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

Average Error: 9.0 → 0.4

Time: 35.6s

Precision: 64

Math

\[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]

↓

\[\left(z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right) + \left(\log \left({y}^{\frac{1}{3}}\right) \cdot x + x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right)\right)\right)\right) - t\]

C

double f(double x, double y, double z, double t) {
        double r23006842 = x;
        double r23006843 = y;
        double r23006844 = log(r23006843);
        double r23006845 = r23006842 * r23006844;
        double r23006846 = z;
        double r23006847 = 1.0;
        double r23006848 = r23006847 - r23006843;
        double r23006849 = log(r23006848);
        double r23006850 = r23006846 * r23006849;
        double r23006851 = r23006845 + r23006850;
        double r23006852 = t;
        double r23006853 = r23006851 - r23006852;
        return r23006853;
}

↓

double f(double x, double y, double z, double t) {
        double r23006854 = z;
        double r23006855 = 1.0;
        double r23006856 = log(r23006855);
        double r23006857 = y;
        double r23006858 = r23006855 * r23006857;
        double r23006859 = r23006856 - r23006858;
        double r23006860 = 0.5;
        double r23006861 = r23006855 / r23006857;
        double r23006862 = r23006860 / r23006861;
        double r23006863 = r23006862 / r23006861;
        double r23006864 = r23006859 - r23006863;
        double r23006865 = r23006854 * r23006864;
        double r23006866 = 0.3333333333333333;
        double r23006867 = pow(r23006857, r23006866);
        double r23006868 = log(r23006867);
        double r23006869 = x;
        double r23006870 = r23006868 * r23006869;
        double r23006871 = cbrt(r23006857);
        double r23006872 = log(r23006871);
        double r23006873 = cbrt(r23006871);
        double r23006874 = r23006873 * r23006873;
        double r23006875 = r23006873 * r23006874;
        double r23006876 = log(r23006875);
        double r23006877 = r23006872 + r23006876;
        double r23006878 = r23006869 * r23006877;
        double r23006879 = r23006870 + r23006878;
        double r23006880 = r23006865 + r23006879;
        double r23006881 = t;
        double r23006882 = r23006880 - r23006881;
        return r23006882;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.0
Target	0.3
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{\frac{1}{3}}{1.0 \cdot \left(1.0 \cdot 1.0\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.0

   \[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]

2. Taylor expanded around 0 0.3

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)\right)}\right) - t\]

3. Simplified 0.3

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)}\right) - t\]
4. Using strategy rm

5. Applied add-cube-cbrt 0.3

   \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]

6. Applied log-prod 0.4

   \[\leadsto \left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]

7. Applied distribute-lft-in 0.4

   \[\leadsto \left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]

8. Simplified 0.4

   \[\leadsto \left(\left(\color{blue}{x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]
9. Using strategy rm

10. Applied pow1/3 0.4

    \[\leadsto \left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \color{blue}{\left({y}^{\frac{1}{3}}\right)}\right) + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]
11. Using strategy rm

12. Applied add-cube-cbrt 0.4

    \[\leadsto \left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) + x \cdot \log \left({y}^{\frac{1}{3}}\right)\right) + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]

13. Final simplification 0.4

    \[\leadsto \left(z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right) + \left(\log \left({y}^{\frac{1}{3}}\right) \cdot x + x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019163 
(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) (* (/ 1/3 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))