Average Error: 6.4 → 6.7
Time: 10.3s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{x \cdot y}}{\mathsf{fma}\left(z, z, 1\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{x \cdot y}}{\mathsf{fma}\left(z, z, 1\right)}
double f(double x, double y, double z) {
        double r367657 = 1.0;
        double r367658 = x;
        double r367659 = r367657 / r367658;
        double r367660 = y;
        double r367661 = z;
        double r367662 = r367661 * r367661;
        double r367663 = r367657 + r367662;
        double r367664 = r367660 * r367663;
        double r367665 = r367659 / r367664;
        return r367665;
}
↓
double f(double x, double y, double z) {
        double r367666 = 1.0;
        double r367667 = x;
        double r367668 = y;
        double r367669 = r367667 * r367668;
        double r367670 = r367666 / r367669;
        double r367671 = z;
        double r367672 = fma(r367671, r367671, r367666);
        double r367673 = r367670 / r367672;
        return r367673;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.7
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  2. Simplified6.6

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

  4. Applied div-inv6.6

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

  5. Applied associate-/l*6.8

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

  6. Simplified6.7

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

  7. Final simplification6.7

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

Reproduce

herbie shell --seed 2020042 +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)))))