Average Error: 6.9 → 6.9
Time: 8.8s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}
double f(double x, double y, double z) {
        double r310413 = 1.0;
        double r310414 = x;
        double r310415 = r310413 / r310414;
        double r310416 = y;
        double r310417 = z;
        double r310418 = r310417 * r310417;
        double r310419 = r310413 + r310418;
        double r310420 = r310416 * r310419;
        double r310421 = r310415 / r310420;
        return r310421;
}
↓
double f(double x, double y, double z) {
        double r310422 = 1.0;
        double r310423 = x;
        double r310424 = r310422 / r310423;
        double r310425 = y;
        double r310426 = r310424 / r310425;
        double r310427 = z;
        double r310428 = r310427 * r310427;
        double r310429 = r310422 + r310428;
        double r310430 = r310426 / r310429;
        return r310430;
}

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

Original6.9
Target6.2
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ 1.0 x) < 2.4392039633298638e+98 or 2.9963394477074743e+252 < (/ 1.0 x)

    1. Initial program 5.8

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

    3. Applied add-sqr-sqrt5.8

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

    4. Applied associate-*r*5.8

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

    if 2.4392039633298638e+98 < (/ 1.0 x) < 2.9963394477074743e+252

    1. Initial program 15.4

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

    3. Applied associate-/r*14.3

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

    4. Simplified14.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{y}}{x}}}{1 + z \cdot z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.9

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

Reproduce

herbie shell --seed 2019298 
(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))) -inf.bf) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.68074325056725162e305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))