Average Error: 20.2 → 0.1
Time: 18.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y \cdot \frac{x}{x + y}}{x + y}}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{y \cdot \frac{x}{x + y}}{x + y}}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r429267 = x;
        double r429268 = y;
        double r429269 = r429267 * r429268;
        double r429270 = r429267 + r429268;
        double r429271 = r429270 * r429270;
        double r429272 = 1.0;
        double r429273 = r429270 + r429272;
        double r429274 = r429271 * r429273;
        double r429275 = r429269 / r429274;
        return r429275;
}
↓
double f(double x, double y) {
        double r429276 = y;
        double r429277 = x;
        double r429278 = r429277 + r429276;
        double r429279 = r429277 / r429278;
        double r429280 = r429276 * r429279;
        double r429281 = r429280 / r429278;
        double r429282 = 1.0;
        double r429283 = r429278 + r429282;
        double r429284 = r429281 / r429283;
        return r429284;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
Target0.1
Herbie0.1
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 20.2

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

  3. Applied distribute-lft-in20.2

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

  5. Applied times-frac10.8

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

  6. Simplified8.3

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

  8. Applied *-un-lft-identity8.3

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

  9. Applied times-frac0.2

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

  10. Applied associate-*l*0.2

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

  12. Applied associate-*r/0.2

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

  13. Applied associate-*r/0.2

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x))))

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