Average Error: 14.7 → 1.2
Time: 14.9s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)
double f(double x, double y, double z) {
        double r212210 = x;
        double r212211 = y;
        double r212212 = r212210 * r212211;
        double r212213 = z;
        double r212214 = r212213 * r212213;
        double r212215 = 1.0;
        double r212216 = r212213 + r212215;
        double r212217 = r212214 * r212216;
        double r212218 = r212212 / r212217;
        return r212218;
}
↓
double f(double x, double y, double z) {
        double r212219 = x;
        double r212220 = cbrt(r212219);
        double r212221 = r212220 * r212220;
        double r212222 = z;
        double r212223 = r212221 / r212222;
        double r212224 = r212220 / r212222;
        double r212225 = y;
        double r212226 = 1.0;
        double r212227 = r212222 + r212226;
        double r212228 = r212225 / r212227;
        double r212229 = r212224 * r212228;
        double r212230 = r212223 * r212229;
        return r212230;
}

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

Original14.7
Target3.8
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.7

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

  3. Applied times-frac10.8

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

  4. Simplified10.8

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

  6. Applied sqr-pow10.8

    \[\leadsto \frac{x}{\color{blue}{{z}^{\left(\frac{2}{2}\right)} \cdot {z}^{\left(\frac{2}{2}\right)}}} \cdot \frac{y}{z + 1}\]

  7. Applied add-cube-cbrt11.2

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{{z}^{\left(\frac{2}{2}\right)} \cdot {z}^{\left(\frac{2}{2}\right)}} \cdot \frac{y}{z + 1}\]

  8. Applied times-frac6.2

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{{z}^{\left(\frac{2}{2}\right)}} \cdot \frac{\sqrt[3]{x}}{{z}^{\left(\frac{2}{2}\right)}}\right)} \cdot \frac{y}{z + 1}\]

  9. Applied associate-*l*1.2

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{{z}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\sqrt[3]{x}}{{z}^{\left(\frac{2}{2}\right)}} \cdot \frac{y}{z + 1}\right)}\]

  10. Simplified1.2

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{{z}^{\left(\frac{2}{2}\right)}} \cdot \color{blue}{\left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)}\]

  11. Final simplification1.2

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

Reproduce

herbie shell --seed 2019305 +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))))