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 r213496 = x;
        double r213497 = y;
        double r213498 = 1.0;
        double r213499 = r213496 * r213497;
        double r213500 = 2.0;
        double r213501 = r213499 / r213500;
        double r213502 = r213498 + r213501;
        double r213503 = r213497 / r213502;
        double r213504 = r213496 - r213503;
        return r213504;
}
↓
double f(double x, double y) {
        double r213505 = x;
        double r213506 = y;
        double r213507 = 2.0;
        double r213508 = r213505 / r213507;
        double r213509 = 1.0;
        double r213510 = fma(r213508, r213506, r213509);
        double r213511 = r213506 / r213510;
        double r213512 = r213505 - r213511;
        return r213512;
}

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 2019347 +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)))))