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

Average Error: 6.7 → 3.1

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) = -inf.0 \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \le 1.09633384586651499 \cdot 10^{304}\right):\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)} - \frac{1}{\left(y \cdot x\right) \cdot {z}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \sqrt{1 + z \cdot z}}}{x \cdot \sqrt{1 + z \cdot z}}\\ \end{array}\]

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) (y * ((double) (1.0 + ((double) (z * z)))))) <= -inf.0) || !(((double) (y * ((double) (1.0 + ((double) (z * z)))))) <= 1.096333845866515e+304))) {
		VAR = ((double) (((double) (1.0 / ((double) (z * ((double) (z * ((double) (y * x)))))))) - ((double) (1.0 / ((double) (((double) (y * x)) * ((double) pow(z, 4.0))))))));
	} else {
		VAR = ((double) (((double) (1.0 / ((double) (y * ((double) sqrt(((double) (1.0 + ((double) (z * z)))))))))) / ((double) (x * ((double) sqrt(((double) (1.0 + ((double) (z * z))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.7
Target	5.9
Herbie	3.1

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

Derivation

1. Split input into 2 regimes

2. if (* y (+ 1.0 (* z z))) < -inf.0 or 1.09633384586651499e304 < (* y (+ 1.0 (* z z)))

   1. Initial program 18.9

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

   3. Applied associate-/r* 14.5

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

   4. Simplified 14.5

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

   6. Applied add-sqr-sqrt 14.5

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

   7. Applied add-cube-cbrt 14.5

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

   8. Applied times-frac 14.5

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

   9. Applied times-frac 15.9

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

   10. Simplified 15.9

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

   11. Simplified 16.1

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

   13. Applied frac-times 16.1

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

   14. Applied associate-*l/ 16.1

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

   15. Simplified 16.1

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

   16. Taylor expanded around inf 19.1

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

   17. Simplified 7.0

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

   if -inf.0 < (* y (+ 1.0 (* z z))) < 1.09633384586651499e304

   1. Initial program 0.3

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

   3. Applied associate-/r* 2.7

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

   4. Simplified 2.8

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

   6. Applied add-sqr-sqrt 2.8

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

   7. Applied add-cube-cbrt 2.8

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

   8. Applied times-frac 2.8

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

   9. Applied times-frac 1.0

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

   10. Simplified 1.0

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

   11. Simplified 1.0

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

   13. Applied frac-times 1.1

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

   14. Applied associate-*l/ 1.0

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

   15. Simplified 1.0

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) = -inf.0 \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \le 1.09633384586651499 \cdot 10^{304}\right):\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)} - \frac{1}{\left(y \cdot x\right) \cdot {z}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \sqrt{1 + z \cdot z}}}{x \cdot \sqrt{1 + z \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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)))))