Average Error: 6.4 → 6.0
Time: 6.5s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}
double f(double x, double y, double z) {
        double r247749 = 1.0;
        double r247750 = x;
        double r247751 = r247749 / r247750;
        double r247752 = y;
        double r247753 = z;
        double r247754 = r247753 * r247753;
        double r247755 = r247749 + r247754;
        double r247756 = r247752 * r247755;
        double r247757 = r247751 / r247756;
        return r247757;
}
↓
double f(double x, double y, double z) {
        double r247758 = 1.0;
        double r247759 = x;
        double r247760 = r247758 / r247759;
        double r247761 = z;
        double r247762 = fma(r247761, r247761, r247758);
        double r247763 = r247760 / r247762;
        double r247764 = y;
        double r247765 = r247763 / r247764;
        return r247765;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.7
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 6.4

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

  2. Simplified6.0

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

  4. Applied div-inv6.1

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

  6. Applied div-inv6.1

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

  7. Applied associate-*l*6.3

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

  9. Applied un-div-inv6.3

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

  10. Applied associate-*r/6.0

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

  11. Simplified6.0

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

  12. Final simplification6.0

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

Reproduce

herbie shell --seed 2019356 +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)))))