Average Error: 0.2 → 0.1
Time: 14.8s
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 r120177 = x;
        double r120178 = 1.0;
        double r120179 = r120177 + r120178;
        double r120180 = sqrt(r120179);
        double r120181 = r120178 + r120180;
        double r120182 = r120177 / r120181;
        return r120182;
}
↓
double f(double x) {
        double r120183 = x;
        double r120184 = 1.0;
        double r120185 = r120184 + r120183;
        double r120186 = r120183 + r120184;
        double r120187 = sqrt(r120186);
        double r120188 = r120184 * r120187;
        double r120189 = r120185 - r120188;
        double r120190 = r120184 * r120184;
        double r120191 = r120189 + r120190;
        double r120192 = r120183 / r120191;
        double r120193 = r120184 + r120187;
        double r120194 = r120192 / r120193;
        double r120195 = r120187 * r120187;
        double r120196 = r120195 - r120188;
        double r120197 = r120190 + r120196;
        double r120198 = r120194 * r120197;
        return r120198;
}

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)))))