Average Error: 0.2 → 0.1
Time: 15.2s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{\frac{x}{\left(\left(1 + x\right) - 1 \cdot \sqrt{x + 1}\right) + 1 \cdot 1}}{1 + \sqrt{x + 1}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{\frac{x}{\left(\left(1 + x\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 r107412 = x;
        double r107413 = 1.0;
        double r107414 = r107412 + r107413;
        double r107415 = sqrt(r107414);
        double r107416 = r107413 + r107415;
        double r107417 = r107412 / r107416;
        return r107417;
}
↓
double f(double x) {
        double r107418 = x;
        double r107419 = 1.0;
        double r107420 = r107419 + r107418;
        double r107421 = r107418 + r107419;
        double r107422 = sqrt(r107421);
        double r107423 = r107419 * r107422;
        double r107424 = r107420 - r107423;
        double r107425 = r107419 * r107419;
        double r107426 = r107424 + r107425;
        double r107427 = r107418 / r107426;
        double r107428 = r107419 + r107422;
        double r107429 = r107427 / r107428;
        double r107430 = r107422 * r107422;
        double r107431 = r107430 - r107423;
        double r107432 = r107425 + r107431;
        double r107433 = r107429 * r107432;
        return r107433;
}

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-+7.1

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

  4. Applied associate-/r/7.1

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

  6. Applied sum-cubes7.1

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

  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

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))