Average Error: 20.0 → 0.1
Time: 15.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}
double f(double x, double y) {
        double r396231 = x;
        double r396232 = y;
        double r396233 = r396231 * r396232;
        double r396234 = r396231 + r396232;
        double r396235 = r396234 * r396234;
        double r396236 = 1.0;
        double r396237 = r396234 + r396236;
        double r396238 = r396235 * r396237;
        double r396239 = r396233 / r396238;
        return r396239;
}
↓
double f(double x, double y) {
        double r396240 = y;
        double r396241 = x;
        double r396242 = r396241 + r396240;
        double r396243 = 1.0;
        double r396244 = r396242 + r396243;
        double r396245 = r396240 / r396244;
        double r396246 = r396241 / r396242;
        double r396247 = r396245 * r396246;
        double r396248 = r396247 / r396242;
        return r396248;
}

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-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. Simplified7.8

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

  5. Simplified7.8

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

  7. Applied *-un-lft-identity7.8

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

  8. Applied associate-*l*7.8

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019194 
(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))))