Average Error: 6.3 → 6.1
Time: 9.1s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[1 \cdot \frac{\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
1 \cdot \frac{\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x}}{y}
double f(double x, double y, double z) {
        double r214622 = 1.0;
        double r214623 = x;
        double r214624 = r214622 / r214623;
        double r214625 = y;
        double r214626 = z;
        double r214627 = r214626 * r214626;
        double r214628 = r214622 + r214627;
        double r214629 = r214625 * r214628;
        double r214630 = r214624 / r214629;
        return r214630;
}
↓
double f(double x, double y, double z) {
        double r214631 = 1.0;
        double r214632 = 1.0;
        double r214633 = z;
        double r214634 = fma(r214633, r214633, r214631);
        double r214635 = r214632 / r214634;
        double r214636 = x;
        double r214637 = r214635 / r214636;
        double r214638 = y;
        double r214639 = r214637 / r214638;
        double r214640 = r214631 * r214639;
        return r214640;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.3
Target5.6
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.3

    \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
  2. Using strategy rm

  3. Applied div-inv6.3

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

  4. Simplified6.2

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

  6. Applied div-inv6.2

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

  7. Applied associate-*l*6.2

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

  8. Simplified6.1

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

  9. Final simplification6.1

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

Reproduce

herbie shell --seed 2019350 +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)))))