Average Error: 0.0 → 0.0
Time: 5.8s
Precision: 64
\[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}\]
x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}
↓
x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}
double f(double x, double y) {
        double r8208499 = x;
        double r8208500 = y;
        double r8208501 = 1.0;
        double r8208502 = r8208499 * r8208500;
        double r8208503 = 2.0;
        double r8208504 = r8208502 / r8208503;
        double r8208505 = r8208501 + r8208504;
        double r8208506 = r8208500 / r8208505;
        double r8208507 = r8208499 - r8208506;
        return r8208507;
}
↓
double f(double x, double y) {
        double r8208508 = x;
        double r8208509 = y;
        double r8208510 = 2.0;
        double r8208511 = r8208508 / r8208510;
        double r8208512 = 1.0;
        double r8208513 = fma(r8208511, r8208509, r8208512);
        double r8208514 = r8208509 / r8208513;
        double r8208515 = r8208508 - r8208514;
        return r8208515;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019168 +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)))))