Average Error: 0.0 → 0.0
Time: 11.2s
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 r140067 = x;
        double r140068 = y;
        double r140069 = 1.0;
        double r140070 = r140067 * r140068;
        double r140071 = 2.0;
        double r140072 = r140070 / r140071;
        double r140073 = r140069 + r140072;
        double r140074 = r140068 / r140073;
        double r140075 = r140067 - r140074;
        return r140075;
}
↓
double f(double x, double y) {
        double r140076 = x;
        double r140077 = y;
        double r140078 = 2.0;
        double r140079 = r140076 / r140078;
        double r140080 = 1.0;
        double r140081 = fma(r140079, r140077, r140080);
        double r140082 = r140077 / r140081;
        double r140083 = r140076 - r140082;
        return r140083;
}

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 2019304 +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)))))