Average Error: 0.0 → 0.1
Time: 3.1s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{1}{\frac{1 + \frac{x \cdot y}{2}}{y}}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{1}{\frac{1 + \frac{x \cdot y}{2}}{y}}
double f(double x, double y) {
        double r228612 = x;
        double r228613 = y;
        double r228614 = 1.0;
        double r228615 = r228612 * r228613;
        double r228616 = 2.0;
        double r228617 = r228615 / r228616;
        double r228618 = r228614 + r228617;
        double r228619 = r228613 / r228618;
        double r228620 = r228612 - r228619;
        return r228620;
}
↓
double f(double x, double y) {
        double r228621 = x;
        double r228622 = 1.0;
        double r228623 = 1.0;
        double r228624 = y;
        double r228625 = r228621 * r228624;
        double r228626 = 2.0;
        double r228627 = r228625 / r228626;
        double r228628 = r228623 + r228627;
        double r228629 = r228628 / r228624;
        double r228630 = r228622 / r228629;
        double r228631 = r228621 - r228630;
        return r228631;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2020001 
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  :precision binary64
  (- x (/ y (+ 1 (/ (* x y) 2)))))