Average Error: 19.9 → 0.1
Time: 4.1s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \cdot \frac{x}{x + y}\]
\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}}{x + y} \cdot \frac{x}{x + y}
double f(double x, double y) {
        double r484013 = x;
        double r484014 = y;
        double r484015 = r484013 * r484014;
        double r484016 = r484013 + r484014;
        double r484017 = r484016 * r484016;
        double r484018 = 1.0;
        double r484019 = r484016 + r484018;
        double r484020 = r484017 * r484019;
        double r484021 = r484015 / r484020;
        return r484021;
}
↓
double f(double x, double y) {
        double r484022 = y;
        double r484023 = x;
        double r484024 = r484023 + r484022;
        double r484025 = 1.0;
        double r484026 = r484024 + r484025;
        double r484027 = r484022 / r484026;
        double r484028 = r484027 / r484024;
        double r484029 = r484023 / r484024;
        double r484030 = r484028 * r484029;
        return r484030;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
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 19.9

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

  3. Applied times-frac8.0

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

  5. Applied *-un-lft-identity8.0

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

  6. Applied times-frac0.2

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

  8. Applied associate-*l*0.2

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

  9. Simplified0.2

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

  11. Applied associate-*r*0.2

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 
(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))))