Average Error: 0.0 → 0.1
Time: 12.1s
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 r14962615 = x;
        double r14962616 = y;
        double r14962617 = 1.0;
        double r14962618 = r14962615 * r14962616;
        double r14962619 = 2.0;
        double r14962620 = r14962618 / r14962619;
        double r14962621 = r14962617 + r14962620;
        double r14962622 = r14962616 / r14962621;
        double r14962623 = r14962615 - r14962622;
        return r14962623;
}
↓
double f(double x, double y) {
        double r14962624 = x;
        double r14962625 = 1.0;
        double r14962626 = 2.0;
        double r14962627 = r14962624 / r14962626;
        double r14962628 = y;
        double r14962629 = 1.0;
        double r14962630 = fma(r14962627, r14962628, r14962629);
        double r14962631 = r14962630 / r14962628;
        double r14962632 = r14962625 / r14962631;
        double r14962633 = r14962624 - r14962632;
        return r14962633;
}

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 2019172 +o rules:numerics
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))