Average Error: 0.0 → 0.0
Time: 6.8s
Precision: 64
\[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}\]
x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}
↓
x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}
double f(double x, double y) {
        double r11980269 = x;
        double r11980270 = y;
        double r11980271 = 1.0;
        double r11980272 = r11980269 * r11980270;
        double r11980273 = 2.0;
        double r11980274 = r11980272 / r11980273;
        double r11980275 = r11980271 + r11980274;
        double r11980276 = r11980270 / r11980275;
        double r11980277 = r11980269 - r11980276;
        return r11980277;
}
↓
double f(double x, double y) {
        double r11980278 = x;
        double r11980279 = y;
        double r11980280 = 2.0;
        double r11980281 = r11980278 / r11980280;
        double r11980282 = 1.0;
        double r11980283 = fma(r11980281, r11980279, r11980282);
        double r11980284 = r11980279 / r11980283;
        double r11980285 = r11980278 - r11980284;
        return r11980285;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}}\]

  3. Final simplification0.0

    \[\leadsto x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}\]

Reproduce

herbie shell --seed 2019163 +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)))))