Average Error: 19.9 → 0.1
Time: 20.1s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{x \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}}{x + y}
double f(double x, double y) {
        double r359250 = x;
        double r359251 = y;
        double r359252 = r359250 * r359251;
        double r359253 = r359250 + r359251;
        double r359254 = r359253 * r359253;
        double r359255 = 1.0;
        double r359256 = r359253 + r359255;
        double r359257 = r359254 * r359256;
        double r359258 = r359252 / r359257;
        return r359258;
}
↓
double f(double x, double y) {
        double r359259 = x;
        double r359260 = y;
        double r359261 = r359259 + r359260;
        double r359262 = 1.0;
        double r359263 = r359261 + r359262;
        double r359264 = r359260 / r359263;
        double r359265 = r359264 / r359261;
        double r359266 = r359259 * r359265;
        double r359267 = r359266 / r359261;
        return r359267;
}

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

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-frac8.1

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 associate-*l/0.1

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

  9. Applied div-inv0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

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