Average Error: 0.0 → 0.0
Time: 8.5s
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 r255890 = x;
        double r255891 = y;
        double r255892 = 1.0;
        double r255893 = r255890 * r255891;
        double r255894 = 2.0;
        double r255895 = r255893 / r255894;
        double r255896 = r255892 + r255895;
        double r255897 = r255891 / r255896;
        double r255898 = r255890 - r255897;
        return r255898;
}
↓
double f(double x, double y) {
        double r255899 = x;
        double r255900 = y;
        double r255901 = 2.0;
        double r255902 = r255899 / r255901;
        double r255903 = 1.0;
        double r255904 = fma(r255902, r255900, r255903);
        double r255905 = r255900 / r255904;
        double r255906 = r255899 - r255905;
        return r255906;
}

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 2020042 +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)))))