Average Error: 6.6 → 6.7
Time: 10.8s
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 r336267 = 1.0;
        double r336268 = x;
        double r336269 = r336267 / r336268;
        double r336270 = y;
        double r336271 = z;
        double r336272 = r336271 * r336271;
        double r336273 = r336267 + r336272;
        double r336274 = r336270 * r336273;
        double r336275 = r336269 / r336274;
        return r336275;
}
↓
double f(double x, double y, double z) {
        double r336276 = 1.0;
        double r336277 = y;
        double r336278 = r336276 / r336277;
        double r336279 = x;
        double r336280 = r336278 / r336279;
        double r336281 = z;
        double r336282 = r336281 * r336281;
        double r336283 = r336276 + r336282;
        double r336284 = sqrt(r336283);
        double r336285 = r336280 / r336284;
        double r336286 = r336285 / r336284;
        return r336286;
}

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 +o rules:numerics
(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)))))