Average Error: 0.0 → 0.1
Time: 7.2s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{1}{\frac{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}{y}}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{1}{\frac{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}{y}}
double f(double x, double y) {
        double r156695 = x;
        double r156696 = y;
        double r156697 = 1.0;
        double r156698 = r156695 * r156696;
        double r156699 = 2.0;
        double r156700 = r156698 / r156699;
        double r156701 = r156697 + r156700;
        double r156702 = r156696 / r156701;
        double r156703 = r156695 - r156702;
        return r156703;
}
↓
double f(double x, double y) {
        double r156704 = x;
        double r156705 = 1.0;
        double r156706 = 2.0;
        double r156707 = r156704 / r156706;
        double r156708 = y;
        double r156709 = 1.0;
        double r156710 = fma(r156707, r156708, r156709);
        double r156711 = r156710 / r156708;
        double r156712 = r156705 / r156711;
        double r156713 = r156704 - r156712;
        return r156713;
}

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 clear-num0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019208 +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)))))