Average Error: 0.0 → 0.1
Time: 25.4s
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 r11199086 = x;
        double r11199087 = y;
        double r11199088 = 1.0;
        double r11199089 = r11199086 * r11199087;
        double r11199090 = 2.0;
        double r11199091 = r11199089 / r11199090;
        double r11199092 = r11199088 + r11199091;
        double r11199093 = r11199087 / r11199092;
        double r11199094 = r11199086 - r11199093;
        return r11199094;
}
↓
double f(double x, double y) {
        double r11199095 = x;
        double r11199096 = 1.0;
        double r11199097 = 2.0;
        double r11199098 = r11199095 / r11199097;
        double r11199099 = y;
        double r11199100 = 1.0;
        double r11199101 = fma(r11199098, r11199099, r11199100);
        double r11199102 = r11199101 / r11199099;
        double r11199103 = r11199096 / r11199102;
        double r11199104 = r11199095 - r11199103;
        return r11199104;
}

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 2019200 +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)))))