Average Error: 6.3 → 6.0
Time: 4.8s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x}
double f(double x, double y, double z) {
        double r289223 = 1.0;
        double r289224 = x;
        double r289225 = r289223 / r289224;
        double r289226 = y;
        double r289227 = z;
        double r289228 = r289227 * r289227;
        double r289229 = r289223 + r289228;
        double r289230 = r289226 * r289229;
        double r289231 = r289225 / r289230;
        return r289231;
}
↓
double f(double x, double y, double z) {
        double r289232 = 1.0;
        double r289233 = z;
        double r289234 = fma(r289233, r289233, r289232);
        double r289235 = sqrt(r289234);
        double r289236 = r289232 / r289235;
        double r289237 = y;
        double r289238 = r289236 / r289237;
        double r289239 = x;
        double r289240 = r289235 * r289239;
        double r289241 = r289238 / r289240;
        return r289241;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.3
Target5.6
Herbie6.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  2. Simplified6.3

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

  4. Applied add-sqr-sqrt6.3

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

  5. Applied div-inv6.3

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

  6. Applied times-frac6.3

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

  7. Applied associate-/l*6.2

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

  8. Simplified6.3

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

  10. Applied associate-/r*6.0

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

  11. Final simplification6.0

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

Reproduce

herbie shell --seed 2020002 +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)))))