Average Error: 19.2 → 0.2
Time: 4.7s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\left(\frac{\frac{x}{x + y}}{x + y} \cdot y\right) \cdot \frac{1}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\left(\frac{\frac{x}{x + y}}{x + y} \cdot y\right) \cdot \frac{1}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r485368 = x;
        double r485369 = y;
        double r485370 = r485368 * r485369;
        double r485371 = r485368 + r485369;
        double r485372 = r485371 * r485371;
        double r485373 = 1.0;
        double r485374 = r485371 + r485373;
        double r485375 = r485372 * r485374;
        double r485376 = r485370 / r485375;
        return r485376;
}
↓
double f(double x, double y) {
        double r485377 = x;
        double r485378 = y;
        double r485379 = r485377 + r485378;
        double r485380 = r485377 / r485379;
        double r485381 = r485380 / r485379;
        double r485382 = r485381 * r485378;
        double r485383 = 1.0;
        double r485384 = 1.0;
        double r485385 = r485379 + r485384;
        double r485386 = r485383 / r485385;
        double r485387 = r485382 * r485386;
        return r485387;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 19.2

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

  3. Applied times-frac7.6

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

  5. Applied associate-/r*0.1

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

  7. Applied div-inv0.2

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

  8. Applied associate-*r*0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020003 
(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))))