?

Average Error: 30.5 → 0.4
Time: 3.7s
Precision: binary64
Cost: 13248

?

\[1000000000 \leq x \land x \leq 10000000000\]
\[\sqrt{x + 1} - \sqrt{x} \]
\[\frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
(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 30.5

    \[\sqrt{x + 1} - \sqrt{x} \]
  2. Applied egg-rr0.4

    \[\leadsto \color{blue}{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} \]
  3. Simplified0.4

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

    [Start]0.4

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

    *-commutative [=>]0.4

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

    associate-*l/ [=>]0.4

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

    *-lft-identity [=>]0.4

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

    +-commutative [=>]0.4

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

    associate-+l- [=>]0.4

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

    +-inverses [=>]0.4

    \[ \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

    metadata-eval [=>]0.4

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

    +-commutative [=>]0.4

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

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

Alternatives

Alternative 1
Error30.5
Cost13120
\[\sqrt{1 + x} - \sqrt{x} \]
Alternative 2
Error56.1
Cost64
\[1 \]

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(x+1)-sqrt(x)"
  :precision binary64
  :pre (and (<= 1000000000.0 x) (<= x 10000000000.0))
  (- (sqrt (+ x 1.0)) (sqrt x)))