Average Error: 0.0 → 0.1
Time: 12.5s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{1}{\frac{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}{y}}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{1}{\frac{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}{y}}
double f(double x, double y) {
        double r10435705 = x;
        double r10435706 = y;
        double r10435707 = 1.0;
        double r10435708 = r10435705 * r10435706;
        double r10435709 = 2.0;
        double r10435710 = r10435708 / r10435709;
        double r10435711 = r10435707 + r10435710;
        double r10435712 = r10435706 / r10435711;
        double r10435713 = r10435705 - r10435712;
        return r10435713;
}
↓
double f(double x, double y) {
        double r10435714 = x;
        double r10435715 = 1.0;
        double r10435716 = 2.0;
        double r10435717 = r10435714 / r10435716;
        double r10435718 = y;
        double r10435719 = 1.0;
        double r10435720 = fma(r10435717, r10435718, r10435719);
        double r10435721 = r10435720 / r10435718;
        double r10435722 = r10435715 / r10435721;
        double r10435723 = r10435714 - r10435722;
        return r10435723;
}

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

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

  4. Applied clear-num0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 +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)))))