Average Error: 0.0 → 0.0
Time: 12.6s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{y}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}
double f(double x, double y) {
        double r9050414 = x;
        double r9050415 = y;
        double r9050416 = 1.0;
        double r9050417 = r9050414 * r9050415;
        double r9050418 = 2.0;
        double r9050419 = r9050417 / r9050418;
        double r9050420 = r9050416 + r9050419;
        double r9050421 = r9050415 / r9050420;
        double r9050422 = r9050414 - r9050421;
        return r9050422;
}
↓
double f(double x, double y) {
        double r9050423 = x;
        double r9050424 = y;
        double r9050425 = 2.0;
        double r9050426 = r9050423 / r9050425;
        double r9050427 = 1.0;
        double r9050428 = fma(r9050426, r9050424, r9050427);
        double r9050429 = r9050424 / r9050428;
        double r9050430 = r9050423 - r9050429;
        return r9050430;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}}\]

  3. Final simplification0.0

    \[\leadsto x - \frac{y}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))