Average Error: 15.3 → 2.2
Time: 15.8s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)}{z} \cdot 1\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)}{z} \cdot 1
double f(double x, double y, double z) {
        double r210244 = x;
        double r210245 = y;
        double r210246 = r210244 * r210245;
        double r210247 = z;
        double r210248 = r210247 * r210247;
        double r210249 = 1.0;
        double r210250 = r210247 + r210249;
        double r210251 = r210248 * r210250;
        double r210252 = r210246 / r210251;
        return r210252;
}
↓
double f(double x, double y, double z) {
        double r210253 = x;
        double r210254 = cbrt(r210253);
        double r210255 = r210254 * r210254;
        double r210256 = z;
        double r210257 = r210254 / r210256;
        double r210258 = y;
        double r210259 = 1.0;
        double r210260 = r210256 + r210259;
        double r210261 = r210258 / r210260;
        double r210262 = r210257 * r210261;
        double r210263 = r210255 * r210262;
        double r210264 = r210263 / r210256;
        double r210265 = 1.0;
        double r210266 = r210264 * r210265;
        return r210266;
}

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

Original15.3
Target4.2
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.3

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

  3. Applied times-frac11.3

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

  5. Applied *-un-lft-identity11.3

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

  6. Applied times-frac6.0

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

  7. Applied associate-*l*2.6

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

  9. Applied *-un-lft-identity2.6

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

  10. Applied add-cube-cbrt2.6

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

  11. Applied times-frac2.6

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

  12. Applied associate-*l*2.6

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

  13. Simplified2.6

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}}\]
  14. Using strategy rm

  15. Applied *-un-lft-identity2.6

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

  16. Applied add-cube-cbrt3.1

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

  17. Applied times-frac3.1

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

  18. Applied associate-*l*2.2

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

  19. Final simplification2.2

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

Reproduce

herbie shell --seed 2019303 
(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))))