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 r163745 = x;
        double r163746 = y;
        double r163747 = 1.0;
        double r163748 = r163745 * r163746;
        double r163749 = 2.0;
        double r163750 = r163748 / r163749;
        double r163751 = r163747 + r163750;
        double r163752 = r163746 / r163751;
        double r163753 = r163745 - r163752;
        return r163753;
}
↓
double f(double x, double y) {
        double r163754 = x;
        double r163755 = y;
        double r163756 = 2.0;
        double r163757 = r163754 / r163756;
        double r163758 = 1.0;
        double r163759 = fma(r163757, r163755, r163758);
        double r163760 = r163755 / r163759;
        double r163761 = r163754 - r163760;
        return r163761;
}

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)))))