Average Error: 20.0 → 0.1
Time: 4.2s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) + 1}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) + 1}}{x + y}
double f(double x, double y) {
        double r375316 = x;
        double r375317 = y;
        double r375318 = r375316 * r375317;
        double r375319 = r375316 + r375317;
        double r375320 = r375319 * r375319;
        double r375321 = 1.0;
        double r375322 = r375319 + r375321;
        double r375323 = r375320 * r375322;
        double r375324 = r375318 / r375323;
        return r375324;
}
↓
double f(double x, double y) {
        double r375325 = x;
        double r375326 = y;
        double r375327 = r375325 + r375326;
        double r375328 = r375325 / r375327;
        double r375329 = r375328 * r375326;
        double r375330 = 1.0;
        double r375331 = r375327 + r375330;
        double r375332 = r375329 / r375331;
        double r375333 = r375332 / r375327;
        return r375333;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
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 20.0

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

  3. Applied times-frac8.3

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

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020039 +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))))