Average Error: 19.6 → 0.1
Time: 4.5s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y} \cdot \frac{y}{x + y}}{\left(x + y\right) + 1}\]
\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} \cdot \frac{y}{x + y}}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r407256 = x;
        double r407257 = y;
        double r407258 = r407256 * r407257;
        double r407259 = r407256 + r407257;
        double r407260 = r407259 * r407259;
        double r407261 = 1.0;
        double r407262 = r407259 + r407261;
        double r407263 = r407260 * r407262;
        double r407264 = r407258 / r407263;
        return r407264;
}
↓
double f(double x, double y) {
        double r407265 = x;
        double r407266 = y;
        double r407267 = r407265 + r407266;
        double r407268 = r407265 / r407267;
        double r407269 = r407266 / r407267;
        double r407270 = r407268 * r407269;
        double r407271 = 1.0;
        double r407272 = r407267 + r407271;
        double r407273 = r407270 / r407272;
        return r407273;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
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 19.6

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

  3. Applied times-frac7.9

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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-*r/0.2

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

  9. Applied div-inv0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020081 
(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))))