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

Average Error: 0.1 → 0.1

Time: 6.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r363841 = x;
        double r363842 = y;
        double r363843 = r363841 + r363842;
        double r363844 = z;
        double r363845 = r363843 + r363844;
        double r363846 = t;
        double r363847 = log(r363846);
        double r363848 = r363844 * r363847;
        double r363849 = r363845 - r363848;
        double r363850 = a;
        double r363851 = 0.5;
        double r363852 = r363850 - r363851;
        double r363853 = b;
        double r363854 = r363852 * r363853;
        double r363855 = r363849 + r363854;
        return r363855;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r363856 = x;
        double r363857 = z;
        double r363858 = y;
        double r363859 = r363857 + r363858;
        double r363860 = t;
        double r363861 = sqrt(r363860);
        double r363862 = log(r363861);
        double r363863 = cbrt(r363860);
        double r363864 = r363863 * r363863;
        double r363865 = sqrt(r363864);
        double r363866 = log(r363865);
        double r363867 = r363862 + r363866;
        double r363868 = r363857 * r363867;
        double r363869 = r363859 - r363868;
        double r363870 = r363856 + r363869;
        double r363871 = 1.0;
        double r363872 = 0.3333333333333333;
        double r363873 = pow(r363860, r363872);
        double r363874 = r363871 * r363873;
        double r363875 = sqrt(r363874);
        double r363876 = log(r363875);
        double r363877 = r363857 * r363876;
        double r363878 = r363870 - r363877;
        double r363879 = a;
        double r363880 = 0.5;
        double r363881 = r363879 - r363880;
        double r363882 = b;
        double r363883 = r363881 * r363882;
        double r363884 = r363878 + r363883;
        return r363884;
}

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	0.4
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

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

10. Applied sqrt-prod 0.1

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

11. Applied log-prod 0.1

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

12. Applied distribute-lft-in 0.1

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

13. Applied associate--r+ 0.1

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

14. Simplified 0.1

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

16. Applied *-un-lft-identity 0.1

    \[\leadsto \left(\left(x + \left(\left(z + y\right) - z \cdot \left(\log \left(\sqrt{t}\right) + \log \left(\sqrt{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)\right)\right)\right) - z \cdot \log \left(\sqrt{\sqrt[3]{\color{blue}{1 \cdot t}}}\right)\right) + \left(a - 0.5\right) \cdot b\]

17. Applied cbrt-prod 0.1

    \[\leadsto \left(\left(x + \left(\left(z + y\right) - z \cdot \left(\log \left(\sqrt{t}\right) + \log \left(\sqrt{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)\right)\right)\right) - z \cdot \log \left(\sqrt{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{t}}}\right)\right) + \left(a - 0.5\right) \cdot b\]

18. Simplified 0.1

    \[\leadsto \left(\left(x + \left(\left(z + y\right) - z \cdot \left(\log \left(\sqrt{t}\right) + \log \left(\sqrt{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)\right)\right)\right) - z \cdot \log \left(\sqrt{\color{blue}{1} \cdot \sqrt[3]{t}}\right)\right) + \left(a - 0.5\right) \cdot b\]

19. Simplified 0.1

    \[\leadsto \left(\left(x + \left(\left(z + y\right) - z \cdot \left(\log \left(\sqrt{t}\right) + \log \left(\sqrt{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)\right)\right)\right) - z \cdot \log \left(\sqrt{1 \cdot \color{blue}{{t}^{\frac{1}{3}}}}\right)\right) + \left(a - 0.5\right) \cdot b\]

20. Final simplification 0.1

    \[\leadsto \left(\left(x + \left(\left(z + y\right) - z \cdot \left(\log \left(\sqrt{t}\right) + \log \left(\sqrt{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)\right)\right)\right) - z \cdot \log \left(\sqrt{1 \cdot {t}^{\frac{1}{3}}}\right)\right) + \left(a - 0.5\right) \cdot b\]

Reproduce

herbie shell --seed 2020047 
(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)))