Average Error: 0.3 → 0.4
Time: 18.6s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\frac{x}{y} + \left(\left(\left(\log z + \log t \cdot \left(-0.5 + a\right)\right) - t\right) + \log y\right) \]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\frac{x}{y} + \left(\left(\left(\log z + \log t \cdot \left(-0.5 + a\right)\right) - t\right) + \log y\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
 (+ (/ x y) (+ (- (+ (log z) (* (log t) (+ -0.5 a))) t) (log y))))
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 (x / y) + (((log(z) + (log(t) * (-0.5 + a))) - t) + log(y));
}

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
Herbie0.4
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right) \]

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 in x around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2022130 
(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))))