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(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

(FPCore (x)
 :precision binary64
 (/ (- (+ 1.0 x) x) (+ (sqrt (+ 1.0 x)) (sqrt x))))

double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}

double code(double x) {
	return ((1.0 + x) - x) / (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 + x) - x) / (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 + x) - x) / (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 + x) - x) / (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(Float64(Float64(1.0 + x) - x) / 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 + x) - x) / (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[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\sqrt{x + 1} - \sqrt{x}

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

Error?

Try it out?

Derivation?

1. Initial program 99.3%

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

2. Applied egg-rr99.9%

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

   [Start]99.3	\[ \sqrt{x + 1} - \sqrt{x} \]
   flip-- [=>]99.4	\[ \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} \]
   div-inv [=>]99.4	\[ \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} \]
   add-sqr-sqrt [<=]99.6	\[ \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
   add-sqr-sqrt [<=]99.9	\[ \left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
   associate--l+ [=>]99.9	\[ \color{blue}{\left(x + \left(1 - x\right)\right)} \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

3. Simplified99.9%

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

   [Start]99.9	\[ \left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
   associate-*r/ [=>]99.9	\[ \color{blue}{\frac{\left(x + \left(1 - x\right)\right) \cdot 1}{\sqrt{x + 1} + \sqrt{x}}} \]
   *-rgt-identity [=>]99.9	\[ \frac{\color{blue}{x + \left(1 - x\right)}}{\sqrt{x + 1} + \sqrt{x}} \]
   associate-+r- [=>]99.9	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-commutative [=>]99.9	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{\sqrt{x + 1} + \sqrt{x}} \]
   +-commutative [=>]99.9	\[ \frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} \]

4. Final simplification99.9%

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

Alternatives

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(x+1) - sqrt(x)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 1000000000.0))
  (- (sqrt (+ x 1.0)) (sqrt x)))