Average Error: 0.2 → 0.2
Time: 1.8min
Precision: binary64
Cost: 6848
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{x}{1 + \sqrt{x + 1}}\]
\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

Alternatives

Alternative 1
Error0.1
Cost7041
\[\begin{array}{l} \mathbf{if}\;x \leq 0.001349740313582845:\\ \;\;\;\;\frac{x}{2 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array}\]
Alternative 2
Error20.4
Cost576
\[\frac{x}{1 + \left(1 + x \cdot 0.5\right)}\]
Alternative 3
Error20.4
Cost448
\[\frac{x}{2 + x \cdot 0.5}\]
Alternative 4
Error20.9
Cost192
\[\frac{x}{2}\]
Alternative 5
Error59.9
Cost385
\[\begin{array}{l} \mathbf{if}\;x \leq 2.2232913121967634 \cdot 10^{-154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 6
Error60.9
Cost64
\[1\]

Error

Time

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2021040 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))