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

(FPCore (x)
 :precision binary64
 (/
  (+ 1.0 (- (+ x x) (sqrt (+ x (* x x)))))
  (+ (pow (+ 1.0 x) 1.5) (pow x 1.5))))

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

double code(double x) {
	return (1.0 + ((x + x) - sqrt((x + (x * x))))) / (pow((1.0 + x), 1.5) + pow(x, 1.5));
}

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((x + (x * x))))) / (((1.0d0 + x) ** 1.5d0) + (x ** 1.5d0))
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((x + (x * x))))) / (Math.pow((1.0 + x), 1.5) + Math.pow(x, 1.5));
}

def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)

def code(x):
	return (1.0 + ((x + x) - math.sqrt((x + (x * x))))) / (math.pow((1.0 + x), 1.5) + math.pow(x, 1.5))

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(x + x) - sqrt(Float64(x + Float64(x * x))))) / Float64((Float64(1.0 + x) ^ 1.5) + (x ^ 1.5)))
end

function tmp = code(x)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end

function tmp = code(x)
	tmp = (1.0 + ((x + x) - sqrt((x + (x * x))))) / (((1.0 + x) ^ 1.5) + (x ^ 1.5));
end

code[x_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

code[x_] := N[(N[(1.0 + N[(N[(x + x), $MachinePrecision] - N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(1.0 + x), $MachinePrecision], 1.5], $MachinePrecision] + N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

\frac{1 + \left(\left(x + x\right) - \sqrt{x + x \cdot x}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}}

Error?

Try it out?

Derivation?

1. Initial program 0.5

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

2. Applied egg-rr0.1

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

3. Simplified0.1

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

   [Start]0.1	\[ \left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
   associate-*r/ [=>]0.1	\[ \color{blue}{\frac{\left(x + \left(1 - x\right)\right) \cdot 1}{\sqrt{x + 1} + \sqrt{x}}} \]
   *-rgt-identity [=>]0.1	\[ \frac{\color{blue}{x + \left(1 - x\right)}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-commutative [=>]0.1	\[ \frac{\color{blue}{\left(1 - x\right) + x}}{\sqrt{x + 1} + \sqrt{x}} \]
   associate-+l- [=>]0.1	\[ \frac{\color{blue}{1 - \left(x - x\right)}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-inverses [=>]0.1	\[ \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]
   metadata-eval [=>]0.1	\[ \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-commutative [=>]0.1	\[ \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} \]

4. Applied egg-rr0.1

   \[\leadsto \color{blue}{\frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot 1 + \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + x\right) - \sqrt{x + x \cdot x}\right)} \]

5. Simplified0.1

   \[\leadsto \color{blue}{\frac{1 + \left(\left(x + x\right) - \sqrt{x + x \cdot x}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}}} \]
   Proof

   [Start]0.1	\[ \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot 1 + \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + x\right) - \sqrt{x + x \cdot x}\right) \]
   distribute-lft-out [=>]0.1	\[ \color{blue}{\frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \left(1 + \left(\left(x + x\right) - \sqrt{x + x \cdot x}\right)\right)} \]
   associate--l+ [<=]0.1	\[ \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(1 + \left(x + x\right)\right) - \sqrt{x + x \cdot x}\right)} \]
   associate-*l/ [=>]0.1	\[ \color{blue}{\frac{1 \cdot \left(\left(1 + \left(x + x\right)\right) - \sqrt{x + x \cdot x}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}}} \]
   metadata-eval [<=]0.1	\[ \frac{\color{blue}{\frac{1}{1}} \cdot \left(\left(1 + \left(x + x\right)\right) - \sqrt{x + x \cdot x}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]
   associate-/r/ [<=]0.1	\[ \frac{\color{blue}{\frac{1}{\frac{1}{\left(1 + \left(x + x\right)\right) - \sqrt{x + x \cdot x}}}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]
   remove-double-div [=>]0.1	\[ \frac{\color{blue}{\left(1 + \left(x + x\right)\right) - \sqrt{x + x \cdot x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]
   associate--l+ [=>]0.1	\[ \frac{\color{blue}{1 + \left(\left(x + x\right) - \sqrt{x + x \cdot x}\right)}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

6. Final simplification0.1

   \[\leadsto \frac{1 + \left(\left(x + x\right) - \sqrt{x + x \cdot x}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

Alternatives

Error

Reproduce?

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