Average Error: 6.5 → 6.5
Time: 2.4s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{y \cdot \mathsf{fma}\left(z, z, 1\right)}}{x}}{\frac{1}{1}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{y \cdot \mathsf{fma}\left(z, z, 1\right)}}{x}}{\frac{1}{1}}
double f(double x, double y, double z) {
        double r334443 = 1.0;
        double r334444 = x;
        double r334445 = r334443 / r334444;
        double r334446 = y;
        double r334447 = z;
        double r334448 = r334447 * r334447;
        double r334449 = r334443 + r334448;
        double r334450 = r334446 * r334449;
        double r334451 = r334445 / r334450;
        return r334451;
}
↓
double f(double x, double y, double z) {
        double r334452 = 1.0;
        double r334453 = y;
        double r334454 = z;
        double r334455 = 1.0;
        double r334456 = fma(r334454, r334454, r334455);
        double r334457 = r334453 * r334456;
        double r334458 = r334452 / r334457;
        double r334459 = x;
        double r334460 = r334458 / r334459;
        double r334461 = r334452 / r334455;
        double r334462 = r334460 / r334461;
        return r334462;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.5
Target5.8
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  3. Applied *-un-lft-identity6.5

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

  4. Applied *-un-lft-identity6.5

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

  5. Applied times-frac6.5

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

  6. Applied associate-/l*6.8

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

  7. Simplified6.8

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

  9. Applied div-inv6.8

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

  10. Applied associate-/r*6.8

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

  11. Simplified6.5

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

  12. Final simplification6.5

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

Reproduce

herbie shell --seed 2020064 +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)))))