Average Error: 19.9 → 0.4
Time: 12.7s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{\left(x + y\right) + 1}{y}}\]
\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{x}{x + y}}{\left(x + y\right) \cdot \frac{\left(x + y\right) + 1}{y}}
double f(double x, double y) {
        double r327398 = x;
        double r327399 = y;
        double r327400 = r327398 * r327399;
        double r327401 = r327398 + r327399;
        double r327402 = r327401 * r327401;
        double r327403 = 1.0;
        double r327404 = r327401 + r327403;
        double r327405 = r327402 * r327404;
        double r327406 = r327400 / r327405;
        return r327406;
}
↓
double f(double x, double y) {
        double r327407 = x;
        double r327408 = y;
        double r327409 = r327407 + r327408;
        double r327410 = r327407 / r327409;
        double r327411 = 1.0;
        double r327412 = r327409 + r327411;
        double r327413 = r327412 / r327408;
        double r327414 = r327409 * r327413;
        double r327415 = r327410 / r327414;
        return r327415;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.1
Herbie0.4
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.9

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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 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 div-inv0.2

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

  8. Applied associate-*l*0.2

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

  10. Applied clear-num0.2

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019298 
(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))))