Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 21.6s

Precision: 64

Math

\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]

↓

\[\left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - \left(\left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}}\right)\right) \cdot z + z \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}}}\right) + \log \left(\sqrt[3]{\sqrt{t}}\right)\right)\right)\right) + \left(a - 0.5\right) \cdot b\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r280872 = x;
        double r280873 = y;
        double r280874 = r280872 + r280873;
        double r280875 = z;
        double r280876 = r280874 + r280875;
        double r280877 = t;
        double r280878 = log(r280877);
        double r280879 = r280875 * r280878;
        double r280880 = r280876 - r280879;
        double r280881 = a;
        double r280882 = 0.5;
        double r280883 = r280881 - r280882;
        double r280884 = b;
        double r280885 = r280883 * r280884;
        double r280886 = r280880 + r280885;
        return r280886;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r280887 = x;
        double r280888 = y;
        double r280889 = r280887 + r280888;
        double r280890 = z;
        double r280891 = r280889 + r280890;
        double r280892 = t;
        double r280893 = sqrt(r280892);
        double r280894 = log(r280893);
        double r280895 = r280894 * r280890;
        double r280896 = r280891 - r280895;
        double r280897 = 2.0;
        double r280898 = cbrt(r280893);
        double r280899 = r280898 * r280898;
        double r280900 = cbrt(r280899);
        double r280901 = log(r280900);
        double r280902 = r280897 * r280901;
        double r280903 = r280902 * r280890;
        double r280904 = cbrt(r280898);
        double r280905 = log(r280904);
        double r280906 = r280897 * r280905;
        double r280907 = log(r280898);
        double r280908 = r280906 + r280907;
        double r280909 = r280890 * r280908;
        double r280910 = r280903 + r280909;
        double r280911 = r280896 - r280910;
        double r280912 = a;
        double r280913 = 0.5;
        double r280914 = r280912 - r280913;
        double r280915 = b;
        double r280916 = r280914 * r280915;
        double r280917 = r280911 + r280916;
        return r280917;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	0.1
Target	0.3
Herbie	0.1

\[\left(\left(x + y\right) + \frac{\left(1 - {\left(\log t\right)}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b\]

Derivation

1. Initial program 0.1

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.1

   \[\leadsto \left(\left(\left(x + y\right) + z\right) - z \cdot \log \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}\right) + \left(a - 0.5\right) \cdot b\]

4. Applied log-prod 0.1

   \[\leadsto \left(\left(\left(x + y\right) + z\right) - z \cdot \color{blue}{\left(\log \left(\sqrt{t}\right) + \log \left(\sqrt{t}\right)\right)}\right) + \left(a - 0.5\right) \cdot b\]

5. Applied distribute-lft-in 0.1

   \[\leadsto \left(\left(\left(x + y\right) + z\right) - \color{blue}{\left(z \cdot \log \left(\sqrt{t}\right) + z \cdot \log \left(\sqrt{t}\right)\right)}\right) + \left(a - 0.5\right) \cdot b\]

6. Applied associate--r+ 0.1

   \[\leadsto \color{blue}{\left(\left(\left(\left(x + y\right) + z\right) - z \cdot \log \left(\sqrt{t}\right)\right) - z \cdot \log \left(\sqrt{t}\right)\right)} + \left(a - 0.5\right) \cdot b\]

7. Simplified 0.1

   \[\leadsto \left(\color{blue}{\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right)} - z \cdot \log \left(\sqrt{t}\right)\right) + \left(a - 0.5\right) \cdot b\]
8. Using strategy rm

9. Applied add-cube-cbrt 0.1

   \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - z \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}\right) \cdot \sqrt[3]{\sqrt{t}}\right)}\right) + \left(a - 0.5\right) \cdot b\]

10. Applied log-prod 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - z \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}\right) + \log \left(\sqrt[3]{\sqrt{t}}\right)\right)}\right) + \left(a - 0.5\right) \cdot b\]

11. Applied distribute-lft-in 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - \color{blue}{\left(z \cdot \log \left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}\right) + z \cdot \log \left(\sqrt[3]{\sqrt{t}}\right)\right)}\right) + \left(a - 0.5\right) \cdot b\]

12. Simplified 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - \left(\color{blue}{z \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt{t}}\right)\right)} + z \cdot \log \left(\sqrt[3]{\sqrt{t}}\right)\right)\right) + \left(a - 0.5\right) \cdot b\]
13. Using strategy rm

14. Applied add-cube-cbrt 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - \left(z \cdot \left(2 \cdot \log \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}\right) \cdot \sqrt[3]{\sqrt{t}}}}\right)\right) + z \cdot \log \left(\sqrt[3]{\sqrt{t}}\right)\right)\right) + \left(a - 0.5\right) \cdot b\]

15. Applied cbrt-prod 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - \left(z \cdot \left(2 \cdot \log \color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{t}}}\right)}\right) + z \cdot \log \left(\sqrt[3]{\sqrt{t}}\right)\right)\right) + \left(a - 0.5\right) \cdot b\]

16. Applied log-prod 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - \left(z \cdot \left(2 \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}}\right) + \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}}}\right)\right)}\right) + z \cdot \log \left(\sqrt[3]{\sqrt{t}}\right)\right)\right) + \left(a - 0.5\right) \cdot b\]

17. Applied distribute-lft-in 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - \left(z \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}}\right) + 2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}}}\right)\right)} + z \cdot \log \left(\sqrt[3]{\sqrt{t}}\right)\right)\right) + \left(a - 0.5\right) \cdot b\]

18. Applied distribute-rgt-in 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - \left(\color{blue}{\left(\left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}}\right)\right) \cdot z + \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}}}\right)\right) \cdot z\right)} + z \cdot \log \left(\sqrt[3]{\sqrt{t}}\right)\right)\right) + \left(a - 0.5\right) \cdot b\]

19. Applied associate-+l+ 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - \color{blue}{\left(\left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}}\right)\right) \cdot z + \left(\left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}}}\right)\right) \cdot z + z \cdot \log \left(\sqrt[3]{\sqrt{t}}\right)\right)\right)}\right) + \left(a - 0.5\right) \cdot b\]

20. Simplified 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - \left(\left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}}\right)\right) \cdot z + \color{blue}{z \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}}}\right) + \log \left(\sqrt[3]{\sqrt{t}}\right)\right)}\right)\right) + \left(a - 0.5\right) \cdot b\]

21. Final simplification 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt{t}\right) \cdot z\right) - \left(\left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}}\right)\right) \cdot z + z \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{t}}}\right) + \log \left(\sqrt[3]{\sqrt{t}}\right)\right)\right)\right) + \left(a - 0.5\right) \cdot b\]

Reproduce

herbie shell --seed 2019208 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))