Average Error: 14.8 → 4.9
Time: 6.2s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{x}{z \cdot \left(z + 1\right)} \cdot \frac{y}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{x}{z \cdot \left(z + 1\right)} \cdot \frac{y}{z}
double f(double x, double y, double z) {
        double r279406 = x;
        double r279407 = y;
        double r279408 = r279406 * r279407;
        double r279409 = z;
        double r279410 = r279409 * r279409;
        double r279411 = 1.0;
        double r279412 = r279409 + r279411;
        double r279413 = r279410 * r279412;
        double r279414 = r279408 / r279413;
        return r279414;
}
↓
double f(double x, double y, double z) {
        double r279415 = x;
        double r279416 = z;
        double r279417 = 1.0;
        double r279418 = r279416 + r279417;
        double r279419 = r279416 * r279418;
        double r279420 = r279415 / r279419;
        double r279421 = y;
        double r279422 = r279421 / r279416;
        double r279423 = r279420 * r279422;
        return r279423;
}

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
Herbie4.9
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

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 add-cube-cbrt11.3

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{z \cdot z} \cdot \frac{y}{z + 1}\]

  6. Applied times-frac6.4

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\sqrt[3]{x}}{z}\right)} \cdot \frac{y}{z + 1}\]

  7. Applied associate-*l*1.2

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)}\]

  8. Final simplification4.9

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

Reproduce

herbie shell --seed 2019304 
(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))))