Average Error: 0.3 → 0.3
Time: 16.4s
Precision: binary64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\log \left(x + y\right) + a \cdot \log t\right) - \left(t + \left(\log t \cdot 0.5 + \log \left(\frac{1}{z}\right)\right)\right)\]
\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) + a \cdot \log t\right) - \left(t + \left(\log t \cdot 0.5 + \log \left(\frac{1}{z}\right)\right)\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)) (* a (log t))) (+ t (+ (* (log t) 0.5) (log (/ 1.0 z))))))
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) + (a * log(t))) - (t + ((log(t) * 0.5) + log(1.0 / z)));
}

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

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. Taylor expanded around inf 0.3

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2021044 
(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))))