Average Error: 15.5 → 2.6
Time: 14.6s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}
double f(double x, double y, double z) {
        double r221264 = x;
        double r221265 = y;
        double r221266 = r221264 * r221265;
        double r221267 = z;
        double r221268 = r221267 * r221267;
        double r221269 = 1.0;
        double r221270 = r221267 + r221269;
        double r221271 = r221268 * r221270;
        double r221272 = r221266 / r221271;
        return r221272;
}
↓
double f(double x, double y, double z) {
        double r221273 = y;
        double r221274 = z;
        double r221275 = 1.0;
        double r221276 = r221274 + r221275;
        double r221277 = r221273 / r221276;
        double r221278 = x;
        double r221279 = r221278 / r221274;
        double r221280 = r221277 * r221279;
        double r221281 = r221280 / r221274;
        return r221281;
}

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

Original15.5
Target4.2
Herbie2.6
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.5

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

  3. Applied times-frac11.4

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

  5. Applied *-un-lft-identity11.4

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

  6. Applied times-frac6.1

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

  7. Applied associate-*l*2.7

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

  9. Applied associate-*l/5.8

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

  10. Applied associate-*r/5.8

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

  11. Simplified2.6

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

  12. Final simplification2.6

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

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