Average Error: 19.4 → 0.1
Time: 16.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) + 1.0}}{y + x}\]
\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{x}{y + x} \cdot \frac{y}{\left(y + x\right) + 1.0}}{y + x}
double f(double x, double y) {
        double r16227388 = x;
        double r16227389 = y;
        double r16227390 = r16227388 * r16227389;
        double r16227391 = r16227388 + r16227389;
        double r16227392 = r16227391 * r16227391;
        double r16227393 = 1.0;
        double r16227394 = r16227391 + r16227393;
        double r16227395 = r16227392 * r16227394;
        double r16227396 = r16227390 / r16227395;
        return r16227396;
}
↓
double f(double x, double y) {
        double r16227397 = x;
        double r16227398 = y;
        double r16227399 = r16227398 + r16227397;
        double r16227400 = r16227397 / r16227399;
        double r16227401 = 1.0;
        double r16227402 = r16227399 + r16227401;
        double r16227403 = r16227398 / r16227402;
        double r16227404 = r16227400 * r16227403;
        double r16227405 = r16227404 / r16227399;
        return r16227405;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.4
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 19.4

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

  3. Applied times-frac7.6

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}}\]
  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 associate-*l/0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019164 +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))))