Average Error: 6.2 → 6.0
Time: 3.4s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}
double f(double x, double y, double z) {
        double r242164 = 1.0;
        double r242165 = x;
        double r242166 = r242164 / r242165;
        double r242167 = y;
        double r242168 = z;
        double r242169 = r242168 * r242168;
        double r242170 = r242164 + r242169;
        double r242171 = r242167 * r242170;
        double r242172 = r242166 / r242171;
        return r242172;
}
↓
double f(double x, double y, double z) {
        double r242173 = 1.0;
        double r242174 = y;
        double r242175 = r242173 / r242174;
        double r242176 = z;
        double r242177 = fma(r242176, r242176, r242173);
        double r242178 = sqrt(r242177);
        double r242179 = x;
        double r242180 = r242178 * r242179;
        double r242181 = r242178 * r242180;
        double r242182 = r242175 / r242181;
        return r242182;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.2
Target5.5
Herbie6.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.2

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

  2. Simplified5.9

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

  4. Applied *-un-lft-identity5.9

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

  5. Applied div-inv5.9

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

  6. Applied times-frac5.9

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

  7. Applied associate-/l*6.2

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

  8. Simplified6.2

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

  10. Applied associate-/r*6.0

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

  11. Simplified6.0

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

  13. Applied add-sqr-sqrt6.0

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

  14. Applied associate-*l*6.0

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

  15. Final simplification6.0

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

Reproduce

herbie shell --seed 2020027 +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)))))