Average Error: 0.0 → 0.0
Time: 4.1s
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 r274410 = x;
        double r274411 = y;
        double r274412 = 1.0;
        double r274413 = r274410 * r274411;
        double r274414 = 2.0;
        double r274415 = r274413 / r274414;
        double r274416 = r274412 + r274415;
        double r274417 = r274411 / r274416;
        double r274418 = r274410 - r274417;
        return r274418;
}
↓
double f(double x, double y) {
        double r274419 = x;
        double r274420 = y;
        double r274421 = 2.0;
        double r274422 = r274419 / r274421;
        double r274423 = 1.0;
        double r274424 = fma(r274422, r274420, r274423);
        double r274425 = r274420 / r274424;
        double r274426 = r274419 - r274425;
        return r274426;
}

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 2020047 +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)))))