Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Average Error: 20.2 → 0.1

Time: 3.1s

Precision: 64

Math

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

↓

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

C

double code(double x, double y) {
	return ((x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0)));
}

↓

double code(double x, double y) {
	return ((y / (x + y)) * ((x / (x + y)) / ((x + y) + 1.0)));
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.2
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 20.2

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

3. Applied associate-*l* 20.2

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

4. Applied *-commutative 20.2

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

5. Applied times-frac 3.9

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

7. Applied associate-/r* 0.1

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

8. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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))))