Average Error: 0.0 → 0.0
Time: 14.7s
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 r168303 = x;
        double r168304 = y;
        double r168305 = 1.0;
        double r168306 = r168303 * r168304;
        double r168307 = 2.0;
        double r168308 = r168306 / r168307;
        double r168309 = r168305 + r168308;
        double r168310 = r168304 / r168309;
        double r168311 = r168303 - r168310;
        return r168311;
}
↓
double f(double x, double y) {
        double r168312 = x;
        double r168313 = y;
        double r168314 = 2.0;
        double r168315 = r168312 / r168314;
        double r168316 = 1.0;
        double r168317 = fma(r168315, r168313, r168316);
        double r168318 = r168313 / r168317;
        double r168319 = r168312 - r168318;
        return r168319;
}

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 2019323 +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)))))