod 0.4

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

11. Applied log-prod 0.4

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

12. Applied distribute-rgt-in 0.4

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

13. Applied distribute-lft-in 0.4

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

14. Applied associate-+l+ 0.4

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

15. Simplified 0.4

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

16. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019298 
(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))