Average Error: 6.4 → 6.2
Time: 12.9s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{\sqrt{1 + z \cdot z} \cdot x}}{y}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{\sqrt{1 + z \cdot z} \cdot x}}{y}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r353269 = 1.0;
        double r353270 = x;
        double r353271 = r353269 / r353270;
        double r353272 = y;
        double r353273 = z;
        double r353274 = r353273 * r353273;
        double r353275 = r353269 + r353274;
        double r353276 = r353272 * r353275;
        double r353277 = r353271 / r353276;
        return r353277;
}
↓
double f(double x, double y, double z) {
        double r353278 = 1.0;
        double r353279 = z;
        double r353280 = r353279 * r353279;
        double r353281 = r353278 + r353280;
        double r353282 = sqrt(r353281);
        double r353283 = x;
        double r353284 = r353282 * r353283;
        double r353285 = r353278 / r353284;
        double r353286 = y;
        double r353287 = r353285 / r353286;
        double r353288 = r353287 / r353282;
        return r353288;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.4
Target5.8
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  3. Applied *-un-lft-identity6.4

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

  4. Applied add-sqr-sqrt6.4

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

  5. Applied times-frac6.4

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

  6. Applied times-frac6.4

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

  7. Simplified6.4

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

  9. Applied add-sqr-sqrt6.4

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

  10. Applied *-un-lft-identity6.4

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

  11. Applied *-un-lft-identity6.4

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

  12. Applied sqrt-prod6.4

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

  13. Applied times-frac6.4

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

  14. Applied times-frac6.4

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

  15. Applied associate-*r*6.0

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

  16. Simplified6.0

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

  18. Applied pow16.0

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

  19. Applied pow16.0

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

  20. Applied pow-prod-down6.0

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

  21. Simplified6.2

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

  22. Final simplification6.2

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

Reproduce

herbie shell --seed 2019326 
(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)))))