Average Error: 14.8 → 2.6
Time: 12.0s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{1}{z} \cdot \left(\frac{x}{z} \cdot {\left(\frac{z + 1}{y}\right)}^{-1}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{1}{z} \cdot \left(\frac{x}{z} \cdot {\left(\frac{z + 1}{y}\right)}^{-1}\right)
double f(double x, double y, double z) {
        double r236835 = x;
        double r236836 = y;
        double r236837 = r236835 * r236836;
        double r236838 = z;
        double r236839 = r236838 * r236838;
        double r236840 = 1.0;
        double r236841 = r236838 + r236840;
        double r236842 = r236839 * r236841;
        double r236843 = r236837 / r236842;
        return r236843;
}
↓
double f(double x, double y, double z) {
        double r236844 = 1.0;
        double r236845 = z;
        double r236846 = r236844 / r236845;
        double r236847 = x;
        double r236848 = r236847 / r236845;
        double r236849 = 1.0;
        double r236850 = r236845 + r236849;
        double r236851 = y;
        double r236852 = r236850 / r236851;
        double r236853 = -1.0;
        double r236854 = pow(r236852, r236853);
        double r236855 = r236848 * r236854;
        double r236856 = r236846 * r236855;
        return r236856;
}

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

Original14.8
Target4.1
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 14.8

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

  3. Applied times-frac10.9

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

  5. Applied *-un-lft-identity10.9

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

  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*2.5

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

  9. Applied clear-num2.6

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

  11. Applied inv-pow2.6

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

  12. Final simplification2.6

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

Reproduce

herbie shell --seed 2019209 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.618281453230708) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

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