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

Average Error: 15.1 → 1.8

Time: 2.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 6.83050355077890696 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sqrt{1}}{1} \cdot \frac{\frac{x}{z}}{\frac{z}{\frac{y}{z + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{z} \cdot \left(\frac{\sqrt{x}}{z} \cdot \frac{y}{z + 1}\right)\\ \end{array}\]

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 temp;
	if ((x <= 6.830503550778907e-289)) {
		temp = ((sqrt(1.0) / 1.0) * ((x / z) / (z / (y / (z + 1.0)))));
	} else {
		temp = ((sqrt(x) / z) * ((sqrt(x) / z) * (y / (z + 1.0))));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.1
Target	4.0
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 2 regimes

2. if x < 6.830503550778907e-289

   1. Initial program 15.5

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

   3. Applied times-frac 11.3

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

   5. Applied *-un-lft-identity 11.3

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

   6. Applied times-frac 6.1

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

   7. Applied associate-*l* 2.6

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

   9. Applied *-un-lft-identity 2.6

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

   10. Applied add-sqr-sqrt 2.6

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

   11. Applied times-frac 2.6

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

   12. Applied associate-*l* 2.6

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

   13. Simplified 2.6

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

   15. Applied associate-/l* 3.1

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

   if 6.830503550778907e-289 < x

   1. Initial program 14.8

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

   3. Applied times-frac 10.8

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

   5. Applied add-sqr-sqrt 10.9

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

   6. Applied times-frac 6.1

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

   7. Applied associate-*l* 0.5

      \[\leadsto \color{blue}{\frac{\sqrt{x}}{z} \cdot \left(\frac{\sqrt{x}}{z} \cdot \frac{y}{z + 1}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 6.83050355077890696 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sqrt{1}}{1} \cdot \frac{\frac{x}{z}}{\frac{z}{\frac{y}{z + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{z} \cdot \left(\frac{\sqrt{x}}{z} \cdot \frac{y}{z + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(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))))