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

Average Error: 6.3 → 5.9

Time: 9.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r293269 = 1.0;
        double r293270 = x;
        double r293271 = r293269 / r293270;
        double r293272 = y;
        double r293273 = z;
        double r293274 = r293273 * r293273;
        double r293275 = r293269 + r293274;
        double r293276 = r293272 * r293275;
        double r293277 = r293271 / r293276;
        return r293277;
}

↓

double f(double x, double y, double z) {
        double r293278 = 1.0;
        double r293279 = sqrt(r293278);
        double r293280 = x;
        double r293281 = r293279 / r293280;
        double r293282 = r293279 * r293281;
        double r293283 = z;
        double r293284 = fma(r293283, r293283, r293278);
        double r293285 = sqrt(r293284);
        double r293286 = r293282 / r293285;
        double r293287 = y;
        double r293288 = r293286 / r293287;
        double r293289 = r293288 / r293285;
        return r293289;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	5.7
Herbie	5.9

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

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-sqr-sqrt 6.3

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

8. Applied times-frac 6.3

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

9. Applied times-frac 6.3

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

10. Applied times-frac 5.9

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

11. Simplified 5.9

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

13. Applied pow1 5.9

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

14. Applied pow1 5.9

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

15. Applied pow-prod-down 5.9

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

16. Simplified 5.9

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

17. Final simplification 5.9

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

Reproduce

herbie shell --seed 2020001 +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)))))