Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.2 → 0.3

Time: 11.8s

Precision: 64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.2
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 0.2

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.2

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

4. Applied log-prod 0.2

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

5. Applied distribute-lft-in 0.2

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

6. Applied associate-+r+ 0.3

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

7. Simplified 0.3

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

9. Applied add-cube-cbrt 0.3

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

10. Applied log-prod 0.3

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

11. Applied distribute-lft-in 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020092 +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))))