Average Error: 0.0 → 0.1
Time: 13.7s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{1}{0.5 \cdot x + \frac{1}{y}}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{1}{0.5 \cdot x + \frac{1}{y}}
double f(double x, double y) {
        double r232016 = x;
        double r232017 = y;
        double r232018 = 1.0;
        double r232019 = r232016 * r232017;
        double r232020 = 2.0;
        double r232021 = r232019 / r232020;
        double r232022 = r232018 + r232021;
        double r232023 = r232017 / r232022;
        double r232024 = r232016 - r232023;
        return r232024;
}
↓
double f(double x, double y) {
        double r232025 = x;
        double r232026 = 1.0;
        double r232027 = 0.5;
        double r232028 = r232027 * r232025;
        double r232029 = 1.0;
        double r232030 = y;
        double r232031 = r232029 / r232030;
        double r232032 = r232028 + r232031;
        double r232033 = r232026 / r232032;
        double r232034 = r232025 - r232033;
        return r232034;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{0.5 \cdot x + 1 \cdot \frac{1}{y}}}\]

  5. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{0.5 \cdot x + \frac{1}{y}}}\]

  6. Final simplification0.1

    \[\leadsto x - \frac{1}{0.5 \cdot x + \frac{1}{y}}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  :precision binary64
  (- x (/ y (+ 1 (/ (* x y) 2)))))