Average Error: 0.2 → 0.2
Time: 1.9s
Precision: binary64
\[\frac{x}{1 + \sqrt{x + 1}} \]
↓
\[\frac{x}{1 + \sqrt{x + 1}} \]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{x}{1 + \sqrt{x + 1}}

(FPCore (x) :precision binary64 (/ x (+ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x) :precision binary64 (/ x (+ 1.0 (sqrt (+ x 1.0)))))
double code(double x) {
	return x / (1.0 + sqrt(x + 1.0));
}
↓
double code(double x) {
	return x / (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. Applied *-un-lft-identity_binary640.2

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

  3. Applied sqrt-prod_binary640.2

    \[\leadsto \frac{x}{1 + \color{blue}{\sqrt{1} \cdot \sqrt{x + 1}}} \]

  4. Applied add-sqr-sqrt_binary640.2

    \[\leadsto \frac{x}{\color{blue}{\sqrt{1} \cdot \sqrt{1}} + \sqrt{1} \cdot \sqrt{x + 1}} \]

  5. Applied distribute-lft-out_binary640.2

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

  6. Applied associate-/r*_binary640.2

    \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{1}}}{\sqrt{1} + \sqrt{x + 1}}} \]

  7. Simplified0.2

    \[\leadsto \frac{\color{blue}{x}}{\sqrt{1} + \sqrt{x + 1}} \]

  8. Final simplification0.2

    \[\leadsto \frac{x}{1 + \sqrt{x + 1}} \]

Reproduce

herbie shell --seed 2021275 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))