Average Error: 0.0 → 0.1
Time: 13.9s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - y \cdot \frac{1}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - y \cdot \frac{1}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}
double f(double x, double y) {
        double r154274 = x;
        double r154275 = y;
        double r154276 = 1.0;
        double r154277 = r154274 * r154275;
        double r154278 = 2.0;
        double r154279 = r154277 / r154278;
        double r154280 = r154276 + r154279;
        double r154281 = r154275 / r154280;
        double r154282 = r154274 - r154281;
        return r154282;
}
↓
double f(double x, double y) {
        double r154283 = x;
        double r154284 = y;
        double r154285 = 1.0;
        double r154286 = 2.0;
        double r154287 = r154283 / r154286;
        double r154288 = 1.0;
        double r154289 = fma(r154287, r154284, r154288);
        double r154290 = r154285 / r154289;
        double r154291 = r154284 * r154290;
        double r154292 = r154283 - r154291;
        return r154292;
}

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. Using strategy rm

  4. Applied div-inv0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019325 +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)))))