Average Error: 0.3 → 0.3
Time: 21.7s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\left(\frac{x}{y} + \left(\log z + \mathsf{fma}\left(\log t, a, \log y\right)\right)\right) - \mathsf{fma}\left(0.5, \log t + \sqrt[3]{{\left(\frac{x}{y}\right)}^{6}}, t\right) \]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\frac{x}{y} + \left(\log z + \mathsf{fma}\left(\log t, a, \log y\right)\right)\right) - \mathsf{fma}\left(0.5, \log t + \sqrt[3]{{\left(\frac{x}{y}\right)}^{6}}, 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
 (-
  (+ (/ x y) (+ (log z) (fma (log t) a (log y))))
  (fma 0.5 (+ (log t) (cbrt (pow (/ x y) 6.0))) 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 ((x / y) + (log(z) + fma(log(t), a, log(y)))) - fma(0.5, (log(t) + cbrt(pow((x / y), 6.0))), 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 x around 0 7.7

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

  4. Simplified0.3

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

  5. Applied add-cbrt-cube_binary640.3

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

  6. Simplified0.3

    \[\leadsto \left(\frac{x}{y} + \left(\log z + \mathsf{fma}\left(\log t, a, \log y\right)\right)\right) - \mathsf{fma}\left(0.5, \sqrt[3]{\color{blue}{{\left(\frac{x}{y}\right)}^{6}}} + \log t, t\right) \]

  7. Final simplification0.3

    \[\leadsto \left(\frac{x}{y} + \left(\log z + \mathsf{fma}\left(\log t, a, \log y\right)\right)\right) - \mathsf{fma}\left(0.5, \log t + \sqrt[3]{{\left(\frac{x}{y}\right)}^{6}}, t\right) \]

Reproduce

herbie shell --seed 2022097 
(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))))