Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 31.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r190743 = x;
        double r190744 = y;
        double r190745 = r190743 + r190744;
        double r190746 = log(r190745);
        double r190747 = z;
        double r190748 = log(r190747);
        double r190749 = r190746 + r190748;
        double r190750 = t;
        double r190751 = r190749 - r190750;
        double r190752 = a;
        double r190753 = 0.5;
        double r190754 = r190752 - r190753;
        double r190755 = log(r190750);
        double r190756 = r190754 * r190755;
        double r190757 = r190751 + r190756;
        return r190757;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r190758 = a;
        double r190759 = 0.5;
        double r190760 = r190758 - r190759;
        double r190761 = t;
        double r190762 = cbrt(r190761);
        double r190763 = cbrt(r190762);
        double r190764 = log(r190763);
        double r190765 = 2.0;
        double r190766 = log(r190762);
        double r190767 = r190765 * r190766;
        double r190768 = r190762 * r190762;
        double r190769 = cbrt(r190768);
        double r190770 = log(r190769);
        double r190771 = r190767 + r190770;
        double r190772 = r190764 + r190771;
        double r190773 = r190760 * r190772;
        double r190774 = x;
        double r190775 = y;
        double r190776 = r190774 + r190775;
        double r190777 = log(r190776);
        double r190778 = r190773 + r190777;
        double r190779 = z;
        double r190780 = log(r190779);
        double r190781 = r190780 - r190761;
        double r190782 = r190778 + r190781;
        return r190782;
}

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

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

9. Applied cbrt-prod 0.3

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

10. Applied log-prod 0.3

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

11. Applied distribute-rgt-in 0.3

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

12. Applied associate-+r+ 0.3

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

13. Simplified 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019294 
(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))))