Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Average Error: 14.8 → 1.8

Time: 2.1s

Precision: 64

Math

\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -6.0290209966276913 \cdot 10^{162}:\\ \;\;\;\;\frac{\frac{y}{z + 1}}{\frac{z}{x} \cdot z}\\ \mathbf{elif}\;x \cdot y \le -1.619015216158531 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{\frac{z}{x}}}{z + 1}\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((x * y) / ((z * z) * (z + 1.0)));
}

↓

double code(double x, double y, double z) {
	double VAR;
	if (((x * y) <= -6.029020996627691e+162)) {
		VAR = ((y / (z + 1.0)) / ((z / x) * z));
	} else {
		double VAR_1;
		if (((x * y) <= -1.6190152161585307e-110)) {
			VAR_1 = (((x * y) / z) / (z * (z + 1.0)));
		} else {
			VAR_1 = (((y / z) / (z / x)) / (z + 1.0));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	14.8
Target	3.9
Herbie	1.8

\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* x y) < -6.029020996627691e+162

   1. Initial program 34.4

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
   2. Using strategy rm

   3. Applied associate-*l* 34.4

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}}\]

   4. Applied times-frac 10.0

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}}\]
   5. Using strategy rm

   6. Applied *-commutative 10.0

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\left(z + 1\right) \cdot z}}\]

   7. Applied associate-/r* 1.8

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z + 1}}{z}}\]

   8. Applied clear-num 1.9

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{\frac{y}{z + 1}}{z}\]

   9. Applied frac-times 1.5

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{z + 1}}{\frac{z}{x} \cdot z}}\]

   10. Simplified 1.5

       \[\leadsto \frac{\color{blue}{\frac{y}{z + 1}}}{\frac{z}{x} \cdot z}\]

   if -6.029020996627691e+162 < (* x y) < -1.6190152161585307e-110

   1. Initial program 3.4

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
   2. Using strategy rm

   3. Applied associate-*l* 3.4

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}}\]

   4. Applied associate-/r* 0.2

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z}}{z \cdot \left(z + 1\right)}}\]

   if -1.6190152161585307e-110 < (* x y)

   1. Initial program 15.2

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
   2. Using strategy rm

   3. Applied associate-*l* 15.2

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}}\]

   4. Applied times-frac 3.9

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}}\]
   5. Using strategy rm

   6. Applied associate-/r* 2.8

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z + 1}}\]

   7. Applied associate-*r/ 2.3

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1}}\]
   8. Using strategy rm

   9. Applied clear-num 2.4

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{y}{z}}{z + 1}\]

   10. Applied associate-*l/ 2.2

       \[\leadsto \frac{\color{blue}{\frac{1 \cdot \frac{y}{z}}{\frac{z}{x}}}}{z + 1}\]

   11. Simplified 2.2

       \[\leadsto \frac{\frac{\color{blue}{\frac{y}{z}}}{\frac{z}{x}}}{z + 1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -6.0290209966276913 \cdot 10^{162}:\\ \;\;\;\;\frac{\frac{y}{z + 1}}{\frac{z}{x} \cdot z}\\ \mathbf{elif}\;x \cdot y \le -1.619015216158531 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{\frac{z}{x}}}{z + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1))))