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

Average Error: 9.6 → 0.4

Time: 26.0s

Precision: 64

Math

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

↓

\[\left(\left(x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2\right) + x \cdot \left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{9}}\right) \cdot 2 + \log \left(\sqrt[3]{y}\right)\right)\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]

C

double f(double x, double y, double z, double t) {
        double r261981 = x;
        double r261982 = y;
        double r261983 = log(r261982);
        double r261984 = r261981 * r261983;
        double r261985 = z;
        double r261986 = 1.0;
        double r261987 = r261986 - r261982;
        double r261988 = log(r261987);
        double r261989 = r261985 * r261988;
        double r261990 = r261984 + r261989;
        double r261991 = t;
        double r261992 = r261990 - r261991;
        return r261992;
}

↓

double f(double x, double y, double z, double t) {
        double r261993 = x;
        double r261994 = y;
        double r261995 = cbrt(r261994);
        double r261996 = r261995 * r261995;
        double r261997 = cbrt(r261996);
        double r261998 = log(r261997);
        double r261999 = 2.0;
        double r262000 = r261998 * r261999;
        double r262001 = r261993 * r262000;
        double r262002 = 1.0;
        double r262003 = r262002 / r261994;
        double r262004 = -0.1111111111111111;
        double r262005 = pow(r262003, r262004);
        double r262006 = log(r262005);
        double r262007 = r262006 * r261999;
        double r262008 = log(r261995);
        double r262009 = r262007 + r262008;
        double r262010 = r261993 * r262009;
        double r262011 = r262001 + r262010;
        double r262012 = z;
        double r262013 = 1.0;
        double r262014 = log(r262013);
        double r262015 = r262013 * r261994;
        double r262016 = r262014 - r262015;
        double r262017 = r262012 * r262016;
        double r262018 = 0.5;
        double r262019 = pow(r261994, r261999);
        double r262020 = r262012 * r262019;
        double r262021 = pow(r262013, r261999);
        double r262022 = r262020 / r262021;
        double r262023 = r262018 * r262022;
        double r262024 = r262017 - r262023;
        double r262025 = r262011 + r262024;
        double r262026 = t;
        double r262027 = r262025 - r262026;
        return r262027;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.6
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 9.6

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

2. Taylor expanded around 0 0.3

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

3. Simplified 0.3

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

5. Applied add-cube-cbrt 0.3

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

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-lft-in 0.4

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

8. Simplified 0.4

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

9. Simplified 0.4

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

11. Applied add-cube-cbrt 0.4

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

12. Applied cbrt-prod 0.4

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

13. Applied log-prod 0.4

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

14. Applied distribute-rgt-in 0.4

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

15. Applied distribute-lft-in 0.4

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

16. Applied associate-+l+ 0.4

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

17. Simplified 0.4

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

18. Taylor expanded around inf 0.4

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

19. Final simplification 0.4

    \[\leadsto \left(\left(x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2\right) + x \cdot \left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{9}}\right) \cdot 2 + \log \left(\sqrt[3]{y}\right)\right)\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]

Reproduce

herbie shell --seed 2019304 
(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))