Average Error: 0.0 → 0.1
Time: 12.5s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{1}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{1}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}
double f(double x, double y) {
        double r183170 = x;
        double r183171 = y;
        double r183172 = 1.0;
        double r183173 = r183170 * r183171;
        double r183174 = 2.0;
        double r183175 = r183173 / r183174;
        double r183176 = r183172 + r183175;
        double r183177 = r183171 / r183176;
        double r183178 = r183170 - r183177;
        return r183178;
}
↓
double f(double x, double y) {
        double r183179 = x;
        double r183180 = 1.0;
        double r183181 = 0.5;
        double r183182 = 1.0;
        double r183183 = y;
        double r183184 = r183182 / r183183;
        double r183185 = fma(r183181, r183179, r183184);
        double r183186 = r183180 / r183185;
        double r183187 = r183179 - r183186;
        return r183187;
}

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. Using strategy rm

  3. Applied clear-num0.1

    \[\leadsto x - \color{blue}{\frac{1}{\frac{1 + \frac{x \cdot y}{2}}{y}}}\]

  4. Simplified0.1

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

  5. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{0.5 \cdot x + 1 \cdot \frac{1}{y}}}\]

  6. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}}\]

  7. Final simplification0.1

    \[\leadsto x - \frac{1}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  :precision binary64
  (- x (/ y (+ 1 (/ (* x y) 2)))))