Average Error: 0.0 → 0.0
Time: 10.5s
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 r215678 = x;
        double r215679 = y;
        double r215680 = 1.0;
        double r215681 = r215678 * r215679;
        double r215682 = 2.0;
        double r215683 = r215681 / r215682;
        double r215684 = r215680 + r215683;
        double r215685 = r215679 / r215684;
        double r215686 = r215678 - r215685;
        return r215686;
}
↓
double f(double x, double y) {
        double r215687 = x;
        double r215688 = y;
        double r215689 = 2.0;
        double r215690 = r215687 / r215689;
        double r215691 = 1.0;
        double r215692 = fma(r215690, r215688, r215691);
        double r215693 = r215688 / r215692;
        double r215694 = r215687 - r215693;
        return r215694;
}

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 2019306 +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)))))