Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 12.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r379938 = x;
        double r379939 = y;
        double r379940 = r379938 + r379939;
        double r379941 = log(r379940);
        double r379942 = z;
        double r379943 = log(r379942);
        double r379944 = r379941 + r379943;
        double r379945 = t;
        double r379946 = r379944 - r379945;
        double r379947 = a;
        double r379948 = 0.5;
        double r379949 = r379947 - r379948;
        double r379950 = log(r379945);
        double r379951 = r379949 * r379950;
        double r379952 = r379946 + r379951;
        return r379952;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r379953 = x;
        double r379954 = y;
        double r379955 = r379953 + r379954;
        double r379956 = log(r379955);
        double r379957 = z;
        double r379958 = log(r379957);
        double r379959 = r379956 + r379958;
        double r379960 = t;
        double r379961 = r379959 - r379960;
        double r379962 = 2.0;
        double r379963 = cbrt(r379960);
        double r379964 = r379963 * r379963;
        double r379965 = cbrt(r379964);
        double r379966 = log(r379965);
        double r379967 = r379962 * r379966;
        double r379968 = a;
        double r379969 = 0.5;
        double r379970 = r379968 - r379969;
        double r379971 = r379967 * r379970;
        double r379972 = cbrt(r379963);
        double r379973 = log(r379972);
        double r379974 = r379973 * r379962;
        double r379975 = log(r379963);
        double r379976 = r379974 + r379975;
        double r379977 = r379970 * r379976;
        double r379978 = r379971 + r379977;
        double r379979 = r379961 + r379978;
        return r379979;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.3
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 0.3

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

3. Applied add-cube-cbrt 0.3

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

4. Applied log-prod 0.3

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

5. Applied distribute-lft-in 0.3

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

6. Simplified 0.3

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

8. Applied add-cube-cbrt 0.3

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

9. Applied cbrt-prod 0.3

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

10. Applied log-prod 0.3

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

11. Applied distribute-lft-in 0.3

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

12. Applied distribute-rgt-in 0.3

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

13. Applied associate-+l+ 0.3

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

14. Simplified 0.3

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

15. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020001 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

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