Average Error: 0.0 → 0.0
Time: 11.5s
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 r14919475 = x;
        double r14919476 = y;
        double r14919477 = 1.0;
        double r14919478 = r14919475 * r14919476;
        double r14919479 = 2.0;
        double r14919480 = r14919478 / r14919479;
        double r14919481 = r14919477 + r14919480;
        double r14919482 = r14919476 / r14919481;
        double r14919483 = r14919475 - r14919482;
        return r14919483;
}
↓
double f(double x, double y) {
        double r14919484 = x;
        double r14919485 = y;
        double r14919486 = 2.0;
        double r14919487 = r14919484 / r14919486;
        double r14919488 = 1.0;
        double r14919489 = fma(r14919487, r14919485, r14919488);
        double r14919490 = r14919485 / r14919489;
        double r14919491 = r14919484 - r14919490;
        return r14919491;
}

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 2019174 +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)))))