Average Error: 0.3 → 0.3
Time: 19.2s
Precision: binary64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\begin{array}{l} t_1 := \log \left(\sqrt{z}\right)\\ \log \left(y + x\right) + \left(\left(t_1 + \mathsf{fma}\left(\log t, a - 0.5, t_1\right)\right) - t\right) \end{array} \]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\begin{array}{l}
t_1 := \log \left(\sqrt{z}\right)\\
\log \left(y + x\right) + \left(\left(t_1 + \mathsf{fma}\left(\log t, a - 0.5, t_1\right)\right) - t\right)
\end{array}

(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
 (let* ((t_1 (log (sqrt z))))
   (+ (log (+ y x)) (- (+ t_1 (fma (log t) (- a 0.5) t_1)) t))))
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) {
	double t_1 = log(sqrt(z));
	return log((y + x)) + ((t_1 + fma(log(t), (a - 0.5), t_1)) - t);
}

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. Simplified0.3

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

  3. Taylor expanded in a around 0 0.3

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

  4. Simplified0.3

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

  5. Applied add-sqr-sqrt_binary640.3

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

  6. Applied log-prod_binary640.3

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

  7. Applied associate-+l+_binary640.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2022125 
(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))))