Average Error: 14.6 → 2.6
Time: 15.6s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{1}{z + 1} \cdot \frac{\frac{y}{z}}{\frac{z}{x}}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{1}{z + 1} \cdot \frac{\frac{y}{z}}{\frac{z}{x}}
double f(double x, double y, double z) {
        double r200254 = x;
        double r200255 = y;
        double r200256 = r200254 * r200255;
        double r200257 = z;
        double r200258 = r200257 * r200257;
        double r200259 = 1.0;
        double r200260 = r200257 + r200259;
        double r200261 = r200258 * r200260;
        double r200262 = r200256 / r200261;
        return r200262;
}
↓
double f(double x, double y, double z) {
        double r200263 = 1.0;
        double r200264 = z;
        double r200265 = 1.0;
        double r200266 = r200264 + r200265;
        double r200267 = r200263 / r200266;
        double r200268 = y;
        double r200269 = r200268 / r200264;
        double r200270 = x;
        double r200271 = r200264 / r200270;
        double r200272 = r200269 / r200271;
        double r200273 = r200267 * r200272;
        return r200273;
}

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

Original14.6
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 14.6

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

  3. Applied associate-/r*13.2

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

  4. Simplified2.7

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

  6. Applied clear-num2.9

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

  7. Simplified2.9

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

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

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

  10. Applied associate-*l*2.9

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

  11. Simplified2.8

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

  13. Applied associate-/r/2.6

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

  14. Final simplification2.6

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))