Average Error: 15.2 → 3.0
Time: 9.4s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \le 1.58735393053565291 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{\frac{{\left(\sqrt[3]{x}\right)}^{3}}{z}}{\frac{z}{y}}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot y\right)}{z}}{z + 1}\\ \end{array}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \le 1.58735393053565291 \cdot 10^{-279}:\\
\;\;\;\;\frac{\frac{\frac{{\left(\sqrt[3]{x}\right)}^{3}}{z}}{\frac{z}{y}}}{z + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot y\right)}{z}}{z + 1}\\

\end{array}
double f(double x, double y, double z) {
        double r319385 = x;
        double r319386 = y;
        double r319387 = r319385 * r319386;
        double r319388 = z;
        double r319389 = r319388 * r319388;
        double r319390 = 1.0;
        double r319391 = r319388 + r319390;
        double r319392 = r319389 * r319391;
        double r319393 = r319387 / r319392;
        return r319393;
}
↓
double f(double x, double y, double z) {
        double r319394 = z;
        double r319395 = r319394 * r319394;
        double r319396 = 1.0;
        double r319397 = r319394 + r319396;
        double r319398 = r319395 * r319397;
        double r319399 = 1.587353930535653e-279;
        bool r319400 = r319398 <= r319399;
        double r319401 = x;
        double r319402 = cbrt(r319401);
        double r319403 = 3.0;
        double r319404 = pow(r319402, r319403);
        double r319405 = r319404 / r319394;
        double r319406 = y;
        double r319407 = r319394 / r319406;
        double r319408 = r319405 / r319407;
        double r319409 = r319408 / r319397;
        double r319410 = cbrt(r319402);
        double r319411 = pow(r319410, r319403);
        double r319412 = r319411 * r319402;
        double r319413 = r319402 / r319394;
        double r319414 = r319413 * r319406;
        double r319415 = r319412 * r319414;
        double r319416 = r319415 / r319394;
        double r319417 = r319416 / r319397;
        double r319418 = r319400 ? r319409 : r319417;
        return r319418;
}

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
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* z z) (+ z 1.0)) < 1.587353930535653e-279

    1. Initial program 22.5

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

    3. Applied times-frac17.8

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

    5. Applied add-cube-cbrt18.0

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

    6. Applied times-frac8.8

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

    7. Applied associate-*l*1.4

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

    9. Applied associate-*r/1.3

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

    10. Applied associate-*r/1.4

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

    11. Simplified2.9

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

    if 1.587353930535653e-279 < (* (* z z) (+ z 1.0))

    1. Initial program 9.4

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

    3. Applied times-frac5.9

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

    5. Applied add-cube-cbrt6.4

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

    6. Applied times-frac4.8

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

    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

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

    11. Applied associate-*r/1.4

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

    12. Applied associate-*r/1.4

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

    13. Simplified3.1

      \[\leadsto \frac{\color{blue}{\frac{\left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot y\right)}{z}}}{z + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \le 1.58735393053565291 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{\frac{{\left(\sqrt[3]{x}\right)}^{3}}{z}}{\frac{z}{y}}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot y\right)}{z}}{z + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))