Average Error: 19.3 → 0.1
Time: 18.5s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{\frac{y}{y + \left(x + 1.0\right)}}{y + x} \cdot \frac{x}{y + x}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}
↓
\frac{\frac{y}{y + \left(x + 1.0\right)}}{y + x} \cdot \frac{x}{y + x}
double f(double x, double y) {
        double r19777406 = x;
        double r19777407 = y;
        double r19777408 = r19777406 * r19777407;
        double r19777409 = r19777406 + r19777407;
        double r19777410 = r19777409 * r19777409;
        double r19777411 = 1.0;
        double r19777412 = r19777409 + r19777411;
        double r19777413 = r19777410 * r19777412;
        double r19777414 = r19777408 / r19777413;
        return r19777414;
}
↓
double f(double x, double y) {
        double r19777415 = y;
        double r19777416 = x;
        double r19777417 = 1.0;
        double r19777418 = r19777416 + r19777417;
        double r19777419 = r19777415 + r19777418;
        double r19777420 = r19777415 / r19777419;
        double r19777421 = r19777415 + r19777416;
        double r19777422 = r19777420 / r19777421;
        double r19777423 = r19777416 / r19777421;
        double r19777424 = r19777422 * r19777423;
        return r19777424;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.3
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 19.3

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
  2. Using strategy rm

  3. Applied times-frac8.0

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}}\]
  4. Using strategy rm

  5. Applied associate-/r*0.2

    \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(x + y\right) + 1.0}\]
  6. Using strategy rm

  7. Applied div-inv0.2

    \[\leadsto \color{blue}{\left(\frac{x}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}\]

  8. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \left(\frac{1}{x + y} \cdot \frac{y}{\left(x + y\right) + 1.0}\right)}\]

  9. Simplified0.1

    \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{\left(1.0 + x\right) + y}}{x + y}}\]

  10. Final simplification0.1

    \[\leadsto \frac{\frac{y}{y + \left(x + 1.0\right)}}{y + x} \cdot \frac{x}{y + x}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

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

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