Average Error: 0.5 → 0.3
Time: 1.4s
Precision: binary64
\[\sqrt{x - 1} \cdot \sqrt{x} \]
↓
\[x + \mathsf{fma}\left(-0.0625, {x}^{-2}, -0.5 + \frac{-0.125}{x}\right) \]
(FPCore (x) :precision binary64 (* (sqrt (- x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (+ x (fma -0.0625 (pow x -2.0) (+ -0.5 (/ -0.125 x)))))
double code(double x) {
	return sqrt((x - 1.0)) * sqrt(x);
}
↓
double code(double x) {
	return x + fma(-0.0625, pow(x, -2.0), (-0.5 + (-0.125 / x)));
}

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

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

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

  4. Applied egg-rr0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2022150 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))