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

Average Error: 15.2 → 2.3

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.7270956690352767 \cdot 10^{-186}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z + 1}}{z}}{z}\\ \mathbf{elif}\;x \cdot y \le 1.1176947562290144 \cdot 10^{258}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((((double) (x * y)) <= -1.7270956690352767e-186)) {
		VAR = ((double) (((double) (((double) (x * ((double) (y / ((double) (z + 1.0)))))) / z)) / z));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= 1.1176947562290144e+258)) {
			VAR_1 = ((double) (((double) (((double) (x / z)) * y)) / ((double) (z * ((double) (z + 1.0))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (x / z)) / z)) * ((double) (y / ((double) (z + 1.0))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.2
Target	4.1
Herbie	2.3

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -1.7270956690352767e-186

   1. Initial program 12.4

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

   3. Applied times-frac 10.3

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

   5. Applied add-cube-cbrt 10.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{z \cdot z} \cdot \frac{y}{z + 1}\]

   6. Applied times-frac 7.2

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\sqrt[3]{x}}{z}\right)} \cdot \frac{y}{z + 1}\]

   7. Applied associate-*l* 1.5

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)}\]
   8. Using strategy rm

   9. Applied associate-*l/ 1.5

      \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \color{blue}{\frac{\sqrt[3]{x} \cdot \frac{y}{z + 1}}{z}}\]

   10. Applied associate-*r/ 1.9

       \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\sqrt[3]{x} \cdot \frac{y}{z + 1}\right)}{z}}\]

   11. Simplified 2.1

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

   if -1.7270956690352767e-186 < (* x y) < 1.1176947562290144e258

   1. Initial program 13.3

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

   3. Applied times-frac 11.4

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

   5. Applied *-un-lft-identity 11.4

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

   6. Applied times-frac 6.2

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

   7. Applied associate-*l* 2.5

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

   9. Applied associate-*r/ 2.3

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

   10. Applied frac-times 2.5

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

   11. Simplified 2.5

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

   if 1.1176947562290144e258 < (* x y)

   1. Initial program 51.1

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

   3. Applied times-frac 15.2

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

   5. Applied associate-/r* 1.3

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.7270956690352767 \cdot 10^{-186}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z + 1}}{z}}{z}\\ \mathbf{elif}\;x \cdot y \le 1.1176947562290144 \cdot 10^{258}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(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) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))