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

Average Error: 6.4 → 2.4

Time: 3.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) = -inf.0 \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \le 3.7697834214281562 \cdot 10^{307}\right):\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)} - \frac{1}{\left(y \cdot x\right) \cdot {z}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot \left(y \cdot \sqrt{1 + z \cdot z}\right)}\\ \end{array}\]

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) (y * ((double) (1.0 + ((double) (z * z)))))) <= -inf.0) || !(((double) (y * ((double) (1.0 + ((double) (z * z)))))) <= 3.769783421428156e+307))) {
		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 / x)) / ((double) (((double) sqrt(((double) (1.0 + ((double) (z * z)))))) * ((double) (y * ((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	6.4
Target	5.7
Herbie	2.4

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

Derivation

1. Split input into 2 regimes

2. if (* y (+ 1.0 (* z z))) < -inf.0 or 3.7697834214281562e307 < (* 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 add-sqr-sqrt 18.9

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

   4. Applied associate-*r* 18.9

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

   6. Applied *-un-lft-identity 18.9

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

   7. Applied *-un-lft-identity 18.9

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

   8. Applied times-frac 18.9

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

   9. Applied times-frac 16.0

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

   10. Simplified 16.0

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

   11. Simplified 16.0

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

   13. Applied associate-*r/ 16.0

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

   14. Simplified 16.0

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

   15. Taylor expanded around inf 18.9

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

   16. Simplified 6.6

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

   if -inf.0 < (* y (+ 1.0 (* z z))) < 3.7697834214281562e307

   1. Initial program 0.3

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

   3. Applied add-sqr-sqrt 0.3

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

   4. Applied associate-*r* 0.3

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) = -inf.0 \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \le 3.7697834214281562 \cdot 10^{307}\right):\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)} - \frac{1}{\left(y \cdot x\right) \cdot {z}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot \left(y \cdot \sqrt{1 + z \cdot z}\right)}\\ \end{array}\]

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)))))