Average Error: 0.5 → 0.3
Time: 1.7s
Precision: binary64
\[\sqrt{x - 1} \cdot \sqrt{x} \]
↓
\[x - \left(0.5 + \mathsf{fma}\left(0.0625, {x}^{-2}, \frac{0.125}{x}\right)\right) \]
(FPCore (x) :precision binary64 (* (sqrt (- x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (- x (+ 0.5 (fma 0.0625 (pow x -2.0) (/ 0.125 x)))))
double code(double x) {
	return sqrt((x - 1.0)) * sqrt(x);
}
↓
double code(double x) {
	return x - (0.5 + fma(0.0625, pow(x, -2.0), (0.125 / x)));
}

function code(x)
	return Float64(sqrt(Float64(x - 1.0)) * sqrt(x))
end
↓
function code(x)
	return Float64(x - Float64(0.5 + fma(0.0625, (x ^ -2.0), Float64(0.125 / 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[(0.0625 * N[Power[x, -2.0], $MachinePrecision] + N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \mathsf{fma}\left(0.0625, {x}^{-2}, \frac{0.125}{x}\right)\right)

Error

Bits error versus x

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

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

  4. Applied egg-rr0.3

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

  5. Final simplification0.3

    \[\leadsto x - \left(0.5 + \mathsf{fma}\left(0.0625, {x}^{-2}, \frac{0.125}{x}\right)\right) \]

Reproduce

herbie shell --seed 2022146 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))