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

Average Error: 0.1 → 0.1

Time: 6.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r119017 = x;
        double r119018 = y;
        double r119019 = log(r119018);
        double r119020 = r119017 * r119019;
        double r119021 = r119020 - r119018;
        double r119022 = z;
        double r119023 = r119021 - r119022;
        double r119024 = t;
        double r119025 = log(r119024);
        double r119026 = r119023 + r119025;
        return r119026;
}

↓

double f(double x, double y, double z, double t) {
        double r119027 = x;
        double r119028 = 1.0;
        double r119029 = y;
        double r119030 = 0.6666666666666666;
        double r119031 = pow(r119029, r119030);
        double r119032 = r119028 * r119031;
        double r119033 = log(r119032);
        double r119034 = r119027 * r119033;
        double r119035 = 0.3333333333333333;
        double r119036 = log(r119029);
        double r119037 = r119036 * r119027;
        double r119038 = r119035 * r119037;
        double r119039 = r119038 - r119029;
        double r119040 = z;
        double r119041 = r119039 - r119040;
        double r119042 = t;
        double r119043 = log(r119042);
        double r119044 = r119041 + r119043;
        double r119045 = r119034 + r119044;
        return r119045;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Initial program 0.1

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

3. Applied add-cube-cbrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Applied associate--l+ 0.1

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

7. Applied associate--l+ 0.1

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

8. Applied associate-+l+ 0.1

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

9. Simplified 0.1

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

11. Applied *-un-lft-identity 0.1

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

12. Applied cbrt-prod 0.1

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

13. Applied *-un-lft-identity 0.1

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

14. Applied cbrt-prod 0.1

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

15. Applied swap-sqr 0.1

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

16. Simplified 0.1

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

17. Simplified 0.1

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

19. Applied pow1/3 0.1

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

20. Applied log-pow 0.1

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

21. Applied associate-*l* 0.1

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

22. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019354 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))