Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 11.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r314833 = x;
        double r314834 = y;
        double r314835 = r314833 + r314834;
        double r314836 = log(r314835);
        double r314837 = z;
        double r314838 = log(r314837);
        double r314839 = r314836 + r314838;
        double r314840 = t;
        double r314841 = r314839 - r314840;
        double r314842 = a;
        double r314843 = 0.5;
        double r314844 = r314842 - r314843;
        double r314845 = log(r314840);
        double r314846 = r314844 * r314845;
        double r314847 = r314841 + r314846;
        return r314847;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r314848 = x;
        double r314849 = y;
        double r314850 = r314848 + r314849;
        double r314851 = log(r314850);
        double r314852 = z;
        double r314853 = log(r314852);
        double r314854 = r314851 + r314853;
        double r314855 = t;
        double r314856 = r314854 - r314855;
        double r314857 = a;
        double r314858 = 0.5;
        double r314859 = r314857 - r314858;
        double r314860 = 2.0;
        double r314861 = cbrt(r314855);
        double r314862 = log(r314861);
        double r314863 = r314860 * r314862;
        double r314864 = cbrt(r314861);
        double r314865 = r314864 * r314864;
        double r314866 = log(r314865);
        double r314867 = r314863 + r314866;
        double r314868 = r314859 * r314867;
        double r314869 = log(r314864);
        double r314870 = r314869 * r314859;
        double r314871 = r314868 + r314870;
        double r314872 = r314856 + r314871;
        return r314872;
}

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

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

9. Applied log-prod 0.3

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

10. Applied distribute-rgt-in 0.3

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

11. Applied associate-+r+ 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020036 
(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))))