Average Error: 0.0 → 0.0
Time: 4.8s
Precision: 64
\[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}\]
x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}
↓
x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}
double f(double x, double y) {
        double r3846347 = x;
        double r3846348 = y;
        double r3846349 = 1.0;
        double r3846350 = r3846347 * r3846348;
        double r3846351 = 2.0;
        double r3846352 = r3846350 / r3846351;
        double r3846353 = r3846349 + r3846352;
        double r3846354 = r3846348 / r3846353;
        double r3846355 = r3846347 - r3846354;
        return r3846355;
}
↓
double f(double x, double y) {
        double r3846356 = x;
        double r3846357 = y;
        double r3846358 = 2.0;
        double r3846359 = r3846356 / r3846358;
        double r3846360 = 1.0;
        double r3846361 = fma(r3846359, r3846357, r3846360);
        double r3846362 = r3846357 / r3846361;
        double r3846363 = r3846356 - r3846362;
        return r3846363;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}}\]

  3. Final simplification0.0

    \[\leadsto x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}\]

Reproduce

herbie shell --seed 2019156 +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)))))