Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 12.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r398932 = x;
        double r398933 = y;
        double r398934 = r398932 + r398933;
        double r398935 = log(r398934);
        double r398936 = z;
        double r398937 = log(r398936);
        double r398938 = r398935 + r398937;
        double r398939 = t;
        double r398940 = r398938 - r398939;
        double r398941 = a;
        double r398942 = 0.5;
        double r398943 = r398941 - r398942;
        double r398944 = log(r398939);
        double r398945 = r398943 * r398944;
        double r398946 = r398940 + r398945;
        return r398946;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r398947 = x;
        double r398948 = y;
        double r398949 = r398947 + r398948;
        double r398950 = log(r398949);
        double r398951 = z;
        double r398952 = sqrt(r398951);
        double r398953 = log(r398952);
        double r398954 = r398950 + r398953;
        double r398955 = r398954 + r398953;
        double r398956 = t;
        double r398957 = r398955 - r398956;
        double r398958 = a;
        double r398959 = 0.5;
        double r398960 = r398958 - r398959;
        double r398961 = sqrt(r398956);
        double r398962 = log(r398961);
        double r398963 = r398960 * r398962;
        double r398964 = cbrt(r398956);
        double r398965 = r398964 * r398964;
        double r398966 = sqrt(r398965);
        double r398967 = log(r398966);
        double r398968 = r398960 * r398967;
        double r398969 = sqrt(r398964);
        double r398970 = log(r398969);
        double r398971 = r398960 * r398970;
        double r398972 = r398968 + r398971;
        double r398973 = r398963 + r398972;
        double r398974 = r398957 + r398973;
        return r398974;
}

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-sqr-sqrt 0.3

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

4. Applied log-prod 0.3

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

5. Applied distribute-lft-in 0.3

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

7. Applied add-cube-cbrt 0.3

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

8. Applied sqrt-prod 0.3

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

9. Applied log-prod 0.3

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

10. Applied distribute-lft-in 0.3

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

12. Applied add-sqr-sqrt 0.3

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

13. Applied log-prod 0.3

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

14. Applied associate-+r+ 0.3

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

15. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019353 
(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))))