Average Error: 0.0 → 0.1
Time: 9.8s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - y \cdot \frac{1}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - y \cdot \frac{1}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}
double f(double x, double y) {
        double r258575 = x;
        double r258576 = y;
        double r258577 = 1.0;
        double r258578 = r258575 * r258576;
        double r258579 = 2.0;
        double r258580 = r258578 / r258579;
        double r258581 = r258577 + r258580;
        double r258582 = r258576 / r258581;
        double r258583 = r258575 - r258582;
        return r258583;
}
↓
double f(double x, double y) {
        double r258584 = x;
        double r258585 = y;
        double r258586 = 1.0;
        double r258587 = 2.0;
        double r258588 = r258584 / r258587;
        double r258589 = 1.0;
        double r258590 = fma(r258588, r258585, r258589);
        double r258591 = r258586 / r258590;
        double r258592 = r258585 * r258591;
        double r258593 = r258584 - r258592;
        return r258593;
}

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 div-inv0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020046 +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)))))