Average Error: 15.1 → 2.6
Time: 15.8s
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 r234514 = x;
        double r234515 = y;
        double r234516 = r234514 * r234515;
        double r234517 = z;
        double r234518 = r234517 * r234517;
        double r234519 = 1.0;
        double r234520 = r234517 + r234519;
        double r234521 = r234518 * r234520;
        double r234522 = r234516 / r234521;
        return r234522;
}
↓
double f(double x, double y, double z) {
        double r234523 = y;
        double r234524 = z;
        double r234525 = 1.0;
        double r234526 = r234524 + r234525;
        double r234527 = r234523 / r234526;
        double r234528 = x;
        double r234529 = r234528 / r234524;
        double r234530 = r234527 * r234529;
        double r234531 = r234530 / r234524;
        return r234531;
}

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

Original15.1
Target4.3
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 15.1

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

  3. Applied times-frac11.1

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

  5. Applied *-un-lft-identity11.1

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

  6. Applied times-frac6.0

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

  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/5.7

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

  10. Applied associate-*r/5.7

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

  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 2019325 
(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))))