Average Error: 30.1 → 0.2
Time: 6.9s
Precision: binary64
Cost: 13248
\[-1.79 \cdot 10^{+308} \leq x \land x \leq 1.79 \cdot 10^{+308}\]
\[\sqrt{x + 1} - \sqrt{x} \]
\[\frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
(FPCore (x) :precision binary64 (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))
double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}
double code(double x) {
	return 1.0 / (sqrt((1.0 + x)) + sqrt(x));
}
real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 1.0d0)) - sqrt(x)
end function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))
end function
public static double code(double x) {
	return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}
public static double code(double x) {
	return 1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
}
def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)
def code(x):
	return 1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))
function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end
function code(x)
	return Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))
end
function tmp = code(x)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end
function tmp = code(x)
	tmp = 1.0 / (sqrt((1.0 + x)) + sqrt(x));
end
code[x_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]
code[x_] := N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\sqrt{x + 1} - \sqrt{x}
\frac{1}{\sqrt{1 + x} + \sqrt{x}}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.1

    \[\sqrt{x + 1} - \sqrt{x} \]
  2. Applied egg-rr29.5

    \[\leadsto \color{blue}{\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}} \]
  3. Taylor expanded in x around 0 0.2

    \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}} \]
  4. Final simplification0.2

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

Alternatives

Alternative 1
Error0.3
Cost26308
\[\begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error1.0
Cost6980
\[\begin{array}{l} \mathbf{if}\;x \leq 0.0007865835300043417:\\ \;\;\;\;1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
Alternative 3
Error2.1
Cost6852
\[\begin{array}{l} \mathbf{if}\;x \leq 0.0007865835300043417:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
Alternative 4
Error31.3
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(x+1)-sqrt(x)"
  :precision binary64
  :pre (and (<= -1.79e+308 x) (<= x 1.79e+308))
  (- (sqrt (+ x 1.0)) (sqrt x)))