Average Error: 0.0 → 0.0
Time: 10.5s
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 r195563 = x;
        double r195564 = y;
        double r195565 = 1.0;
        double r195566 = r195563 * r195564;
        double r195567 = 2.0;
        double r195568 = r195566 / r195567;
        double r195569 = r195565 + r195568;
        double r195570 = r195564 / r195569;
        double r195571 = r195563 - r195570;
        return r195571;
}
↓
double f(double x, double y) {
        double r195572 = x;
        double r195573 = y;
        double r195574 = 2.0;
        double r195575 = r195574 / r195572;
        double r195576 = r195573 / r195575;
        double r195577 = 1.0;
        double r195578 = r195576 + r195577;
        double r195579 = r195573 / r195578;
        double r195580 = r195572 - r195579;
        return r195580;
}

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)))))