Average Error: 14.9 → 4.1
Time: 5.8s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)}^{3}}{z + 1} \cdot \frac{y}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)}^{3}}{z + 1} \cdot \frac{y}{z}
double f(double x, double y, double z) {
        double r215262 = x;
        double r215263 = y;
        double r215264 = r215262 * r215263;
        double r215265 = z;
        double r215266 = r215265 * r215265;
        double r215267 = 1.0;
        double r215268 = r215265 + r215267;
        double r215269 = r215266 * r215268;
        double r215270 = r215264 / r215269;
        return r215270;
}
↓
double f(double x, double y, double z) {
        double r215271 = x;
        double r215272 = cbrt(r215271);
        double r215273 = z;
        double r215274 = cbrt(r215273);
        double r215275 = r215272 / r215274;
        double r215276 = 3.0;
        double r215277 = pow(r215275, r215276);
        double r215278 = 1.0;
        double r215279 = r215273 + r215278;
        double r215280 = r215277 / r215279;
        double r215281 = y;
        double r215282 = r215281 / r215273;
        double r215283 = r215280 * r215282;
        return r215283;
}

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

Original14.9
Target4.0
Herbie4.1
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.9

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

  3. Applied times-frac11.1

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

  5. Applied add-cube-cbrt11.5

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

  6. Applied times-frac6.6

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

  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. Using strategy rm

  9. Applied add-cube-cbrt1.4

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

  10. Applied times-frac1.4

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

  11. Applied associate-*l*1.3

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

  12. Final simplification4.1

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

Reproduce

herbie shell --seed 2019308 
(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))))