Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Average Error: 6.4 → 3.8

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 2.01365834726829764 \cdot 10^{300}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(z \cdot z + 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)} - \frac{1}{x \cdot \left(y \cdot {z}^{4}\right)}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((((double) (z * z)) <= 2.0136583472682976e+300)) {
		VAR = ((double) (((double) (1.0 / ((double) (x * ((double) (((double) (z * z)) + 1.0)))))) / y));
	} else {
		VAR = ((double) (((double) (1.0 / ((double) (x * ((double) (z * ((double) (z * y)))))))) - ((double) (1.0 / ((double) (x * ((double) (y * ((double) pow(z, 4.0))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	5.7
Herbie	3.8

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -inf.0:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* z z) < 2.01365834726829764e300

   1. Initial program 2.3

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

   3. Applied *-un-lft-identity 2.3

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

   4. Applied *-un-lft-identity 2.3

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

   5. Applied times-frac 2.3

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

   6. Applied times-frac 2.0

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

   7. Simplified 2.0

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

   8. Simplified 2.2

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

   10. Applied associate-*l/ 2.1

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

   11. Simplified 2.1

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

   if 2.01365834726829764e300 < (* z z)

   1. Initial program 17.5

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

   3. Applied *-un-lft-identity 17.5

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

   4. Applied *-un-lft-identity 17.5

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

   5. Applied times-frac 17.5

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

   6. Applied times-frac 17.7

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

   7. Simplified 17.7

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

   8. Simplified 17.7

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

   10. Applied associate-*l/ 17.7

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

   11. Simplified 17.7

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

   12. Taylor expanded around inf 17.5

       \[\leadsto \color{blue}{1 \cdot \frac{1}{x \cdot \left({z}^{2} \cdot y\right)} - 1 \cdot \frac{1}{x \cdot \left({z}^{4} \cdot y\right)}}\]

   13. Simplified 8.0

       \[\leadsto \color{blue}{\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)} - \frac{1}{x \cdot \left(y \cdot {z}^{4}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 2.01365834726829764 \cdot 10^{300}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(z \cdot z + 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)} - \frac{1}{x \cdot \left(y \cdot {z}^{4}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

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