Average Error: 0.0 → 0.0
Time: 9.1s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(x, \frac{y}{2}, 1\right)}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{y}{\mathsf{fma}\left(x, \frac{y}{2}, 1\right)}
double f(double x, double y) {
        double r9030102 = x;
        double r9030103 = y;
        double r9030104 = 1.0;
        double r9030105 = r9030102 * r9030103;
        double r9030106 = 2.0;
        double r9030107 = r9030105 / r9030106;
        double r9030108 = r9030104 + r9030107;
        double r9030109 = r9030103 / r9030108;
        double r9030110 = r9030102 - r9030109;
        return r9030110;
}
↓
double f(double x, double y) {
        double r9030111 = x;
        double r9030112 = y;
        double r9030113 = 2.0;
        double r9030114 = r9030112 / r9030113;
        double r9030115 = 1.0;
        double r9030116 = fma(r9030111, r9030114, r9030115);
        double r9030117 = r9030112 / r9030116;
        double r9030118 = r9030111 - r9030117;
        return r9030118;
}

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

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, \frac{y}{2}, 1\right)}}\]

  3. Final simplification0.0

    \[\leadsto x - \frac{y}{\mathsf{fma}\left(x, \frac{y}{2}, 1\right)}\]

Reproduce

herbie shell --seed 2019174 +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)))))