Average Error: 0.0 → 0.0
Time: 8.6s
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 r237008 = x;
        double r237009 = y;
        double r237010 = 1.0;
        double r237011 = r237008 * r237009;
        double r237012 = 2.0;
        double r237013 = r237011 / r237012;
        double r237014 = r237010 + r237013;
        double r237015 = r237009 / r237014;
        double r237016 = r237008 - r237015;
        return r237016;
}
↓
double f(double x, double y) {
        double r237017 = x;
        double r237018 = y;
        double r237019 = 2.0;
        double r237020 = r237017 / r237019;
        double r237021 = 1.0;
        double r237022 = fma(r237020, r237018, r237021);
        double r237023 = r237018 / r237022;
        double r237024 = r237017 - r237023;
        return r237024;
}

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 2019198 +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)))))