Average Error: 6.1 → 6.3
Time: 4.7s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{\frac{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}{\sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{\frac{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}{\sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}}
double f(double x, double y, double z) {
        double r278767 = 1.0;
        double r278768 = x;
        double r278769 = r278767 / r278768;
        double r278770 = y;
        double r278771 = z;
        double r278772 = r278771 * r278771;
        double r278773 = r278767 + r278772;
        double r278774 = r278770 * r278773;
        double r278775 = r278769 / r278774;
        return r278775;
}
↓
double f(double x, double y, double z) {
        double r278776 = 1.0;
        double r278777 = z;
        double r278778 = fma(r278777, r278777, r278776);
        double r278779 = sqrt(r278778);
        double r278780 = r278776 / r278779;
        double r278781 = sqrt(r278780);
        double r278782 = y;
        double r278783 = x;
        double r278784 = r278779 * r278783;
        double r278785 = r278782 * r278784;
        double r278786 = r278785 / r278781;
        double r278787 = r278781 / r278786;
        return r278787;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.1
Target5.6
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.1

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

  2. Simplified6.4

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

  4. Applied add-sqr-sqrt6.4

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

  5. Applied div-inv6.4

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

  6. Applied times-frac6.4

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

  7. Applied associate-/l*6.2

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

  8. Simplified6.3

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

  10. Applied add-sqr-sqrt6.3

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

  11. Applied associate-/l*6.3

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

  12. Final simplification6.3

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

Reproduce

herbie shell --seed 2020081 +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)))))