Average Error: 0.0 → 0.0
Time: 7.1s
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 r7876121 = x;
        double r7876122 = y;
        double r7876123 = 1.0;
        double r7876124 = r7876121 * r7876122;
        double r7876125 = 2.0;
        double r7876126 = r7876124 / r7876125;
        double r7876127 = r7876123 + r7876126;
        double r7876128 = r7876122 / r7876127;
        double r7876129 = r7876121 - r7876128;
        return r7876129;
}
↓
double f(double x, double y) {
        double r7876130 = x;
        double r7876131 = y;
        double r7876132 = 2.0;
        double r7876133 = r7876131 / r7876132;
        double r7876134 = 1.0;
        double r7876135 = fma(r7876130, r7876133, r7876134);
        double r7876136 = r7876131 / r7876135;
        double r7876137 = r7876130 - r7876136;
        return r7876137;
}

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 2019192 +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)))))