Average Error: 0.2 → 0.1
Time: 7.7s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\left(\frac{1}{\left(\left(1 + x\right) - 1 \cdot \sqrt{x + 1}\right) + 1 \cdot 1} \cdot \frac{x}{1 + \sqrt{x + 1}}\right) \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\left(\frac{1}{\left(\left(1 + x\right) - 1 \cdot \sqrt{x + 1}\right) + 1 \cdot 1} \cdot \frac{x}{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)
double f(double x) {
        double r123060 = x;
        double r123061 = 1.0;
        double r123062 = r123060 + r123061;
        double r123063 = sqrt(r123062);
        double r123064 = r123061 + r123063;
        double r123065 = r123060 / r123064;
        return r123065;
}
↓
double f(double x) {
        double r123066 = 1.0;
        double r123067 = 1.0;
        double r123068 = x;
        double r123069 = r123067 + r123068;
        double r123070 = r123068 + r123067;
        double r123071 = sqrt(r123070);
        double r123072 = r123067 * r123071;
        double r123073 = r123069 - r123072;
        double r123074 = r123067 * r123067;
        double r123075 = r123073 + r123074;
        double r123076 = r123066 / r123075;
        double r123077 = r123067 + r123071;
        double r123078 = r123068 / r123077;
        double r123079 = r123076 * r123078;
        double r123080 = r123071 * r123071;
        double r123081 = r123080 - r123072;
        double r123082 = r123074 + r123081;
        double r123083 = r123079 * r123082;
        return r123083;
}

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-+7.0

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

  4. Applied associate-/r/7.0

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

  6. Applied sum-cubes7.0

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

  7. Applied *-un-lft-identity7.0

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

  8. Applied times-frac0.2

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))