Average Error: 0.0 → 0.0
Time: 11.8s
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 r10836151 = x;
        double r10836152 = y;
        double r10836153 = 1.0;
        double r10836154 = r10836151 * r10836152;
        double r10836155 = 2.0;
        double r10836156 = r10836154 / r10836155;
        double r10836157 = r10836153 + r10836156;
        double r10836158 = r10836152 / r10836157;
        double r10836159 = r10836151 - r10836158;
        return r10836159;
}
↓
double f(double x, double y) {
        double r10836160 = x;
        double r10836161 = y;
        double r10836162 = 2.0;
        double r10836163 = r10836160 / r10836162;
        double r10836164 = 1.0;
        double r10836165 = fma(r10836163, r10836161, r10836164);
        double r10836166 = r10836161 / r10836165;
        double r10836167 = r10836160 - r10836166;
        return r10836167;
}

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. Using strategy rm

  4. Applied div-inv0.1

    \[\leadsto x - \color{blue}{y \cdot \frac{1}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}}\]
  5. Using strategy rm

  6. Applied un-div-inv0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019164 +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)))))