Average Error: 0.0 → 0.1
Time: 12.5s
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 r191629 = x;
        double r191630 = y;
        double r191631 = 1.0;
        double r191632 = r191629 * r191630;
        double r191633 = 2.0;
        double r191634 = r191632 / r191633;
        double r191635 = r191631 + r191634;
        double r191636 = r191630 / r191635;
        double r191637 = r191629 - r191636;
        return r191637;
}
↓
double f(double x, double y) {
        double r191638 = x;
        double r191639 = 1.0;
        double r191640 = 0.5;
        double r191641 = 1.0;
        double r191642 = y;
        double r191643 = r191641 / r191642;
        double r191644 = fma(r191640, r191638, r191643);
        double r191645 = r191639 / r191644;
        double r191646 = r191638 - r191645;
        return r191646;
}

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. Using strategy rm

  3. Applied clear-num0.1

    \[\leadsto x - \color{blue}{\frac{1}{\frac{1 + \frac{x \cdot y}{2}}{y}}}\]

  4. Simplified0.1

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

  5. Taylor expanded around 0 0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 +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)))))