Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 6.5s

Precision: binary64

Math

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

↓

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

C

double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (((double) (((double) log(((double) (x + y)))) + ((double) log(z)))) - t)) + ((double) (((double) (a - 0.5)) * ((double) log(t))))));
}

↓

double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (((double) (((double) log(((double) (x + y)))) + ((double) log(z)))) - t)) + ((double) (((double) (((double) (a - 0.5)) * ((double) (((double) (((double) log(((double) cbrt(t)))) * 2.0)) + ((double) log(((double) cbrt(((double) pow(((double) cbrt(t)), 2.0)))))))))) + ((double) (((double) (a - 0.5)) * ((double) log(((double) cbrt(((double) cbrt(t))))))))))));
}

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

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

8. Applied add-cube-cbrt 0.3

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

9. Applied cbrt-prod 0.3

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

10. Applied log-prod 0.3

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

11. Applied distribute-lft-in 0.3

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

12. Applied associate-+r+ 0.3

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

13. Simplified 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020184 
(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))))