Average Error: 0.0 → 0.0
Time: 6.7s
Precision: 64
\[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(x, \frac{y}{2.0}, 1.0\right)}\]
x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}
↓
x - \frac{y}{\mathsf{fma}\left(x, \frac{y}{2.0}, 1.0\right)}
double f(double x, double y) {
        double r9961093 = x;
        double r9961094 = y;
        double r9961095 = 1.0;
        double r9961096 = r9961093 * r9961094;
        double r9961097 = 2.0;
        double r9961098 = r9961096 / r9961097;
        double r9961099 = r9961095 + r9961098;
        double r9961100 = r9961094 / r9961099;
        double r9961101 = r9961093 - r9961100;
        return r9961101;
}
↓
double f(double x, double y) {
        double r9961102 = x;
        double r9961103 = y;
        double r9961104 = 2.0;
        double r9961105 = r9961103 / r9961104;
        double r9961106 = 1.0;
        double r9961107 = fma(r9961102, r9961105, r9961106);
        double r9961108 = r9961103 / r9961107;
        double r9961109 = r9961102 - r9961108;
        return r9961109;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +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)))))