Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 38.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r231732 = x;
        double r231733 = y;
        double r231734 = r231732 + r231733;
        double r231735 = log(r231734);
        double r231736 = z;
        double r231737 = log(r231736);
        double r231738 = r231735 + r231737;
        double r231739 = t;
        double r231740 = r231738 - r231739;
        double r231741 = a;
        double r231742 = 0.5;
        double r231743 = r231741 - r231742;
        double r231744 = log(r231739);
        double r231745 = r231743 * r231744;
        double r231746 = r231740 + r231745;
        return r231746;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r231747 = 2.0;
        double r231748 = t;
        double r231749 = cbrt(r231748);
        double r231750 = log(r231749);
        double r231751 = r231747 * r231750;
        double r231752 = a;
        double r231753 = 0.5;
        double r231754 = r231752 - r231753;
        double r231755 = z;
        double r231756 = cbrt(r231755);
        double r231757 = log(r231756);
        double r231758 = x;
        double r231759 = y;
        double r231760 = r231758 + r231759;
        double r231761 = log(r231760);
        double r231762 = fma(r231747, r231757, r231761);
        double r231763 = r231757 - r231748;
        double r231764 = r231762 + r231763;
        double r231765 = fma(r231751, r231754, r231764);
        double r231766 = 1.0;
        double r231767 = r231766 / r231748;
        double r231768 = -0.3333333333333333;
        double r231769 = pow(r231767, r231768);
        double r231770 = log(r231769);
        double r231771 = r231754 * r231770;
        double r231772 = r231765 + r231771;
        return r231772;
}

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. 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

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

9. Applied add-cube-cbrt 0.3

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

10. Applied log-prod 0.3

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

11. Applied associate--l+ 0.3

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

12. Applied associate-+r+ 0.3

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

13. Simplified 0.3

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

14. Taylor expanded around inf 0.3

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

15. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019322 +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))))