Average Error: 0.5 → 0.3
Time: 1.7s
Precision: binary64
\[\sqrt{x - 1} \cdot \sqrt{x} \]
↓
\[\left(x + -0.5\right) + \frac{1}{x} \cdot \left(-0.125 + \frac{-0.0625}{x}\right) \]
(FPCore (x) :precision binary64 (* (sqrt (- x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (+ (+ x -0.5) (* (/ 1.0 x) (+ -0.125 (/ -0.0625 x)))))
double code(double x) {
	return sqrt((x - 1.0)) * sqrt(x);
}
↓
double code(double x) {
	return (x + -0.5) + ((1.0 / x) * (-0.125 + (-0.0625 / 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)) + ((1.0d0 / x) * ((-0.125d0) + ((-0.0625d0) / 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) + ((1.0 / x) * (-0.125 + (-0.0625 / x)));
}

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

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

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

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

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

Error

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

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2022210 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))