Average Error: 0.5 → 0.3
Time: 1.3s
Precision: binary64
\[\sqrt{x - 1} \cdot \sqrt{x} \]
↓
\[x - \left(0.5 + \frac{0.125 + \frac{0.0625}{x}}{x}\right) \]
(FPCore (x) :precision binary64 (* (sqrt (- x 1.0)) (sqrt x)))
↓
(FPCore (x) :precision binary64 (- x (+ 0.5 (/ (+ 0.125 (/ 0.0625 x)) x))))
double code(double x) {
	return sqrt((x - 1.0)) * sqrt(x);
}
↓
double code(double x) {
	return x - (0.5 + ((0.125 + (0.0625 / x)) / 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 = x - (0.5d0 + ((0.125d0 + (0.0625d0 / x)) / 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 x - (0.5 + ((0.125 + (0.0625 / x)) / x));
}

def code(x):
	return math.sqrt((x - 1.0)) * math.sqrt(x)
↓
def code(x):
	return x - (0.5 + ((0.125 + (0.0625 / x)) / x))

function code(x)
	return Float64(sqrt(Float64(x - 1.0)) * sqrt(x))
end
↓
function code(x)
	return Float64(x - Float64(0.5 + Float64(Float64(0.125 + Float64(0.0625 / x)) / x)))
end

function tmp = code(x)
	tmp = sqrt((x - 1.0)) * sqrt(x);
end
↓
function tmp = code(x)
	tmp = x - (0.5 + ((0.125 + (0.0625 / x)) / x));
end

code[x_] := N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(x - N[(0.5 + N[(N[(0.125 + N[(0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \frac{0.125 + \frac{0.0625}{x}}{x}\right)

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  2. Taylor expanded in x around inf 0.3

    \[\leadsto \color{blue}{x - \left(0.5 + \left(0.125 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

  3. Simplified0.3

    \[\leadsto \color{blue}{x - \left(0.5 + \frac{0.125 + \frac{0.0625}{x}}{x}\right)} \]

  4. Final simplification0.3

    \[\leadsto x - \left(0.5 + \frac{0.125 + \frac{0.0625}{x}}{x}\right) \]

Reproduce

herbie shell --seed 2022155 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))