Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.2 → 0.2

Time: 12.5s

Precision: 64

Math

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

↓

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

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(t), (a - 0.5), (1.0 * ((log((x + y)) + log(z)) - 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	0.2

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

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. Simplified 0.2

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

4. Applied *-un-lft-identity 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020056 +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))))