Average Error: 0.0 → 0.0
Time: 10.9s
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 r249663 = x;
        double r249664 = y;
        double r249665 = 1.0;
        double r249666 = r249663 * r249664;
        double r249667 = 2.0;
        double r249668 = r249666 / r249667;
        double r249669 = r249665 + r249668;
        double r249670 = r249664 / r249669;
        double r249671 = r249663 - r249670;
        return r249671;
}
↓
double f(double x, double y) {
        double r249672 = x;
        double r249673 = y;
        double r249674 = 2.0;
        double r249675 = r249672 / r249674;
        double r249676 = 1.0;
        double r249677 = fma(r249675, r249673, r249676);
        double r249678 = r249673 / r249677;
        double r249679 = r249672 - r249678;
        return r249679;
}

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 2020045 +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)))))