Average Error: 6.5 → 3.3
Time: 52.6s
Precision: 64
\[\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 5.459495421423262 \cdot 10^{+291}:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\sqrt[3]{x} \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt[3]{x}\right)} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1.0}{x}}{z \cdot \left(y \cdot z\right)}\\ \end{array}\]
\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 5.459495421423262 \cdot 10^{+291}:\\
\;\;\;\;\frac{\frac{1.0}{y}}{\sqrt[3]{x} \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt[3]{x}\right)} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1.0}{x}}{z \cdot \left(y \cdot z\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r15916513 = 1.0;
        double r15916514 = x;
        double r15916515 = r15916513 / r15916514;
        double r15916516 = y;
        double r15916517 = z;
        double r15916518 = r15916517 * r15916517;
        double r15916519 = r15916513 + r15916518;
        double r15916520 = r15916516 * r15916519;
        double r15916521 = r15916515 / r15916520;
        return r15916521;
}
↓
double f(double x, double y, double z) {
        double r15916522 = z;
        double r15916523 = r15916522 * r15916522;
        double r15916524 = 5.459495421423262e+291;
        bool r15916525 = r15916523 <= r15916524;
        double r15916526 = 1.0;
        double r15916527 = y;
        double r15916528 = r15916526 / r15916527;
        double r15916529 = x;
        double r15916530 = cbrt(r15916529);
        double r15916531 = fma(r15916522, r15916522, r15916526);
        double r15916532 = sqrt(r15916531);
        double r15916533 = r15916532 * r15916530;
        double r15916534 = r15916530 * r15916533;
        double r15916535 = r15916528 / r15916534;
        double r15916536 = 1.0;
        double r15916537 = r15916536 / r15916530;
        double r15916538 = r15916537 / r15916532;
        double r15916539 = r15916535 * r15916538;
        double r15916540 = r15916526 / r15916529;
        double r15916541 = r15916527 * r15916522;
        double r15916542 = r15916522 * r15916541;
        double r15916543 = r15916540 / r15916542;
        double r15916544 = r15916525 ? r15916539 : r15916543;
        return r15916544;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.5
Target5.9
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1.0 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\left(1.0 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1.0 + z \cdot z\right) \lt 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1.0}{x}}{\left(1.0 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\left(1.0 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 5.459495421423262e+291

    1. Initial program 1.9

      \[\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}\]
    2. Using strategy rm

    3. Applied div-inv1.9

      \[\leadsto \frac{\color{blue}{1.0 \cdot \frac{1}{x}}}{y \cdot \left(1.0 + z \cdot z\right)}\]

    4. Applied times-frac1.9

      \[\leadsto \color{blue}{\frac{1.0}{y} \cdot \frac{\frac{1}{x}}{1.0 + z \cdot z}}\]

    5. Simplified1.9

      \[\leadsto \frac{1.0}{y} \cdot \color{blue}{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1.0\right)}}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt1.9

      \[\leadsto \frac{1.0}{y} \cdot \frac{\frac{1}{x}}{\color{blue}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt{\mathsf{fma}\left(z, z, 1.0\right)}}}\]

    8. Applied add-cube-cbrt2.7

      \[\leadsto \frac{1.0}{y} \cdot \frac{\frac{1}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt{\mathsf{fma}\left(z, z, 1.0\right)}}\]

    9. Applied *-un-lft-identity2.7

      \[\leadsto \frac{1.0}{y} \cdot \frac{\frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt{\mathsf{fma}\left(z, z, 1.0\right)}}\]

    10. Applied times-frac2.7

      \[\leadsto \frac{1.0}{y} \cdot \frac{\color{blue}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{1}{\sqrt[3]{x}}}}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt{\mathsf{fma}\left(z, z, 1.0\right)}}\]

    11. Applied times-frac2.7

      \[\leadsto \frac{1.0}{y} \cdot \color{blue}{\left(\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)}} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)}}\right)}\]

    12. Applied associate-*r*1.2

      \[\leadsto \color{blue}{\left(\frac{1.0}{y} \cdot \frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)}}\right) \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)}}}\]

    13. Simplified1.2

      \[\leadsto \color{blue}{\frac{\frac{1.0}{y}}{\left(\sqrt{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)}}\]

    if 5.459495421423262e+291 < (* z z)

    1. Initial program 18.1

      \[\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}\]
    2. Using strategy rm

    3. Applied div-inv18.1

      \[\leadsto \frac{\color{blue}{1.0 \cdot \frac{1}{x}}}{y \cdot \left(1.0 + z \cdot z\right)}\]

    4. Applied times-frac18.0

      \[\leadsto \color{blue}{\frac{1.0}{y} \cdot \frac{\frac{1}{x}}{1.0 + z \cdot z}}\]

    5. Simplified18.0

      \[\leadsto \frac{1.0}{y} \cdot \color{blue}{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1.0\right)}}\]

    6. Taylor expanded around inf 18.2

      \[\leadsto \color{blue}{\frac{1.0}{x \cdot \left({z}^{2} \cdot y\right)}}\]

    7. Simplified8.6

      \[\leadsto \color{blue}{\frac{\frac{1.0}{x}}{\left(y \cdot z\right) \cdot z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 5.459495421423262 \cdot 10^{+291}:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\sqrt[3]{x} \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt[3]{x}\right)} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1.0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1.0}{x}}{z \cdot \left(y \cdot z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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)))))