Average Error: 0.0 → 0.0
Time: 3.1s
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 r214848 = x;
        double r214849 = y;
        double r214850 = 1.0;
        double r214851 = r214848 * r214849;
        double r214852 = 2.0;
        double r214853 = r214851 / r214852;
        double r214854 = r214850 + r214853;
        double r214855 = r214849 / r214854;
        double r214856 = r214848 - r214855;
        return r214856;
}
↓
double f(double x, double y) {
        double r214857 = x;
        double r214858 = y;
        double r214859 = 2.0;
        double r214860 = r214857 / r214859;
        double r214861 = 1.0;
        double r214862 = fma(r214860, r214858, r214861);
        double r214863 = r214858 / r214862;
        double r214864 = r214857 - r214863;
        return r214864;
}

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 2020047 +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)))))