Average Error: 6.7 → 6.3
Time: 26.6s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{y}}}{\mathsf{fma}\left(z, z, 1\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{y}}}{\mathsf{fma}\left(z, z, 1\right)}
double f(double x, double y, double z) {
        double r257037 = 1.0;
        double r257038 = x;
        double r257039 = r257037 / r257038;
        double r257040 = y;
        double r257041 = z;
        double r257042 = r257041 * r257041;
        double r257043 = r257037 + r257042;
        double r257044 = r257040 * r257043;
        double r257045 = r257039 / r257044;
        return r257045;
}
↓
double f(double x, double y, double z) {
        double r257046 = 1.0;
        double r257047 = sqrt(r257046);
        double r257048 = x;
        double r257049 = cbrt(r257048);
        double r257050 = r257049 * r257049;
        double r257051 = r257047 / r257050;
        double r257052 = y;
        double r257053 = cbrt(r257052);
        double r257054 = r257053 * r257053;
        double r257055 = r257051 / r257054;
        double r257056 = r257047 / r257049;
        double r257057 = r257056 / r257053;
        double r257058 = z;
        double r257059 = fma(r257058, r257058, r257046);
        double r257060 = r257057 / r257059;
        double r257061 = r257055 * r257060;
        return r257061;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.7
Target6.1
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.7

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

  2. Simplified6.8

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

  4. Applied *-un-lft-identity6.8

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

  5. Applied add-cube-cbrt7.4

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

  6. Applied add-cube-cbrt7.6

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

  7. Applied add-sqr-sqrt7.6

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

  8. Applied times-frac7.6

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

  9. Applied times-frac7.6

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

  10. Applied times-frac6.3

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

  11. Simplified6.3

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

  12. Final simplification6.3

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

Reproduce

herbie shell --seed 2019323 +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)))))