Average Error: 15.6 → 2.6
Time: 3.5s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\left(\frac{y}{z + 1} \cdot \frac{x}{z}\right) \cdot \frac{1}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\left(\frac{y}{z + 1} \cdot \frac{x}{z}\right) \cdot \frac{1}{z}
double f(double x, double y, double z) {
        double r286413 = x;
        double r286414 = y;
        double r286415 = r286413 * r286414;
        double r286416 = z;
        double r286417 = r286416 * r286416;
        double r286418 = 1.0;
        double r286419 = r286416 + r286418;
        double r286420 = r286417 * r286419;
        double r286421 = r286415 / r286420;
        return r286421;
}
↓
double f(double x, double y, double z) {
        double r286422 = y;
        double r286423 = z;
        double r286424 = 1.0;
        double r286425 = r286423 + r286424;
        double r286426 = r286422 / r286425;
        double r286427 = x;
        double r286428 = r286427 / r286423;
        double r286429 = r286426 * r286428;
        double r286430 = 1.0;
        double r286431 = r286430 / r286423;
        double r286432 = r286429 * r286431;
        return r286432;
}

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

Original15.6
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 15.6

    \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
  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*2.6

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

  9. Applied div-inv2.6

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

  11. Applied pow12.6

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

  12. Applied pow12.6

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

  13. Applied pow-prod-down2.6

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

  14. Simplified2.6

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

  15. Final simplification2.6

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

Reproduce

herbie shell --seed 2019354 
(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))))