Average Error: 0.2 → 0.3
Time: 16.3s
Precision: binary64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\begin{array}{l} t_1 := \log \left(\sqrt{z}\right)\\ \left(\log \left(y + x\right) + \left(t_1 + \mathsf{fma}\left(\log t, -0.5, t_1\right)\right)\right) - \mathsf{fma}\left(a, -\log t, t\right) \end{array} \]
\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)\\
\left(\log \left(y + x\right) + \left(t_1 + \mathsf{fma}\left(\log t, -0.5, t_1\right)\right)\right) - \mathsf{fma}\left(a, -\log t, 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) -0.5 t_1))) (fma a (- (log t)) 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), -0.5, t_1))) - fma(a, -log(t), t);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.2
Target0.2
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 0.2

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

  2. Taylor expanded in t around inf 0.3

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

  3. Simplified0.2

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

  4. Applied add-sqr-sqrt_binary640.2

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

  5. Applied log-prod_binary640.2

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

  6. Applied associate-+l+_binary640.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2022088 
(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))))