Average Error: 0.2 → 0.1
Time: 5.6s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{\frac{x}{\left(\left(1 \cdot 1 + \left(x + 1\right)\right) - 1 \cdot \sqrt{x + 1}\right) \cdot 1}}{1 + \sqrt{x + 1}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{\frac{x}{\left(\left(1 \cdot 1 + \left(x + 1\right)\right) - 1 \cdot \sqrt{x + 1}\right) \cdot 1}}{1 + \sqrt{x + 1}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)
double f(double x) {
        double r104820 = x;
        double r104821 = 1.0;
        double r104822 = r104820 + r104821;
        double r104823 = sqrt(r104822);
        double r104824 = r104821 + r104823;
        double r104825 = r104820 / r104824;
        return r104825;
}
↓
double f(double x) {
        double r104826 = x;
        double r104827 = 1.0;
        double r104828 = r104827 * r104827;
        double r104829 = r104826 + r104827;
        double r104830 = r104828 + r104829;
        double r104831 = sqrt(r104829);
        double r104832 = r104827 * r104831;
        double r104833 = r104830 - r104832;
        double r104834 = 1.0;
        double r104835 = r104833 * r104834;
        double r104836 = r104826 / r104835;
        double r104837 = r104827 + r104831;
        double r104838 = r104836 / r104837;
        double r104839 = r104831 * r104831;
        double r104840 = r104839 - r104832;
        double r104841 = r104828 + r104840;
        double r104842 = r104838 * r104841;
        return r104842;
}

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 flip3-+7.6

    \[\leadsto \frac{x}{\color{blue}{\frac{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}}{1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)}}}\]

  4. Applied associate-/r/7.6

    \[\leadsto \color{blue}{\frac{x}{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)}\]
  5. Using strategy rm

  6. Applied sum-cubes7.6

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

  7. Applied associate-/r*0.2

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020002 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))