Average Error: 15.1 → 2.6
Time: 14.7s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{1}{\frac{z + 1}{y}}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{1}{\frac{z + 1}{y}}\right)
double f(double x, double y, double z) {
        double r192921 = x;
        double r192922 = y;
        double r192923 = r192921 * r192922;
        double r192924 = z;
        double r192925 = r192924 * r192924;
        double r192926 = 1.0;
        double r192927 = r192924 + r192926;
        double r192928 = r192925 * r192927;
        double r192929 = r192923 / r192928;
        return r192929;
}
↓
double f(double x, double y, double z) {
        double r192930 = 1.0;
        double r192931 = z;
        double r192932 = r192930 / r192931;
        double r192933 = x;
        double r192934 = r192933 / r192931;
        double r192935 = 1.0;
        double r192936 = r192931 + r192935;
        double r192937 = y;
        double r192938 = r192936 / r192937;
        double r192939 = r192930 / r192938;
        double r192940 = r192934 * r192939;
        double r192941 = r192932 * r192940;
        return r192941;
}

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
Target4.0
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-frac11.2

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

  4. Simplified11.2

    \[\leadsto \color{blue}{\frac{x}{{z}^{2}}} \cdot \frac{y}{z + 1}\]
  5. Using strategy rm

  6. Applied unpow211.2

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

  7. Applied *-un-lft-identity11.2

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

  8. Applied times-frac6.1

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

  9. Applied associate-*l*2.6

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

  11. Applied clear-num2.6

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

  12. Final simplification2.6

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

Reproduce

herbie shell --seed 2019235 +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 249.618281453230708) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

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