Average Error: 6.7 → 6.2
Time: 8.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}
double f(double x, double y, double z) {
        double r376171 = 1.0;
        double r376172 = x;
        double r376173 = r376171 / r376172;
        double r376174 = y;
        double r376175 = z;
        double r376176 = r376175 * r376175;
        double r376177 = r376171 + r376176;
        double r376178 = r376174 * r376177;
        double r376179 = r376173 / r376178;
        return r376179;
}
↓
double f(double x, double y, double z) {
        double r376180 = 1.0;
        double r376181 = x;
        double r376182 = r376180 / r376181;
        double r376183 = z;
        double r376184 = fma(r376183, r376183, r376180);
        double r376185 = sqrt(r376184);
        double r376186 = r376182 / r376185;
        double r376187 = 1.0;
        double r376188 = y;
        double r376189 = r376187 / r376188;
        double r376190 = r376189 / r376185;
        double r376191 = r376186 * r376190;
        return r376191;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.7
Target6.1
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.7

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

  2. Simplified6.9

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

  4. Applied add-sqr-sqrt6.9

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

  5. Applied div-inv6.9

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

  6. Applied times-frac6.2

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

  7. Final simplification6.2

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

Reproduce

herbie shell --seed 2020046 +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)))))