Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 12.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r344908 = x;
        double r344909 = y;
        double r344910 = r344908 + r344909;
        double r344911 = log(r344910);
        double r344912 = z;
        double r344913 = log(r344912);
        double r344914 = r344911 + r344913;
        double r344915 = t;
        double r344916 = r344914 - r344915;
        double r344917 = a;
        double r344918 = 0.5;
        double r344919 = r344917 - r344918;
        double r344920 = log(r344915);
        double r344921 = r344919 * r344920;
        double r344922 = r344916 + r344921;
        return r344922;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r344923 = x;
        double r344924 = y;
        double r344925 = r344923 + r344924;
        double r344926 = log(r344925);
        double r344927 = z;
        double r344928 = log(r344927);
        double r344929 = r344926 + r344928;
        double r344930 = t;
        double r344931 = r344929 - r344930;
        double r344932 = cbrt(r344930);
        double r344933 = r344932 * r344932;
        double r344934 = cbrt(r344933);
        double r344935 = log(r344934);
        double r344936 = 2.0;
        double r344937 = r344935 * r344936;
        double r344938 = a;
        double r344939 = 0.5;
        double r344940 = r344938 - r344939;
        double r344941 = r344937 * r344940;
        double r344942 = cbrt(r344932);
        double r344943 = log(r344942);
        double r344944 = r344943 * r344936;
        double r344945 = log(r344932);
        double r344946 = r344944 + r344945;
        double r344947 = r344940 * r344946;
        double r344948 = r344941 + r344947;
        double r344949 = r344931 + r344948;
        return r344949;
}

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-rgt-in 0.3

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

12. Applied distribute-rgt-in 0.3

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

13. Applied associate-+l+ 0.3

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

14. Simplified 0.3

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

15. Final simplification 0.3

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

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))))