Average Error: 6.5 → 6.1
Time: 22.7s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{\frac{1}{\sqrt[3]{x}}}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{\frac{1}{\sqrt[3]{x}}}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}
double f(double x, double y, double z) {
        double r298062 = 1.0;
        double r298063 = x;
        double r298064 = r298062 / r298063;
        double r298065 = y;
        double r298066 = z;
        double r298067 = r298066 * r298066;
        double r298068 = r298062 + r298067;
        double r298069 = r298065 * r298068;
        double r298070 = r298064 / r298069;
        return r298070;
}
↓
double f(double x, double y, double z) {
        double r298071 = 1.0;
        double r298072 = x;
        double r298073 = cbrt(r298072);
        double r298074 = r298073 * r298073;
        double r298075 = r298071 / r298074;
        double r298076 = z;
        double r298077 = 1.0;
        double r298078 = fma(r298076, r298076, r298077);
        double r298079 = sqrt(r298078);
        double r298080 = r298075 / r298079;
        double r298081 = r298077 / r298073;
        double r298082 = y;
        double r298083 = r298081 / r298082;
        double r298084 = r298083 / r298079;
        double r298085 = r298080 * r298084;
        return r298085;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.5
Target5.9
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 6.5

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

  2. Simplified6.5

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

  4. Applied add-sqr-sqrt6.5

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

  5. Applied *-un-lft-identity6.5

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

  6. Applied add-cube-cbrt7.1

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

  7. Applied *-un-lft-identity7.1

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

  8. Applied times-frac7.1

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

  9. Applied times-frac7.1

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

  10. Applied times-frac6.1

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

  11. Simplified6.1

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

  12. Final simplification6.1

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

Reproduce

herbie shell --seed 2019305 +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))) -inf.bf) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.68074325056725162e305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))