Average Error: 19.7 → 0.2
Time: 3.5s
Precision: binary64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x}{x + y} \cdot \left(\frac{1}{x + y} \cdot \frac{y}{y + \left(x + 1\right)}\right)\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{x}{x + y} \cdot \left(\frac{1}{x + y} \cdot \frac{y}{y + \left(x + 1\right)}\right)
(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (* (/ x (+ x y)) (* (/ 1.0 (+ x y)) (/ y (+ y (+ x 1.0))))))
double code(double x, double y) {
	return (((double) (x * y)) / ((double) (((double) (((double) (x + y)) * ((double) (x + y)))) * ((double) (((double) (x + y)) + 1.0)))));
}
↓
double code(double x, double y) {
	return ((double) ((x / ((double) (x + y))) * ((double) ((1.0 / ((double) (x + y))) * (y / ((double) (y + ((double) (x + 1.0)))))))));
}

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

Derivation

  1. Initial program Error: 19.7 bits

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

  2. SimplifiedError: 11.3 bits

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

  4. Applied *-un-lft-identityError: 11.3 bits

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

  5. Applied times-fracError: 5.8 bits

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

  6. Applied associate-*r*Error: 4.0 bits

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

  7. SimplifiedError: 3.9 bits

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

  9. Applied *-un-lft-identityError: 3.9 bits

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

  10. Applied times-fracError: 0.2 bits

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

  11. SimplifiedError: 0.2 bits

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

  12. SimplifiedError: 0.2 bits

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

  13. Final simplificationError: 0.2 bits

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

Reproduce

herbie shell --seed 2020204 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))