Average Error: 6.4 → 5.2
Time: 4.4s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -2.2032784662325649 \cdot 10^{-5} \lor \neg \left(\frac{1}{x} \le 1.47733744568406066 \cdot 10^{-203}\right):\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z, 1\right)} \cdot \frac{\frac{1}{y}}{x}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{x} \le -2.2032784662325649 \cdot 10^{-5} \lor \neg \left(\frac{1}{x} \le 1.47733744568406066 \cdot 10^{-203}\right):\\
\;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot x}}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r263741 = 1.0;
        double r263742 = x;
        double r263743 = r263741 / r263742;
        double r263744 = y;
        double r263745 = z;
        double r263746 = r263745 * r263745;
        double r263747 = r263741 + r263746;
        double r263748 = r263744 * r263747;
        double r263749 = r263743 / r263748;
        return r263749;
}
↓
double f(double x, double y, double z) {
        double r263750 = 1.0;
        double r263751 = x;
        double r263752 = r263750 / r263751;
        double r263753 = -2.203278466232565e-05;
        bool r263754 = r263752 <= r263753;
        double r263755 = 1.4773374456840607e-203;
        bool r263756 = r263752 <= r263755;
        double r263757 = !r263756;
        bool r263758 = r263754 || r263757;
        double r263759 = z;
        double r263760 = fma(r263759, r263759, r263750);
        double r263761 = r263760 * r263751;
        double r263762 = r263750 / r263761;
        double r263763 = y;
        double r263764 = r263762 / r263763;
        double r263765 = 1.0;
        double r263766 = r263765 / r263760;
        double r263767 = r263750 / r263763;
        double r263768 = r263767 / r263751;
        double r263769 = r263766 * r263768;
        double r263770 = r263758 ? r263764 : r263769;
        return r263770;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.7
Herbie5.2
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ 1.0 x) < -2.203278466232565e-05 or 1.4773374456840607e-203 < (/ 1.0 x)

    1. Initial program 9.4

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

    2. Simplified7.4

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

    4. Applied div-inv7.4

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

    5. Applied associate-/l*7.5

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

    6. Simplified7.4

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

    if -2.203278466232565e-05 < (/ 1.0 x) < 1.4773374456840607e-203

    1. Initial program 1.2

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

    2. Simplified3.9

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

    4. Applied div-inv3.9

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

    5. Applied associate-/l*4.2

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

    6. Simplified4.2

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

    8. Applied *-un-lft-identity4.2

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

    9. Applied *-un-lft-identity4.2

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

    10. Applied times-frac3.9

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

    11. Applied times-frac1.2

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

    12. Simplified1.2

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

    13. Simplified1.2

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

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -2.2032784662325649 \cdot 10^{-5} \lor \neg \left(\frac{1}{x} \le 1.47733744568406066 \cdot 10^{-203}\right):\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z, 1\right)} \cdot \frac{\frac{1}{y}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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)))))