Average Error: 6.5 → 3.9
Time: 10.1s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.307370655589997521046711449010723357855 \cdot 10^{-279}:\\ \;\;\;\;\frac{1}{y \cdot \mathsf{fma}\left(x \cdot z, z, 1 \cdot x\right)}\\ \mathbf{elif}\;y \le 5.552809287377704998861487931028038747128 \cdot 10^{-266}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\mathsf{fma}\left(x \cdot z, z, 1 \cdot x\right)}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.307370655589997521046711449010723357855 \cdot 10^{-279}:\\
\;\;\;\;\frac{1}{y \cdot \mathsf{fma}\left(x \cdot z, z, 1 \cdot x\right)}\\

\mathbf{elif}\;y \le 5.552809287377704998861487931028038747128 \cdot 10^{-266}:\\
\;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{\mathsf{fma}\left(z, z, 1\right)}\\

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

\end{array}
double f(double x, double y, double z) {
        double r11506466 = 1.0;
        double r11506467 = x;
        double r11506468 = r11506466 / r11506467;
        double r11506469 = y;
        double r11506470 = z;
        double r11506471 = r11506470 * r11506470;
        double r11506472 = r11506466 + r11506471;
        double r11506473 = r11506469 * r11506472;
        double r11506474 = r11506468 / r11506473;
        return r11506474;
}
↓
double f(double x, double y, double z) {
        double r11506475 = y;
        double r11506476 = -3.3073706555899975e-279;
        bool r11506477 = r11506475 <= r11506476;
        double r11506478 = 1.0;
        double r11506479 = x;
        double r11506480 = z;
        double r11506481 = r11506479 * r11506480;
        double r11506482 = r11506478 * r11506479;
        double r11506483 = fma(r11506481, r11506480, r11506482);
        double r11506484 = r11506475 * r11506483;
        double r11506485 = r11506478 / r11506484;
        double r11506486 = 5.552809287377705e-266;
        bool r11506487 = r11506475 <= r11506486;
        double r11506488 = r11506478 / r11506479;
        double r11506489 = r11506488 / r11506475;
        double r11506490 = fma(r11506480, r11506480, r11506478);
        double r11506491 = r11506489 / r11506490;
        double r11506492 = r11506478 / r11506475;
        double r11506493 = r11506492 / r11506483;
        double r11506494 = r11506487 ? r11506491 : r11506493;
        double r11506495 = r11506477 ? r11506485 : r11506494;
        return r11506495;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.5
Target5.7
Herbie3.9
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -3.3073706555899975e-279

    1. Initial program 6.6

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

    3. Applied div-inv6.6

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

    4. Applied associate-/l*6.8

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

    5. Simplified6.8

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

    6. Taylor expanded around inf 6.8

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

    7. Simplified3.8

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

    if -3.3073706555899975e-279 < y < 5.552809287377705e-266

    1. Initial program 16.9

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

    2. Simplified22.7

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

    if 5.552809287377705e-266 < y

    1. Initial program 5.5

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

    3. Applied div-inv5.5

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

    4. Applied associate-/l*5.7

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

    5. Simplified5.6

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

    6. Taylor expanded around inf 5.6

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

    7. Simplified2.5

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

    9. Applied associate-/r*2.3

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

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.307370655589997521046711449010723357855 \cdot 10^{-279}:\\ \;\;\;\;\frac{1}{y \cdot \mathsf{fma}\left(x \cdot z, z, 1 \cdot x\right)}\\ \mathbf{elif}\;y \le 5.552809287377704998861487931028038747128 \cdot 10^{-266}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\mathsf{fma}\left(x \cdot z, z, 1 \cdot x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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)))))