Average Error: 0.0 → 0.0
Time: 3.6s
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 r204338 = x;
        double r204339 = y;
        double r204340 = 1.0;
        double r204341 = r204338 * r204339;
        double r204342 = 2.0;
        double r204343 = r204341 / r204342;
        double r204344 = r204340 + r204343;
        double r204345 = r204339 / r204344;
        double r204346 = r204338 - r204345;
        return r204346;
}
↓
double f(double x, double y) {
        double r204347 = x;
        double r204348 = y;
        double r204349 = 2.0;
        double r204350 = r204347 / r204349;
        double r204351 = 1.0;
        double r204352 = fma(r204350, r204348, r204351);
        double r204353 = r204348 / r204352;
        double r204354 = r204347 - r204353;
        return r204354;
}

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)))))