Average Error: 0.2 → 0.2
Time: 13.8s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{1 + \sqrt{x + 1}}\right)\right)\]
\frac{x}{1 + \sqrt{x + 1}}
↓
x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{1 + \sqrt{x + 1}}\right)\right)
double f(double x) {
        double r108157 = x;
        double r108158 = 1.0;
        double r108159 = r108157 + r108158;
        double r108160 = sqrt(r108159);
        double r108161 = r108158 + r108160;
        double r108162 = r108157 / r108161;
        return r108162;
}
↓
double f(double x) {
        double r108163 = x;
        double r108164 = 1.0;
        double r108165 = 1.0;
        double r108166 = r108163 + r108165;
        double r108167 = sqrt(r108166);
        double r108168 = r108165 + r108167;
        double r108169 = r108164 / r108168;
        double r108170 = expm1(r108169);
        double r108171 = log1p(r108170);
        double r108172 = r108163 * r108171;
        return r108172;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\frac{x}{1 + \sqrt{x + 1}}\]
  2. Using strategy rm

  3. Applied div-inv0.2

    \[\leadsto \color{blue}{x \cdot \frac{1}{1 + \sqrt{x + 1}}}\]
  4. Using strategy rm

  5. Applied log1p-expm1-u0.2

    \[\leadsto x \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{1 + \sqrt{x + 1}}\right)\right)}\]

  6. Final simplification0.2

    \[\leadsto x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{1 + \sqrt{x + 1}}\right)\right)\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))