Average Error: 6.6 → 6.0
Time: 9.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}
double f(double x, double y, double z) {
        double r374678 = 1.0;
        double r374679 = x;
        double r374680 = r374678 / r374679;
        double r374681 = y;
        double r374682 = z;
        double r374683 = r374682 * r374682;
        double r374684 = r374678 + r374683;
        double r374685 = r374681 * r374684;
        double r374686 = r374680 / r374685;
        return r374686;
}
↓
double f(double x, double y, double z) {
        double r374687 = 1.0;
        double r374688 = z;
        double r374689 = fma(r374688, r374688, r374687);
        double r374690 = sqrt(r374689);
        double r374691 = r374687 / r374690;
        double r374692 = y;
        double r374693 = x;
        double r374694 = r374690 * r374693;
        double r374695 = r374692 * r374694;
        double r374696 = r374691 / r374695;
        return r374696;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.6
Target5.9
Herbie6.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  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-sqrt6.3

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

  5. Applied div-inv6.3

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

  6. Applied times-frac6.3

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

  7. Applied associate-/l*6.0

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

  8. Simplified6.0

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

  9. Final simplification6.0

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

Reproduce

herbie shell --seed 2019362 +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)))))