Average Error: 15.2 → 1.3
Time: 18.4s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{y}{z + 1} \cdot \frac{\sqrt[3]{x}}{z}\right)\]
\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{y}{z + 1} \cdot \frac{\sqrt[3]{x}}{z}\right)
double f(double x, double y, double z) {
        double r11921505 = x;
        double r11921506 = y;
        double r11921507 = r11921505 * r11921506;
        double r11921508 = z;
        double r11921509 = r11921508 * r11921508;
        double r11921510 = 1.0;
        double r11921511 = r11921508 + r11921510;
        double r11921512 = r11921509 * r11921511;
        double r11921513 = r11921507 / r11921512;
        return r11921513;
}
↓
double f(double x, double y, double z) {
        double r11921514 = x;
        double r11921515 = cbrt(r11921514);
        double r11921516 = r11921515 * r11921515;
        double r11921517 = z;
        double r11921518 = r11921516 / r11921517;
        double r11921519 = y;
        double r11921520 = 1.0;
        double r11921521 = r11921517 + r11921520;
        double r11921522 = r11921519 / r11921521;
        double r11921523 = r11921515 / r11921517;
        double r11921524 = r11921522 * r11921523;
        double r11921525 = r11921518 * r11921524;
        return r11921525;
}

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

Original15.2
Target4.1
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.2

    \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
  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-frac6.2

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

  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. Final simplification1.3

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))