Average Error: 0.0 → 0.0
Time: 1.6s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{1}{\mathsf{fma}\left(x, 0.5, \frac{1}{y}\right)}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{1}{\mathsf{fma}\left(x, 0.5, \frac{1}{y}\right)}
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 / fma(x, 0.5, (1.0 / y))));
}

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-num0.1

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

  4. Taylor expanded around 0 0.0

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

  5. Simplified0.0

    \[\leadsto x - \frac{1}{\color{blue}{\mathsf{fma}\left(x, 0.5, \frac{1}{y}\right)}}\]

  6. Final simplification0.0

    \[\leadsto x - \frac{1}{\mathsf{fma}\left(x, 0.5, \frac{1}{y}\right)}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  :precision binary64
  (- x (/ y (+ 1 (/ (* x y) 2)))))