?

Average Error: 0.1 → 0.0
Time: 8.5s
Precision: binary64
Cost: 33984

?

\[0 \leq x \land x \leq 1000\]
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
\[\begin{array}{l} t_0 := \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}}\\ t_0 \cdot \left(x - \sqrt{x + x \cdot x}\right) + \left(1 + x\right) \cdot t_0 \end{array} \]
(FPCore (x) :precision binary64 (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))))
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ (pow (+ 1.0 x) 1.5) (pow x 1.5)))))
   (+ (* t_0 (- x (sqrt (+ x (* x x))))) (* (+ 1.0 x) t_0))))
double code(double x) {
	return 1.0 / (sqrt((x + 1.0)) + sqrt(x));
}
double code(double x) {
	double t_0 = 1.0 / (pow((1.0 + x), 1.5) + pow(x, 1.5));
	return (t_0 * (x - sqrt((x + (x * x))))) + ((1.0 + x) * t_0);
}
real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))
end function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (((1.0d0 + x) ** 1.5d0) + (x ** 1.5d0))
    code = (t_0 * (x - sqrt((x + (x * x))))) + ((1.0d0 + x) * t_0)
end function
public static double code(double x) {
	return 1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x));
}
public static double code(double x) {
	double t_0 = 1.0 / (Math.pow((1.0 + x), 1.5) + Math.pow(x, 1.5));
	return (t_0 * (x - Math.sqrt((x + (x * x))))) + ((1.0 + x) * t_0);
}
def code(x):
	return 1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))
def code(x):
	t_0 = 1.0 / (math.pow((1.0 + x), 1.5) + math.pow(x, 1.5))
	return (t_0 * (x - math.sqrt((x + (x * x))))) + ((1.0 + x) * t_0)
function code(x)
	return Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)))
end
function code(x)
	t_0 = Float64(1.0 / Float64((Float64(1.0 + x) ^ 1.5) + (x ^ 1.5)))
	return Float64(Float64(t_0 * Float64(x - sqrt(Float64(x + Float64(x * x))))) + Float64(Float64(1.0 + x) * t_0))
end
function tmp = code(x)
	tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
end
function tmp = code(x)
	t_0 = 1.0 / (((1.0 + x) ^ 1.5) + (x ^ 1.5));
	tmp = (t_0 * (x - sqrt((x + (x * x))))) + ((1.0 + x) * t_0);
end
code[x_] := N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
code[x_] := Block[{t$95$0 = N[(1.0 / N[(N[Power[N[(1.0 + x), $MachinePrecision], 1.5], $MachinePrecision] + N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * N[(x - N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
\begin{array}{l}
t_0 := \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}}\\
t_0 \cdot \left(x - \sqrt{x + x \cdot x}\right) + \left(1 + x\right) \cdot t_0
\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.1

    \[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
  2. Applied egg-rr0.0

    \[\leadsto \color{blue}{\frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \left(x - \sqrt{x + x \cdot x}\right) + \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \left(1 + x\right)} \]
  3. Final simplification0.0

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

Alternatives

Alternative 1
Error0.1
Cost13248
\[\frac{1}{\sqrt{x} + \sqrt{1 + x}} \]
Alternative 2
Error0.1
Cost13120
\[\sqrt{1 + x} - \sqrt{x} \]
Alternative 3
Error1.2
Cost7104
\[\frac{1}{\frac{-1 + \left(x \cdot -0.5 - \sqrt{x}\right)}{-1}} \]
Alternative 4
Error1.2
Cost6976
\[\frac{1}{\sqrt{x} + \left(1 + x \cdot 0.5\right)} \]
Alternative 5
Error1.3
Cost6848
\[\left(1 + x \cdot 0.5\right) - \sqrt{x} \]
Alternative 6
Error3.3
Cost64
\[1 \]

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x)
  :name "1/(sqrt(x + 1) + sqrt(x))"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 1000.0))
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))))