Average Error: 19.7 → 0.5
Time: 9.3s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{1}{\frac{x + y}{\frac{x}{x + y}}} \cdot \frac{y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{1}{\frac{x + y}{\frac{x}{x + y}}} \cdot \frac{y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r498710 = x;
        double r498711 = y;
        double r498712 = r498710 * r498711;
        double r498713 = r498710 + r498711;
        double r498714 = r498713 * r498713;
        double r498715 = 1.0;
        double r498716 = r498713 + r498715;
        double r498717 = r498714 * r498716;
        double r498718 = r498712 / r498717;
        return r498718;
}
↓
double f(double x, double y) {
        double r498719 = 1.0;
        double r498720 = x;
        double r498721 = y;
        double r498722 = r498720 + r498721;
        double r498723 = r498720 / r498722;
        double r498724 = r498722 / r498723;
        double r498725 = r498719 / r498724;
        double r498726 = 1.0;
        double r498727 = r498722 + r498726;
        double r498728 = r498721 / r498727;
        double r498729 = r498725 * r498728;
        return r498729;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.1
Herbie0.5
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied times-frac7.5

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 *-un-lft-identity0.2

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

  8. Applied *-un-lft-identity0.2

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

  9. Applied times-frac0.2

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

  10. Applied associate-/l*0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2020001 
(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))))