Average Error: 0.0 → 0.1
Time: 1.8s
Precision: binary64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{1}{\frac{1}{y} + x \cdot 0.5}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{1}{\frac{1}{y} + x \cdot 0.5}
(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))
↓
(FPCore (x y) :precision binary64 (- x (/ 1.0 (+ (/ 1.0 y) (* x 0.5)))))
double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}
↓
double code(double x, double y) {
	return x - (1.0 / ((1.0 / y) + (x * 0.5)));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
  2. Using strategy rm

  3. Applied clear-num_binary640.1

    \[\leadsto x - \color{blue}{\frac{1}{\frac{1 + \frac{x \cdot y}{2}}{y}}}\]

  4. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{1 + \frac{y \cdot x}{2}}{y}}}\]

  5. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{0.5 \cdot x + \frac{1}{y}}}\]

  6. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{1}{y} + x \cdot 0.5}}\]

  7. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{1}{y} + x \cdot 0.5}\]

Reproduce

herbie shell --seed 2020232 
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  :precision binary64
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))