Average Error: 0.0 → 0.1
Time: 11.9s
Precision: 64
\[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
↓
\[x - \frac{1}{x \cdot 0.5 + \frac{1.0}{y}}\]
x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}
↓
x - \frac{1}{x \cdot 0.5 + \frac{1.0}{y}}
double f(double x, double y) {
        double r11314971 = x;
        double r11314972 = y;
        double r11314973 = 1.0;
        double r11314974 = r11314971 * r11314972;
        double r11314975 = 2.0;
        double r11314976 = r11314974 / r11314975;
        double r11314977 = r11314973 + r11314976;
        double r11314978 = r11314972 / r11314977;
        double r11314979 = r11314971 - r11314978;
        return r11314979;
}
↓
double f(double x, double y) {
        double r11314980 = x;
        double r11314981 = 1.0;
        double r11314982 = 0.5;
        double r11314983 = r11314980 * r11314982;
        double r11314984 = 1.0;
        double r11314985 = y;
        double r11314986 = r11314984 / r11314985;
        double r11314987 = r11314983 + r11314986;
        double r11314988 = r11314981 / r11314987;
        double r11314989 = r11314980 - r11314988;
        return r11314989;
}

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

    \[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
  2. Using strategy rm

  3. Applied clear-num0.1

    \[\leadsto x - \color{blue}{\frac{1}{\frac{1.0 + \frac{x \cdot y}{2.0}}{y}}}\]

  4. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{0.5 \cdot x + 1.0 \cdot \frac{1}{y}}}\]

  5. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{0.5 \cdot x + \frac{1.0}{y}}}\]

  6. Final simplification0.1

    \[\leadsto x - \frac{1}{x \cdot 0.5 + \frac{1.0}{y}}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))