Average Error: 19.5 → 0.1
Time: 41.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x}{y + x} \cdot \frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{x}{y + x} \cdot \frac{\frac{y}{\left(y + x\right) + 1}}{y + x}
double f(double x, double y) {
        double r22877171 = x;
        double r22877172 = y;
        double r22877173 = r22877171 * r22877172;
        double r22877174 = r22877171 + r22877172;
        double r22877175 = r22877174 * r22877174;
        double r22877176 = 1.0;
        double r22877177 = r22877174 + r22877176;
        double r22877178 = r22877175 * r22877177;
        double r22877179 = r22877173 / r22877178;
        return r22877179;
}
↓
double f(double x, double y) {
        double r22877180 = x;
        double r22877181 = y;
        double r22877182 = r22877181 + r22877180;
        double r22877183 = r22877180 / r22877182;
        double r22877184 = 1.0;
        double r22877185 = r22877182 + r22877184;
        double r22877186 = r22877181 / r22877185;
        double r22877187 = r22877186 / r22877182;
        double r22877188 = r22877183 * r22877187;
        return r22877188;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original19.5
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 19.5

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

  3. Applied times-frac8.1

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 div-inv0.2

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

  8. Applied associate-*l*0.2

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 
(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))))