Average Error: 6.1 → 6.2
Time: 12.2s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{1 + z \cdot z}}{y \cdot x}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{1 + z \cdot z}}{y \cdot x}
double f(double x, double y, double z) {
        double r260362 = 1.0;
        double r260363 = x;
        double r260364 = r260362 / r260363;
        double r260365 = y;
        double r260366 = z;
        double r260367 = r260366 * r260366;
        double r260368 = r260362 + r260367;
        double r260369 = r260365 * r260368;
        double r260370 = r260364 / r260369;
        return r260370;
}
↓
double f(double x, double y, double z) {
        double r260371 = 1.0;
        double r260372 = z;
        double r260373 = r260372 * r260372;
        double r260374 = r260371 + r260373;
        double r260375 = r260371 / r260374;
        double r260376 = y;
        double r260377 = x;
        double r260378 = r260376 * r260377;
        double r260379 = r260375 / r260378;
        return r260379;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original6.1
Target5.5
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 6.1

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

  3. Applied *-un-lft-identity6.1

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

  4. Applied add-sqr-sqrt6.1

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

  5. Applied times-frac6.1

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

  6. Applied times-frac6.0

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

  7. Simplified6.0

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

  9. Applied associate-*l/6.0

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

  10. Simplified6.0

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

  12. Applied *-un-lft-identity6.0

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

  13. Applied add-sqr-sqrt6.0

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

  14. Applied div-inv6.0

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

  15. Applied times-frac6.0

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

  16. Applied times-frac5.7

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

  17. Simplified5.7

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

  18. Final simplification6.2

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

Reproduce

herbie shell --seed 2019294 
(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))) -inf.bf) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.68074325056725162e305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

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