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

Average Error: 6.3 → 5.9

Time: 13.4s

Precision: 64

Math

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

↓

\[\frac{\sqrt[3]{1} \cdot \frac{\sqrt[3]{1}}{x}}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\sqrt[3]{1}}} \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}\]

C

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

↓

double code(double x, double y, double z) {
	return (((cbrt(1.0) * (cbrt(1.0) / x)) / (sqrt(fma(z, z, 1.0)) / cbrt(1.0))) * ((1.0 / sqrt(fma(z, z, 1.0))) / y));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	5.6
Herbie	5.9

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

Derivation

1. Initial program 6.3

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

2. Simplified 6.3

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

4. Applied *-un-lft-identity 6.3

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

5. Applied add-sqr-sqrt 6.3

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

6. Applied *-un-lft-identity 6.3

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

7. Applied add-cube-cbrt 6.3

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

8. Applied times-frac 6.3

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

9. Applied times-frac 6.4

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

10. Applied times-frac 6.0

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

11. Simplified 6.0

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

13. Applied *-un-lft-identity 6.0

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

14. Applied div-inv 6.0

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

15. Applied times-frac 6.0

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

16. Applied associate-*r* 6.0

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

17. Simplified 5.9

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

18. Final simplification 5.9

    \[\leadsto \frac{\sqrt[3]{1} \cdot \frac{\sqrt[3]{1}}{x}}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\sqrt[3]{1}}} \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1 (* z z))) #f) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.680743250567252e+305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))