Average Error: 6.5 → 5.3
Time: 43.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 7.386968723849264976414036800702327019396 \cdot 10^{84}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\frac{\mathsf{fma}\left(z, z, 1\right)}{\frac{\frac{1}{x}}{\sqrt[3]{y}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le 7.386968723849264976414036800702327019396 \cdot 10^{84}:\\
\;\;\;\;\frac{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\frac{\mathsf{fma}\left(z, z, 1\right)}{\frac{\frac{1}{x}}{\sqrt[3]{y}}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r10739212 = 1.0;
        double r10739213 = x;
        double r10739214 = r10739212 / r10739213;
        double r10739215 = y;
        double r10739216 = z;
        double r10739217 = r10739216 * r10739216;
        double r10739218 = r10739212 + r10739217;
        double r10739219 = r10739215 * r10739218;
        double r10739220 = r10739214 / r10739219;
        return r10739220;
}
↓
double f(double x, double y, double z) {
        double r10739221 = z;
        double r10739222 = 7.386968723849265e+84;
        bool r10739223 = r10739221 <= r10739222;
        double r10739224 = 1.0;
        double r10739225 = y;
        double r10739226 = cbrt(r10739225);
        double r10739227 = r10739226 * r10739226;
        double r10739228 = r10739224 / r10739227;
        double r10739229 = fma(r10739221, r10739221, r10739224);
        double r10739230 = 1.0;
        double r10739231 = x;
        double r10739232 = r10739230 / r10739231;
        double r10739233 = r10739232 / r10739226;
        double r10739234 = r10739229 / r10739233;
        double r10739235 = r10739228 / r10739234;
        double r10739236 = r10739224 / r10739231;
        double r10739237 = r10739225 * r10739221;
        double r10739238 = r10739221 * r10739237;
        double r10739239 = r10739236 / r10739238;
        double r10739240 = r10739223 ? r10739235 : r10739239;
        return r10739240;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.5
Target5.7
Herbie5.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < 7.386968723849265e+84

    1. Initial program 4.2

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

    2. Simplified4.4

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

    4. Applied add-cube-cbrt5.1

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

    5. Applied div-inv5.1

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

    6. Applied times-frac5.1

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

    7. Applied associate-/l*4.5

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

    if 7.386968723849265e+84 < z

    1. Initial program 15.4

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

    2. Simplified15.7

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

    3. Taylor expanded around inf 15.5

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left({z}^{2} \cdot y\right)}}\]

    4. Simplified8.4

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{z \cdot \left(z \cdot y\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 7.386968723849264976414036800702327019396 \cdot 10^{84}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\frac{\mathsf{fma}\left(z, z, 1\right)}{\frac{\frac{1}{x}}{\sqrt[3]{y}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) -inf.0) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))