Average Error: 15.0 → 1.3
Time: 4.6s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{z} \cdot \frac{y}{z + 1}\right)\]
\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]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{z} \cdot \frac{y}{z + 1}\right)
double f(double x, double y, double z) {
        double r309420 = x;
        double r309421 = y;
        double r309422 = r309420 * r309421;
        double r309423 = z;
        double r309424 = r309423 * r309423;
        double r309425 = 1.0;
        double r309426 = r309423 + r309425;
        double r309427 = r309424 * r309426;
        double r309428 = r309422 / r309427;
        return r309428;
}
↓
double f(double x, double y, double z) {
        double r309429 = x;
        double r309430 = cbrt(r309429);
        double r309431 = r309430 * r309430;
        double r309432 = z;
        double r309433 = r309431 / r309432;
        double r309434 = cbrt(r309431);
        double r309435 = cbrt(r309430);
        double r309436 = r309434 * r309435;
        double r309437 = r309436 / r309432;
        double r309438 = y;
        double r309439 = 1.0;
        double r309440 = r309432 + r309439;
        double r309441 = r309438 / r309440;
        double r309442 = r309437 * r309441;
        double r309443 = r309433 * r309442;
        return r309443;
}

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

Original15.0
Target4.1
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.0

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

  3. Applied times-frac11.0

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

  5. Applied add-cube-cbrt11.4

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

  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*1.3

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

  9. Applied add-cube-cbrt1.3

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

  10. Applied cbrt-prod1.3

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

  11. Final simplification1.3

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

Reproduce

herbie shell --seed 2020033 +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 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1))))