Average Error: 0.0 → 0.1
Time: 7.7s
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 r221161 = x;
        double r221162 = y;
        double r221163 = 1.0;
        double r221164 = r221161 * r221162;
        double r221165 = 2.0;
        double r221166 = r221164 / r221165;
        double r221167 = r221163 + r221166;
        double r221168 = r221162 / r221167;
        double r221169 = r221161 - r221168;
        return r221169;
}
↓
double f(double x, double y) {
        double r221170 = x;
        double r221171 = 1.0;
        double r221172 = 0.5;
        double r221173 = 1.0;
        double r221174 = y;
        double r221175 = r221173 / r221174;
        double r221176 = fma(r221172, r221170, r221175);
        double r221177 = r221171 / r221176;
        double r221178 = r221170 - r221177;
        return r221178;
}

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. 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 2020043 +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)))))