Average Error: 0.2 → 0.1
Time: 3.8s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\left(\frac{1}{\frac{\left(1 \cdot 1 + \left(x + 1\right)\right) - 1 \cdot \sqrt{x + 1}}{1}} \cdot \frac{x}{1 + \sqrt{x + 1}}\right) \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\left(\frac{1}{\frac{\left(1 \cdot 1 + \left(x + 1\right)\right) - 1 \cdot \sqrt{x + 1}}{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 code(double x) {
	return (x / (1.0 + sqrt((x + 1.0))));
}
↓
double code(double x) {
	return (((1.0 / ((((1.0 * 1.0) + (x + 1.0)) - (1.0 * sqrt((x + 1.0)))) / 1.0)) * (x / (1.0 + sqrt((x + 1.0))))) * ((1.0 * 1.0) + ((sqrt((x + 1.0)) * sqrt((x + 1.0))) - (1.0 * sqrt((x + 1.0))))));
}

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/7.2

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

  6. Applied sum-cubes7.1

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

  7. Applied *-un-lft-identity7.1

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

  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

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))