Average Error: 19.5 → 0.2
Time: 9.5s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r438223 = x;
        double r438224 = y;
        double r438225 = r438223 * r438224;
        double r438226 = r438223 + r438224;
        double r438227 = r438226 * r438226;
        double r438228 = 1.0;
        double r438229 = r438226 + r438228;
        double r438230 = r438227 * r438229;
        double r438231 = r438225 / r438230;
        return r438231;
}
↓
double f(double x, double y) {
        double r438232 = x;
        double r438233 = y;
        double r438234 = r438232 + r438233;
        double r438235 = r438232 / r438234;
        double r438236 = r438235 / r438234;
        double r438237 = 1.0;
        double r438238 = r438234 + r438237;
        double r438239 = r438233 / r438238;
        double r438240 = r438236 * r438239;
        return r438240;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.5
Target0.2
Herbie0.2
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.5

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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))))