Average Error: 19.6 → 0.1
Time: 16.3s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{y}{y + x} \cdot x}{y + x}}{1 + \left(y + x\right)}\]
\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{y}{y + x} \cdot x}{y + x}}{1 + \left(y + x\right)}
double f(double x, double y) {
        double r421210 = x;
        double r421211 = y;
        double r421212 = r421210 * r421211;
        double r421213 = r421210 + r421211;
        double r421214 = r421213 * r421213;
        double r421215 = 1.0;
        double r421216 = r421213 + r421215;
        double r421217 = r421214 * r421216;
        double r421218 = r421212 / r421217;
        return r421218;
}
↓
double f(double x, double y) {
        double r421219 = y;
        double r421220 = x;
        double r421221 = r421219 + r421220;
        double r421222 = r421219 / r421221;
        double r421223 = r421222 * r421220;
        double r421224 = r421223 / r421221;
        double r421225 = 1.0;
        double r421226 = r421225 + r421221;
        double r421227 = r421224 / r421226;
        return r421227;
}

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. Simplified19.6

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

  4. Applied times-frac7.8

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

  5. Simplified7.8

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

  6. Simplified0.2

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

  8. Applied associate-*l/0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 
(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))))