Average Error: 0.0 → 0.0
Time: 9.7s
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 r299810 = x;
        double r299811 = y;
        double r299812 = 1.0;
        double r299813 = r299810 * r299811;
        double r299814 = 2.0;
        double r299815 = r299813 / r299814;
        double r299816 = r299812 + r299815;
        double r299817 = r299811 / r299816;
        double r299818 = r299810 - r299817;
        return r299818;
}
↓
double f(double x, double y) {
        double r299819 = x;
        double r299820 = y;
        double r299821 = 2.0;
        double r299822 = r299819 / r299821;
        double r299823 = 1.0;
        double r299824 = fma(r299822, r299820, r299823);
        double r299825 = r299820 / r299824;
        double r299826 = r299819 - r299825;
        return r299826;
}

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 2019350 +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)))))