Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 25.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\mathsf{fma}\left(a - 0.5, 2 \cdot \log \left(\sqrt{\sqrt[3]{t}} \cdot \sqrt{\sqrt[3]{t}}\right), \left(\log z - t\right) + \log \left(x + y\right)\right) + \left(a - 0.5\right) \cdot \log \left({t}^{\frac{1}{3}}\right)\]

C

double f(double x, double y, double z, double t, double a) {
        double r344777 = x;
        double r344778 = y;
        double r344779 = r344777 + r344778;
        double r344780 = log(r344779);
        double r344781 = z;
        double r344782 = log(r344781);
        double r344783 = r344780 + r344782;
        double r344784 = t;
        double r344785 = r344783 - r344784;
        double r344786 = a;
        double r344787 = 0.5;
        double r344788 = r344786 - r344787;
        double r344789 = log(r344784);
        double r344790 = r344788 * r344789;
        double r344791 = r344785 + r344790;
        return r344791;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r344792 = a;
        double r344793 = 0.5;
        double r344794 = r344792 - r344793;
        double r344795 = 2.0;
        double r344796 = t;
        double r344797 = cbrt(r344796);
        double r344798 = sqrt(r344797);
        double r344799 = r344798 * r344798;
        double r344800 = log(r344799);
        double r344801 = r344795 * r344800;
        double r344802 = z;
        double r344803 = log(r344802);
        double r344804 = r344803 - r344796;
        double r344805 = x;
        double r344806 = y;
        double r344807 = r344805 + r344806;
        double r344808 = log(r344807);
        double r344809 = r344804 + r344808;
        double r344810 = fma(r344794, r344801, r344809);
        double r344811 = 0.3333333333333333;
        double r344812 = pow(r344796, r344811);
        double r344813 = log(r344812);
        double r344814 = r344794 * r344813;
        double r344815 = r344810 + r344814;
        return r344815;
}

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	0.2
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. Applied associate-+r+ 0.3

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

7. Simplified 0.3

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

9. Applied pow1/3 0.3

   \[\leadsto \mathsf{fma}\left(a - 0.5, 2 \cdot \log \left(\sqrt[3]{t}\right), \left(\log z - t\right) + \log \left(x + y\right)\right) + \left(a - 0.5\right) \cdot \log \color{blue}{\left({t}^{\frac{1}{3}}\right)}\]
10. Using strategy rm

11. Applied add-sqr-sqrt 0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, 2 \cdot \log \color{blue}{\left(\sqrt{\sqrt[3]{t}} \cdot \sqrt{\sqrt[3]{t}}\right)}, \left(\log z - t\right) + \log \left(x + y\right)\right) + \left(a - 0.5\right) \cdot \log \left({t}^{\frac{1}{3}}\right)\]

12. Final simplification 0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, 2 \cdot \log \left(\sqrt{\sqrt[3]{t}} \cdot \sqrt{\sqrt[3]{t}}\right), \left(\log z - t\right) + \log \left(x + y\right)\right) + \left(a - 0.5\right) \cdot \log \left({t}^{\frac{1}{3}}\right)\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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))))