Average Error: 0.0 → 0.0
Time: 11.2s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{y}{\frac{y}{\frac{2}{x}} + 1}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{y}{\frac{y}{\frac{2}{x}} + 1}
double f(double x, double y) {
        double r214610 = x;
        double r214611 = y;
        double r214612 = 1.0;
        double r214613 = r214610 * r214611;
        double r214614 = 2.0;
        double r214615 = r214613 / r214614;
        double r214616 = r214612 + r214615;
        double r214617 = r214611 / r214616;
        double r214618 = r214610 - r214617;
        return r214618;
}
↓
double f(double x, double y) {
        double r214619 = x;
        double r214620 = y;
        double r214621 = 2.0;
        double r214622 = r214621 / r214619;
        double r214623 = r214620 / r214622;
        double r214624 = 1.0;
        double r214625 = r214623 + r214624;
        double r214626 = r214620 / r214625;
        double r214627 = r214619 - r214626;
        return r214627;
}

Error

Bits error versus x

Bits error versus y

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Results

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Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{x - \frac{y}{\frac{x \cdot y}{2} + 1}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied distribute-lft-out--0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))