Average Error: 0.2 → 0.1
Time: 10.3s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{\frac{x}{\left(\left(x + 1\right) - 1 \cdot \sqrt{x + 1}\right) + 1 \cdot 1}}{1 + \sqrt{x + 1}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{\frac{x}{\left(\left(x + 1\right) - 1 \cdot \sqrt{x + 1}\right) + 1 \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 r103753 = x;
        double r103754 = 1.0;
        double r103755 = r103753 + r103754;
        double r103756 = sqrt(r103755);
        double r103757 = r103754 + r103756;
        double r103758 = r103753 / r103757;
        return r103758;
}
↓
double f(double x) {
        double r103759 = x;
        double r103760 = 1.0;
        double r103761 = r103759 + r103760;
        double r103762 = sqrt(r103761);
        double r103763 = r103760 * r103762;
        double r103764 = r103761 - r103763;
        double r103765 = r103760 * r103760;
        double r103766 = r103764 + r103765;
        double r103767 = r103759 / r103766;
        double r103768 = r103760 + r103762;
        double r103769 = r103767 / r103768;
        double r103770 = r103762 * r103762;
        double r103771 = r103770 - r103763;
        double r103772 = r103765 + r103771;
        double r103773 = r103769 * r103772;
        return r103773;
}

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-+6.7

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

  4. Applied associate-/r/6.7

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

  6. Applied sum-cubes6.7

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

  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

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

  9. Final simplification0.1

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

Reproduce

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