Average Error: 0.2 → 0.1
Time: 5.7s
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 r105097 = x;
        double r105098 = 1.0;
        double r105099 = r105097 + r105098;
        double r105100 = sqrt(r105099);
        double r105101 = r105098 + r105100;
        double r105102 = r105097 / r105101;
        return r105102;
}
↓
double f(double x) {
        double r105103 = x;
        double r105104 = 1.0;
        double r105105 = r105104 * r105104;
        double r105106 = r105103 + r105104;
        double r105107 = r105105 + r105106;
        double r105108 = sqrt(r105106);
        double r105109 = r105104 * r105108;
        double r105110 = r105107 - r105109;
        double r105111 = 1.0;
        double r105112 = r105110 * r105111;
        double r105113 = r105103 / r105112;
        double r105114 = r105104 + r105108;
        double r105115 = r105113 / r105114;
        double r105116 = r105108 * r105108;
        double r105117 = r105116 - r105109;
        double r105118 = r105105 + r105117;
        double r105119 = r105115 * r105118;
        return r105119;
}

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-+7.2

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

  4. Applied associate-/r/7.2

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

  6. Applied sum-cubes7.2

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

  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 2019354 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))