Average Error: 20.0 → 0.5
Time: 21.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\left(\left(\sqrt[3]{\frac{x}{y + x}} \cdot \sqrt[3]{\frac{x}{y + x}}\right) \cdot \frac{\sqrt[3]{\frac{x}{y + x}}}{y + x}\right) \cdot \frac{y}{1 + \left(y + x\right)}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\left(\left(\sqrt[3]{\frac{x}{y + x}} \cdot \sqrt[3]{\frac{x}{y + x}}\right) \cdot \frac{\sqrt[3]{\frac{x}{y + x}}}{y + x}\right) \cdot \frac{y}{1 + \left(y + x\right)}
double f(double x, double y) {
        double r23481222 = x;
        double r23481223 = y;
        double r23481224 = r23481222 * r23481223;
        double r23481225 = r23481222 + r23481223;
        double r23481226 = r23481225 * r23481225;
        double r23481227 = 1.0;
        double r23481228 = r23481225 + r23481227;
        double r23481229 = r23481226 * r23481228;
        double r23481230 = r23481224 / r23481229;
        return r23481230;
}
↓
double f(double x, double y) {
        double r23481231 = x;
        double r23481232 = y;
        double r23481233 = r23481232 + r23481231;
        double r23481234 = r23481231 / r23481233;
        double r23481235 = cbrt(r23481234);
        double r23481236 = r23481235 * r23481235;
        double r23481237 = r23481235 / r23481233;
        double r23481238 = r23481236 * r23481237;
        double r23481239 = 1.0;
        double r23481240 = r23481239 + r23481233;
        double r23481241 = r23481232 / r23481240;
        double r23481242 = r23481238 * r23481241;
        return r23481242;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.1
Herbie0.5
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 20.0

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

  3. Applied times-frac8.1

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

  5. Applied associate-/r*0.2

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

  7. Applied *-un-lft-identity0.2

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

  8. Applied add-cube-cbrt0.5

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

  9. Applied times-frac0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

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