Average Error: 0.0 → 0.0
Time: 13.6s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{1}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{1}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}
double f(double x, double y) {
        double r130688 = x;
        double r130689 = y;
        double r130690 = 1.0;
        double r130691 = r130688 * r130689;
        double r130692 = 2.0;
        double r130693 = r130691 / r130692;
        double r130694 = r130690 + r130693;
        double r130695 = r130689 / r130694;
        double r130696 = r130688 - r130695;
        return r130696;
}
↓
double f(double x, double y) {
        double r130697 = x;
        double r130698 = 1.0;
        double r130699 = 0.5;
        double r130700 = 1.0;
        double r130701 = y;
        double r130702 = r130700 / r130701;
        double r130703 = fma(r130699, r130697, r130702);
        double r130704 = r130698 / r130703;
        double r130705 = r130697 - r130704;
        return r130705;
}

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. Taylor expanded around 0 0.0

    \[\leadsto x - \frac{1}{\color{blue}{0.5 \cdot x + 1 \cdot \frac{1}{y}}}\]

  6. Simplified0.0

    \[\leadsto x - \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}}\]

  7. Final simplification0.0

    \[\leadsto x - \frac{1}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}\]

Reproduce

herbie shell --seed 2019322 +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)))))