Average Error: 6.5 → 6.2
Time: 10.2s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}
double f(double x, double y, double z) {
        double r270258 = 1.0;
        double r270259 = x;
        double r270260 = r270258 / r270259;
        double r270261 = y;
        double r270262 = z;
        double r270263 = r270262 * r270262;
        double r270264 = r270258 + r270263;
        double r270265 = r270261 * r270264;
        double r270266 = r270260 / r270265;
        return r270266;
}
↓
double f(double x, double y, double z) {
        double r270267 = 1.0;
        double r270268 = x;
        double r270269 = r270267 / r270268;
        double r270270 = z;
        double r270271 = fma(r270270, r270270, r270267);
        double r270272 = sqrt(r270271);
        double r270273 = r270269 / r270272;
        double r270274 = r270273 / r270272;
        double r270275 = y;
        double r270276 = r270274 / r270275;
        return r270276;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.5
Target5.9
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 6.5

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

  2. Simplified6.2

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

  4. Applied add-sqr-sqrt6.2

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

  5. Applied associate-/r*6.2

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

  6. Final simplification6.2

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

Reproduce

herbie shell --seed 2020089 +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)))))