Average Error: 6.4 → 5.3
Time: 9.1s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.5940737606410852 \cdot 10^{159} \lor \neg \left(x \le 3.3502160983592637 \cdot 10^{92}\right):\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.5940737606410852 \cdot 10^{159} \lor \neg \left(x \le 3.3502160983592637 \cdot 10^{92}\right):\\
\;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r346191 = 1.0;
        double r346192 = x;
        double r346193 = r346191 / r346192;
        double r346194 = y;
        double r346195 = z;
        double r346196 = r346195 * r346195;
        double r346197 = r346191 + r346196;
        double r346198 = r346194 * r346197;
        double r346199 = r346193 / r346198;
        return r346199;
}
↓
double f(double x, double y, double z) {
        double r346200 = x;
        double r346201 = -7.594073760641085e+159;
        bool r346202 = r346200 <= r346201;
        double r346203 = 3.3502160983592637e+92;
        bool r346204 = r346200 <= r346203;
        double r346205 = !r346204;
        bool r346206 = r346202 || r346205;
        double r346207 = 1.0;
        double r346208 = y;
        double r346209 = r346207 / r346208;
        double r346210 = r346209 / r346200;
        double r346211 = z;
        double r346212 = r346211 * r346211;
        double r346213 = r346207 + r346212;
        double r346214 = r346210 / r346213;
        double r346215 = r346207 / r346200;
        double r346216 = fma(r346211, r346211, r346207);
        double r346217 = r346215 / r346216;
        double r346218 = r346217 / r346208;
        double r346219 = r346206 ? r346214 : r346218;
        return r346219;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.6
Herbie5.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7.594073760641085e+159 or 3.3502160983592637e+92 < x

    1. Initial program 0.4

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
    2. Using strategy rm

    3. Applied associate-/r*0.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}}\]

    4. Simplified0.5

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{y}}{x}}}{1 + z \cdot z}\]

    if -7.594073760641085e+159 < x < 3.3502160983592637e+92

    1. Initial program 9.1

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

    2. Simplified7.5

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.5940737606410852 \cdot 10^{159} \lor \neg \left(x \le 3.3502160983592637 \cdot 10^{92}\right):\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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)))))