Average Error: 0.3 → 0.3
Time: 21.2s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\left(\frac{x}{y} + \mathsf{fma}\left(a, \log t, \log z\right)\right) - \left(t + \left(0.5 \cdot \left(\log t + \frac{x}{y} \cdot \frac{x}{y}\right) - \log y\right)\right) \]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\frac{x}{y} + \mathsf{fma}\left(a, \log t, \log z\right)\right) - \left(t + \left(0.5 \cdot \left(\log t + \frac{x}{y} \cdot \frac{x}{y}\right) - \log y\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
 (-
  (+ (/ x y) (fma a (log t) (log z)))
  (+ t (- (* 0.5 (+ (log t) (* (/ x y) (/ x y)))) (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) + fma(a, log(t), log(z))) - (t + ((0.5 * (log(t) + ((x / y) * (x / y)))) - log(y)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

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. Applied associate--l+_binary640.3

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

  3. Taylor expanded in y around inf 7.8

    \[\leadsto \color{blue}{\left(\frac{x}{y} + \left(\log z + a \cdot \log t\right)\right) - \left(t + \left(0.5 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(0.5 \cdot \log t + \log \left(\frac{1}{y}\right)\right)\right)\right)} \]

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2021280 
(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))))