Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 5.9

Time: 8.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \le -733.828577951272678:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right) + \log z\right)}^{3} - {t}^{3}}{\mathsf{fma}\left(\log \left(x + y\right) + \log z, \left(\log \left(x + y\right) + \log z\right) + t, t \cdot t\right)}\right)\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \le 635.802165622473353:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(\left(x + y\right) \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \log t, \log z\right) + \left(\log y - \mathsf{fma}\left(\log t, 0.5, t\right)\right)\\ \end{array}\]

C

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 VAR;
	if (((log((x + y)) + log(z)) <= -733.8285779512727)) {
		VAR = fma(log(t), (a - 0.5), ((pow((log((x + y)) + log(z)), 3.0) - pow(t, 3.0)) / fma((log((x + y)) + log(z)), ((log((x + y)) + log(z)) + t), (t * t))));
	} else {
		double VAR_1;
		if (((log((x + y)) + log(z)) <= 635.8021656224734)) {
			VAR_1 = fma(log(t), (a - 0.5), (log(((x + y) * z)) - t));
		} else {
			VAR_1 = (fma(a, log(t), log(z)) + (log(y) - fma(log(t), 0.5, t)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.3
Target	0.3
Herbie	5.9

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

Derivation

1. Split input into 3 regimes

2. if (+ (log (+ x y)) (log z)) < -733.8285779512727

   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. Simplified 0.2

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

   4. Applied flip3-- 20.5

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

   5. Simplified 20.5

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

   if -733.8285779512727 < (+ (log (+ x y)) (log z)) < 635.8021656224734

   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. Simplified 0.3

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

   4. Applied sum-log 0.2

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

   if 635.8021656224734 < (+ (log (+ x y)) (log z))

   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. Simplified 0.2

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

   3. Taylor expanded around 0 19.7

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

   4. Simplified 19.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \log t, \log z\right) + \left(\log y - \mathsf{fma}\left(\log t, 0.5, t\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \le -733.828577951272678:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right) + \log z\right)}^{3} - {t}^{3}}{\mathsf{fma}\left(\log \left(x + y\right) + \log z, \left(\log \left(x + y\right) + \log z\right) + t, t \cdot t\right)}\right)\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \le 635.802165622473353:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(\left(x + y\right) \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \log t, \log z\right) + \left(\log y - \mathsf{fma}\left(\log t, 0.5, t\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(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))))