Average Error: 6.2 → 4.3
Time: 47.6s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.388362437333713663189698402008227229826 \cdot 10^{192}:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \le 2189352106627621.75:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.388362437333713663189698402008227229826 \cdot 10^{192}:\\
\;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(z \cdot y\right)}\\

\mathbf{elif}\;z \le 2189352106627621.75:\\
\;\;\;\;\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r16985536 = 1.0;
        double r16985537 = x;
        double r16985538 = r16985536 / r16985537;
        double r16985539 = y;
        double r16985540 = z;
        double r16985541 = r16985540 * r16985540;
        double r16985542 = r16985536 + r16985541;
        double r16985543 = r16985539 * r16985542;
        double r16985544 = r16985538 / r16985543;
        return r16985544;
}
↓
double f(double x, double y, double z) {
        double r16985545 = z;
        double r16985546 = -3.3883624373337137e+192;
        bool r16985547 = r16985545 <= r16985546;
        double r16985548 = 1.0;
        double r16985549 = x;
        double r16985550 = r16985548 / r16985549;
        double r16985551 = y;
        double r16985552 = r16985545 * r16985551;
        double r16985553 = r16985545 * r16985552;
        double r16985554 = r16985550 / r16985553;
        double r16985555 = 2189352106627621.8;
        bool r16985556 = r16985545 <= r16985555;
        double r16985557 = r16985545 * r16985545;
        double r16985558 = r16985548 + r16985557;
        double r16985559 = r16985550 / r16985558;
        double r16985560 = r16985559 / r16985551;
        double r16985561 = r16985549 * r16985553;
        double r16985562 = r16985548 / r16985561;
        double r16985563 = r16985556 ? r16985560 : r16985562;
        double r16985564 = r16985547 ? r16985554 : r16985563;
        return r16985564;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.2
Target5.6
Herbie4.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.3883624373337137e+192

    1. Initial program 14.6

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

    3. Applied add-sqr-sqrt14.6

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

    4. Applied associate-*r*14.6

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

    5. Taylor expanded around inf 14.6

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

    6. Simplified7.8

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

    if -3.3883624373337137e+192 < z < 2189352106627621.8

    1. Initial program 2.8

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

    3. Applied div-inv2.8

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

    4. Applied times-frac2.7

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}}\]
    5. Using strategy rm

    6. Applied associate-*l/2.7

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

    7. Simplified2.7

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

    if 2189352106627621.8 < z

    1. Initial program 11.6

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

    3. Applied div-inv11.6

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

    4. Applied times-frac11.7

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

    5. Taylor expanded around inf 11.7

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

    6. Simplified7.1

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

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.388362437333713663189698402008227229826 \cdot 10^{192}:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \le 2189352106627621.75:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))))