Average Error: 0.3 → 0.2
Time: 9.5s
Precision: binary64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\log \left(x + y\right) + \left(\log z + 0.5 \cdot \log \left(\frac{1}{t}\right)\right)\right) - \left(t + \log \left(\frac{1}{t}\right) \cdot a\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\log \left(x + y\right) + \left(\log z + 0.5 \cdot \log \left(\frac{1}{t}\right)\right)\right) - \left(t + \log \left(\frac{1}{t}\right) \cdot a\right)
(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
↓
(FPCore (x y z t a)
 :precision binary64
 (-
  (+ (log (+ x y)) (+ (log z) (* 0.5 (log (/ 1.0 t)))))
  (+ t (* (log (/ 1.0 t)) a))))
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 (log(x + y) + (log(z) + (0.5 * log(1.0 / t)))) - (t + (log(1.0 / t) * a));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.3
Target0.3
Herbie0.2
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

Derivation

  1. Initial program 0.3

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

  2. Taylor expanded around inf 0.2

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

  3. Final simplification0.2

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))