Average Error: 19.5 → 0.1
Time: 8.9s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{x + y} \cdot y}{x + y}}{\left(x + y\right) + 1}\]
\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{\frac{x}{x + y} \cdot y}{x + y}}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r475359 = x;
        double r475360 = y;
        double r475361 = r475359 * r475360;
        double r475362 = r475359 + r475360;
        double r475363 = r475362 * r475362;
        double r475364 = 1.0;
        double r475365 = r475362 + r475364;
        double r475366 = r475363 * r475365;
        double r475367 = r475361 / r475366;
        return r475367;
}
↓
double f(double x, double y) {
        double r475368 = x;
        double r475369 = y;
        double r475370 = r475368 + r475369;
        double r475371 = r475368 / r475370;
        double r475372 = r475371 * r475369;
        double r475373 = r475372 / r475370;
        double r475374 = 1.0;
        double r475375 = r475370 + r475374;
        double r475376 = r475373 / r475375;
        return r475376;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.5
Target0.2
Herbie0.1
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.5

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

  3. Applied times-frac7.8

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

  5. Applied *-un-lft-identity7.8

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

  6. Applied times-frac0.2

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

  7. Applied associate-*l*0.2

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

  9. Applied associate-*r/0.2

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

  10. Applied associate-*r/0.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 
(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))))