Average Error: 6.5 → 5.8
Time: 6.2s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}{\frac{y}{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}{\frac{y}{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}}
double f(double x, double y, double z) {
        double r301257 = 1.0;
        double r301258 = x;
        double r301259 = r301257 / r301258;
        double r301260 = y;
        double r301261 = z;
        double r301262 = r301261 * r301261;
        double r301263 = r301257 + r301262;
        double r301264 = r301260 * r301263;
        double r301265 = r301259 / r301264;
        return r301265;
}
↓
double f(double x, double y, double z) {
        double r301266 = 1.0;
        double r301267 = sqrt(r301266);
        double r301268 = x;
        double r301269 = cbrt(r301268);
        double r301270 = r301269 * r301269;
        double r301271 = r301267 / r301270;
        double r301272 = z;
        double r301273 = fma(r301272, r301272, r301266);
        double r301274 = cbrt(r301273);
        double r301275 = r301274 * r301274;
        double r301276 = r301271 / r301275;
        double r301277 = y;
        double r301278 = r301267 / r301269;
        double r301279 = r301278 / r301274;
        double r301280 = r301277 / r301279;
        double r301281 = r301276 / r301280;
        return r301281;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.5
Target5.7
Herbie5.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  2. Simplified6.5

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

  4. Applied add-cube-cbrt6.6

    \[\leadsto \frac{\frac{\frac{1}{x}}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}}{y}\]

  5. Applied add-cube-cbrt7.1

    \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}{y}\]

  6. Applied add-sqr-sqrt7.1

    \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}{y}\]

  7. Applied times-frac7.1

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{1}}{\sqrt[3]{x}}}}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}{y}\]

  8. Applied times-frac7.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}}{y}\]

  9. Applied associate-/l*5.8

    \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}{\frac{y}{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}}}\]

  10. Final simplification5.8

    \[\leadsto \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}{\frac{y}{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}}\]

Reproduce

herbie shell --seed 2020025 +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)))))