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

Average Error: 9.8 → 0.4

Time: 17.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r591889 = x;
        double r591890 = y;
        double r591891 = log(r591890);
        double r591892 = r591889 * r591891;
        double r591893 = z;
        double r591894 = 1.0;
        double r591895 = r591894 - r591890;
        double r591896 = log(r591895);
        double r591897 = r591893 * r591896;
        double r591898 = r591892 + r591897;
        double r591899 = t;
        double r591900 = r591898 - r591899;
        return r591900;
}

↓

double f(double x, double y, double z, double t) {
        double r591901 = y;
        double r591902 = cbrt(r591901);
        double r591903 = r591902 * r591902;
        double r591904 = log(r591903);
        double r591905 = x;
        double r591906 = r591904 * r591905;
        double r591907 = 1.0;
        double r591908 = 0.6666666666666666;
        double r591909 = pow(r591901, r591908);
        double r591910 = r591907 / r591909;
        double r591911 = -0.3333333333333333;
        double r591912 = pow(r591910, r591911);
        double r591913 = r591907 / r591902;
        double r591914 = pow(r591913, r591911);
        double r591915 = r591912 * r591914;
        double r591916 = log(r591915);
        double r591917 = r591905 * r591916;
        double r591918 = z;
        double r591919 = 1.0;
        double r591920 = log(r591919);
        double r591921 = r591919 * r591901;
        double r591922 = r591920 - r591921;
        double r591923 = r591918 * r591922;
        double r591924 = 0.5;
        double r591925 = 2.0;
        double r591926 = pow(r591901, r591925);
        double r591927 = r591918 * r591926;
        double r591928 = pow(r591919, r591925);
        double r591929 = r591927 / r591928;
        double r591930 = r591924 * r591929;
        double r591931 = r591923 - r591930;
        double r591932 = r591917 + r591931;
        double r591933 = r591906 + r591932;
        double r591934 = t;
        double r591935 = r591933 - r591934;
        return r591935;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.8
Target	0.3
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.8

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

2. Taylor expanded around 0 0.4

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

3. Simplified 0.4

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

5. Applied add-cube-cbrt 0.4

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

6. Applied log-prod 0.4

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

7. Applied distribute-rgt-in 0.4

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

8. Applied associate-+l+ 0.4

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

9. Simplified 0.4

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

10. Taylor expanded around inf 0.4

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

12. Applied add-cube-cbrt 0.4

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

13. Applied add-cube-cbrt 0.4

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

14. Applied times-frac 0.4

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

15. Applied unpow-prod-down 0.4

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

16. Simplified 0.4

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

17. Simplified 0.4

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

18. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020047 
(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) (* (/ 0.3333333333333333 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

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