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 r273873 = x;
        double r273874 = y;
        double r273875 = r273873 * r273874;
        double r273876 = z;
        double r273877 = r273876 * r273876;
        double r273878 = 1.0;
        double r273879 = r273876 + r273878;
        double r273880 = r273877 * r273879;
        double r273881 = r273875 / r273880;
        return r273881;
}
↓
double f(double x, double y, double z) {
        double r273882 = y;
        double r273883 = z;
        double r273884 = 1.0;
        double r273885 = r273883 + r273884;
        double r273886 = r273882 / r273885;
        double r273887 = x;
        double r273888 = r273887 / r273883;
        double r273889 = r273886 * r273888;
        double r273890 = 1.0;
        double r273891 = r273890 / r273883;
        double r273892 = r273889 * r273891;
        return r273892;
}

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 +o rules:numerics
(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))))