Average Error: 6.6 → 6.7
Time: 19.1s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{\frac{1}{y}}{x}}{\sqrt{1 + z \cdot z}}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{\frac{1}{y}}{x}}{\sqrt{1 + z \cdot z}}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r419308 = 1.0;
        double r419309 = x;
        double r419310 = r419308 / r419309;
        double r419311 = y;
        double r419312 = z;
        double r419313 = r419312 * r419312;
        double r419314 = r419308 + r419313;
        double r419315 = r419311 * r419314;
        double r419316 = r419310 / r419315;
        return r419316;
}
↓
double f(double x, double y, double z) {
        double r419317 = 1.0;
        double r419318 = y;
        double r419319 = r419317 / r419318;
        double r419320 = x;
        double r419321 = r419319 / r419320;
        double r419322 = z;
        double r419323 = r419322 * r419322;
        double r419324 = r419317 + r419323;
        double r419325 = sqrt(r419324);
        double r419326 = r419321 / r419325;
        double r419327 = r419326 / r419325;
        return r419327;
}

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

Original6.6
Target6.0
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  3. Applied associate-/r*6.7

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

  4. Simplified6.7

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

  6. Applied add-sqr-sqrt6.7

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

  7. Applied associate-/r*6.7

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

  8. Final simplification6.7

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

Reproduce

herbie shell --seed 2020057 
(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)))))