Average Error: 0.0 → 0.1
Time: 24.6s
Precision: 64
\[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
↓
\[x - \frac{1}{0.5 \cdot x + \frac{1.0}{y}}\]
x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}
↓
x - \frac{1}{0.5 \cdot x + \frac{1.0}{y}}
double f(double x, double y) {
        double r24638101 = x;
        double r24638102 = y;
        double r24638103 = 1.0;
        double r24638104 = r24638101 * r24638102;
        double r24638105 = 2.0;
        double r24638106 = r24638104 / r24638105;
        double r24638107 = r24638103 + r24638106;
        double r24638108 = r24638102 / r24638107;
        double r24638109 = r24638101 - r24638108;
        return r24638109;
}
↓
double f(double x, double y) {
        double r24638110 = x;
        double r24638111 = 1.0;
        double r24638112 = 0.5;
        double r24638113 = r24638112 * r24638110;
        double r24638114 = 1.0;
        double r24638115 = y;
        double r24638116 = r24638114 / r24638115;
        double r24638117 = r24638113 + r24638116;
        double r24638118 = r24638111 / r24638117;
        double r24638119 = r24638110 - r24638118;
        return r24638119;
}

Error

Bits error versus x

Bits error versus y

Try it out

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

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

  6. Final simplification0.1

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

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)))))