Average Error: 0.0 → 0.0
Time: 8.5s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(x, \frac{y}{2}, 1\right)}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{y}{\mathsf{fma}\left(x, \frac{y}{2}, 1\right)}
double f(double x, double y) {
        double r149153 = x;
        double r149154 = y;
        double r149155 = 1.0;
        double r149156 = r149153 * r149154;
        double r149157 = 2.0;
        double r149158 = r149156 / r149157;
        double r149159 = r149155 + r149158;
        double r149160 = r149154 / r149159;
        double r149161 = r149153 - r149160;
        return r149161;
}
↓
double f(double x, double y) {
        double r149162 = x;
        double r149163 = y;
        double r149164 = 2.0;
        double r149165 = r149163 / r149164;
        double r149166 = 1.0;
        double r149167 = fma(r149162, r149165, r149166);
        double r149168 = r149163 / r149167;
        double r149169 = r149162 - r149168;
        return r149169;
}

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

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, \frac{y}{2}, 1\right)}}\]

  3. Final simplification0.0

    \[\leadsto x - \frac{y}{\mathsf{fma}\left(x, \frac{y}{2}, 1\right)}\]

Reproduce

herbie shell --seed 2019196 +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)))))