Average Error: 0.0 → 0.1
Time: 14.7s
Precision: 64
\[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
↓
\[x - \frac{1}{\frac{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}{y}}\]
x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}
↓
x - \frac{1}{\frac{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}{y}}
double f(double x, double y) {
        double r6659392 = x;
        double r6659393 = y;
        double r6659394 = 1.0;
        double r6659395 = r6659392 * r6659393;
        double r6659396 = 2.0;
        double r6659397 = r6659395 / r6659396;
        double r6659398 = r6659394 + r6659397;
        double r6659399 = r6659393 / r6659398;
        double r6659400 = r6659392 - r6659399;
        return r6659400;
}
↓
double f(double x, double y) {
        double r6659401 = x;
        double r6659402 = 1.0;
        double r6659403 = 2.0;
        double r6659404 = r6659401 / r6659403;
        double r6659405 = y;
        double r6659406 = 1.0;
        double r6659407 = fma(r6659404, r6659405, r6659406);
        double r6659408 = r6659407 / r6659405;
        double r6659409 = r6659402 / r6659408;
        double r6659410 = r6659401 - r6659409;
        return r6659410;
}

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. Using strategy rm

  4. Applied clear-num0.1

    \[\leadsto x - \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}{y}}}\]

  5. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}{y}}\]

Reproduce

herbie shell --seed 2019162 +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)))))