Average Error: 15.1 → 3.3
Time: 3.1s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{x}{z} \cdot \frac{\frac{y}{z + 1}}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{x}{z} \cdot \frac{\frac{y}{z + 1}}{z}
double f(double x, double y, double z) {
        double r289867 = x;
        double r289868 = y;
        double r289869 = r289867 * r289868;
        double r289870 = z;
        double r289871 = r289870 * r289870;
        double r289872 = 1.0;
        double r289873 = r289870 + r289872;
        double r289874 = r289871 * r289873;
        double r289875 = r289869 / r289874;
        return r289875;
}
↓
double f(double x, double y, double z) {
        double r289876 = x;
        double r289877 = z;
        double r289878 = r289876 / r289877;
        double r289879 = y;
        double r289880 = 1.0;
        double r289881 = r289877 + r289880;
        double r289882 = r289879 / r289881;
        double r289883 = r289882 / r289877;
        double r289884 = r289878 * r289883;
        return r289884;
}

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
Target4.1
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

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-frac11.3

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

  5. Applied *-un-lft-identity11.3

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

  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*2.6

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

  9. Applied *-un-lft-identity2.6

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

  10. Applied associate-*l*2.6

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

  11. Simplified2.6

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

  13. Applied *-un-lft-identity2.6

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

  14. Applied times-frac3.3

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

  15. Simplified3.3

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

  16. Final simplification3.3

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

Reproduce

herbie shell --seed 2019356 +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))))