Average Error: 0.3 → 0.3
Time: 13.9s
Precision: binary64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(y + x\right) + 3 \cdot \log \left({\left(\frac{1}{z}\right)}^{-0.3333333333333333}\right)\right) - t\right) \]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(y + x\right) + 3 \cdot \log \left({\left(\frac{1}{z}\right)}^{-0.3333333333333333}\right)\right) - t\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
 (fma
  (- a 0.5)
  (log t)
  (- (+ (log (+ y x)) (* 3.0 (log (pow (/ 1.0 z) -0.3333333333333333)))) 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) {
	return fma((a - 0.5), log(t), ((log(y + x) + (3.0 * log(pow((1.0 / z), -0.3333333333333333)))) - 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. Applied add-cube-cbrt_binary640.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(x + y\right) + \log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right) - t\right) \]

  4. Applied log-prod_binary640.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(x + y\right) + \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)}\right) - t\right) \]

  5. Applied associate-+r+_binary640.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \color{blue}{\left(\left(\log \left(x + y\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(\sqrt[3]{z}\right)\right)} - t\right) \]

  6. Simplified0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\color{blue}{\left(\log \left(y + x\right) + 2 \cdot \log \left(\sqrt[3]{z}\right)\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right) \]

  7. Taylor expanded in z around inf 0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \color{blue}{\left(\log \left(y + x\right) + 3 \cdot \log \left({\left(\frac{1}{z}\right)}^{-0.3333333333333333}\right)\right)} - t\right) \]

  8. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(y + x\right) + 3 \cdot \log \left({\left(\frac{1}{z}\right)}^{-0.3333333333333333}\right)\right) - t\right) \]

Reproduce

herbie shell --seed 2022081 
(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))))