Average Error: 19.6 → 0.2
Time: 15.9s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\left(\frac{1}{x + y} \cdot \frac{x}{x + y}\right) \cdot \frac{y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\left(\frac{1}{x + y} \cdot \frac{x}{x + y}\right) \cdot \frac{y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r24851151 = x;
        double r24851152 = y;
        double r24851153 = r24851151 * r24851152;
        double r24851154 = r24851151 + r24851152;
        double r24851155 = r24851154 * r24851154;
        double r24851156 = 1.0;
        double r24851157 = r24851154 + r24851156;
        double r24851158 = r24851155 * r24851157;
        double r24851159 = r24851153 / r24851158;
        return r24851159;
}
↓
double f(double x, double y) {
        double r24851160 = 1.0;
        double r24851161 = x;
        double r24851162 = y;
        double r24851163 = r24851161 + r24851162;
        double r24851164 = r24851160 / r24851163;
        double r24851165 = r24851161 / r24851163;
        double r24851166 = r24851164 * r24851165;
        double r24851167 = 1.0;
        double r24851168 = r24851163 + r24851167;
        double r24851169 = r24851162 / r24851168;
        double r24851170 = r24851166 * r24851169;
        return r24851170;
}

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
Herbie0.2
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

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-frac7.8

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

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

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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))))