Average Error: 0.0 → 0.1
Time: 9.6s
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 r128184 = x;
        double r128185 = y;
        double r128186 = 1.0;
        double r128187 = r128184 * r128185;
        double r128188 = 2.0;
        double r128189 = r128187 / r128188;
        double r128190 = r128186 + r128189;
        double r128191 = r128185 / r128190;
        double r128192 = r128184 - r128191;
        return r128192;
}
↓
double f(double x, double y) {
        double r128193 = x;
        double r128194 = y;
        double r128195 = 1.0;
        double r128196 = 2.0;
        double r128197 = r128193 / r128196;
        double r128198 = 1.0;
        double r128199 = fma(r128197, r128194, r128198);
        double r128200 = r128195 / r128199;
        double r128201 = r128194 * r128200;
        double r128202 = r128193 - r128201;
        return r128202;
}

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 2019235 +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)))))