Average Error: 0.0 → 0.0
Time: 12.2s
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 r231449 = x;
        double r231450 = y;
        double r231451 = 1.0;
        double r231452 = r231449 * r231450;
        double r231453 = 2.0;
        double r231454 = r231452 / r231453;
        double r231455 = r231451 + r231454;
        double r231456 = r231450 / r231455;
        double r231457 = r231449 - r231456;
        return r231457;
}
↓
double f(double x, double y) {
        double r231458 = x;
        double r231459 = y;
        double r231460 = 2.0;
        double r231461 = r231459 / r231460;
        double r231462 = 1.0;
        double r231463 = fma(r231458, r231461, r231462);
        double r231464 = r231459 / r231463;
        double r231465 = r231458 - r231464;
        return r231465;
}

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 2019194 +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)))))