Average Error: 0.0 → 0.1
Time: 13.5s
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 r142109 = x;
        double r142110 = y;
        double r142111 = 1.0;
        double r142112 = r142109 * r142110;
        double r142113 = 2.0;
        double r142114 = r142112 / r142113;
        double r142115 = r142111 + r142114;
        double r142116 = r142110 / r142115;
        double r142117 = r142109 - r142116;
        return r142117;
}
↓
double f(double x, double y) {
        double r142118 = x;
        double r142119 = 1.0;
        double r142120 = 2.0;
        double r142121 = r142118 / r142120;
        double r142122 = y;
        double r142123 = 1.0;
        double r142124 = fma(r142121, r142122, r142123);
        double r142125 = r142124 / r142122;
        double r142126 = r142119 / r142125;
        double r142127 = r142118 - r142126;
        return r142127;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 +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)))))