Average Error: 0.0 → 0.0
Time: 6.1s
Precision: 64
\[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}\]
x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}
↓
x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}
double f(double x, double y) {
        double r9791424 = x;
        double r9791425 = y;
        double r9791426 = 1.0;
        double r9791427 = r9791424 * r9791425;
        double r9791428 = 2.0;
        double r9791429 = r9791427 / r9791428;
        double r9791430 = r9791426 + r9791429;
        double r9791431 = r9791425 / r9791430;
        double r9791432 = r9791424 - r9791431;
        return r9791432;
}
↓
double f(double x, double y) {
        double r9791433 = x;
        double r9791434 = y;
        double r9791435 = 2.0;
        double r9791436 = r9791433 / r9791435;
        double r9791437 = 1.0;
        double r9791438 = fma(r9791436, r9791434, r9791437);
        double r9791439 = r9791434 / r9791438;
        double r9791440 = r9791433 - r9791439;
        return r9791440;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}}\]

  3. Final simplification0.0

    \[\leadsto x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}\]

Reproduce

herbie shell --seed 2019163 +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)))))