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

Average Error: 15.0 → 2.1

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \le -3.73650829724365182 \cdot 10^{144}:\\ \;\;\;\;\left(\sqrt[3]{y} \cdot \left(\frac{\sqrt[3]{y}}{z} \cdot \left(x \cdot \frac{\sqrt[3]{1}}{z}\right)\right)\right) \cdot \frac{\sqrt[3]{y}}{z + 1}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \le 2.3814555202498264 \cdot 10^{-306}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\left(x \cdot \frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{z}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{y}}\right)}^{4}}{z}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{y}}}{z + 1}\\ \end{array}\]

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) (((double) (z * z)) * ((double) (z + 1.0)))) <= -3.736508297243652e+144)) {
		VAR = ((double) (((double) (((double) cbrt(y)) * ((double) (((double) (((double) cbrt(y)) / z)) * ((double) (x * ((double) (((double) cbrt(1.0)) / z)))))))) * ((double) (((double) cbrt(y)) / ((double) (z + 1.0))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (z * z)) * ((double) (z + 1.0)))) <= 2.3814555202498264e-306)) {
			VAR_1 = ((double) (((double) (x * ((double) (y / z)))) / ((double) (z * ((double) (z + 1.0))))));
		} else {
			VAR_1 = ((double) (((double) (((double) cbrt(((double) (((double) cbrt(y)) * ((double) cbrt(y)))))) * ((double) (((double) (((double) (x * ((double) (((double) cbrt(((double) (((double) cbrt(y)) * ((double) cbrt(y)))))) / z)))) * ((double) pow(((double) cbrt(((double) cbrt(y)))), 4.0)))) / z)))) * ((double) (((double) cbrt(((double) cbrt(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	15.0
Target	4.2
Herbie	2.1

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

Derivation

1. Split input into 3 regimes

2. if (* (* z z) (+ z 1.0)) < -3.73650829724365182e144

   1. Initial program 13.4

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

   2. Simplified 10.7

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

   4. Applied add-cube-cbrt 10.8

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

   5. Applied times-frac 6.9

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

   6. Applied associate-*r* 4.5

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

   7. Simplified 4.6

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

   9. Applied *-un-lft-identity 4.6

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

   10. Applied cbrt-prod 4.6

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

   11. Applied times-frac 2.7

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

   12. Applied associate-*r* 0.9

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

   13. Simplified 0.9

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

   if -3.73650829724365182e144 < (* (* z z) (+ z 1.0)) < 2.3814555202498264e-306

   1. Initial program 41.1

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

   2. Simplified 40.7

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

   4. Applied add-cube-cbrt 41.0

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

   5. Applied times-frac 19.3

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

   6. Applied associate-*r* 1.9

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

   7. Simplified 1.9

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

   9. Applied associate-*r/ 3.5

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

   10. Simplified 3.7

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

   if 2.3814555202498264e-306 < (* (* z z) (+ z 1.0))

   1. Initial program 9.3

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

   2. Simplified 8.2

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

   4. Applied add-cube-cbrt 8.7

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

   5. Applied times-frac 6.8

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

   6. Applied associate-*r* 2.9

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

   7. Simplified 2.9

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

   9. Applied add-cube-cbrt 2.9

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

   10. Applied cbrt-prod 3.0

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

   11. Applied times-frac 2.1

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

   12. Applied associate-*r* 1.5

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

   13. Simplified 1.9

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

   15. Applied add-cube-cbrt 2.0

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

   16. Applied cbrt-prod 2.1

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

   17. Applied associate-*l* 2.0

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

   18. Simplified 2.3

       \[\leadsto \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \color{blue}{\frac{\left(x \cdot \frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{z}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{y}}\right)}^{4}}{z}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{y}}}{z + 1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \le -3.73650829724365182 \cdot 10^{144}:\\ \;\;\;\;\left(\sqrt[3]{y} \cdot \left(\frac{\sqrt[3]{y}}{z} \cdot \left(x \cdot \frac{\sqrt[3]{1}}{z}\right)\right)\right) \cdot \frac{\sqrt[3]{y}}{z + 1}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \le 2.3814555202498264 \cdot 10^{-306}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\left(x \cdot \frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{z}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{y}}\right)}^{4}}{z}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{y}}}{z + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))))