Average Error: 6.4 → 5.8
Time: 8.3s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}
double f(double x, double y, double z) {
        double r308755 = 1.0;
        double r308756 = x;
        double r308757 = r308755 / r308756;
        double r308758 = y;
        double r308759 = z;
        double r308760 = r308759 * r308759;
        double r308761 = r308755 + r308760;
        double r308762 = r308758 * r308761;
        double r308763 = r308757 / r308762;
        return r308763;
}
↓
double f(double x, double y, double z) {
        double r308764 = 1.0;
        double r308765 = sqrt(r308764);
        double r308766 = x;
        double r308767 = cbrt(r308766);
        double r308768 = r308767 * r308767;
        double r308769 = r308765 / r308768;
        double r308770 = z;
        double r308771 = fma(r308770, r308770, r308764);
        double r308772 = sqrt(r308771);
        double r308773 = r308769 / r308772;
        double r308774 = r308765 / r308767;
        double r308775 = r308774 / r308772;
        double r308776 = y;
        double r308777 = r308775 / r308776;
        double r308778 = r308773 * r308777;
        return r308778;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.7
Herbie5.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  2. Simplified6.2

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

  4. Applied *-un-lft-identity6.2

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

  5. Applied add-sqr-sqrt6.2

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

  6. Applied add-cube-cbrt6.8

    \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1 \cdot y}\]

  7. Applied add-sqr-sqrt6.8

    \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1 \cdot y}\]

  8. Applied times-frac6.8

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{1}}{\sqrt[3]{x}}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1 \cdot y}\]

  9. Applied times-frac6.8

    \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{1 \cdot y}\]

  10. Applied times-frac5.8

    \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}}\]

  11. Simplified5.8

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}\]

  12. Final simplification5.8

    \[\leadsto \frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}\]

Reproduce

herbie shell --seed 2020020 +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)))))