Average Error: 38.6 → 8.0
Time: 4.0s
Precision: binary64
\[im > 0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq 7.962544944086704 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{0.5}}{{2}^{-0.5} \cdot {re}^{0.5}}\\ \end{array} \]
(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re 7.962544944086704e-55)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (/ (* im (sqrt 0.5)) (* (pow 2.0 -0.5) (pow re 0.5))))))
double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= 7.962544944086704e-55) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * ((im * sqrt(0.5)) / (pow(2.0, -0.5) * pow(re, 0.5)));
	}
	return tmp;
}

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}
↓
public static double code(double re, double im) {
	double tmp;
	if (re <= 7.962544944086704e-55) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * ((im * Math.sqrt(0.5)) / (Math.pow(2.0, -0.5) * Math.pow(re, 0.5)));
	}
	return tmp;
}

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))
↓
def code(re, im):
	tmp = 0
	if re <= 7.962544944086704e-55:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * ((im * math.sqrt(0.5)) / (math.pow(2.0, -0.5) * math.pow(re, 0.5)))
	return tmp

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end
↓
function code(re, im)
	tmp = 0.0
	if (re <= 7.962544944086704e-55)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * Float64(Float64(im * sqrt(0.5)) / Float64((2.0 ^ -0.5) * (re ^ 0.5))));
	end
	return tmp
end

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
end
↓
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 7.962544944086704e-55)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * ((im * sqrt(0.5)) / ((2.0 ^ -0.5) * (re ^ 0.5)));
	end
	tmp_2 = tmp;
end

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[re_, im_] := If[LessEqual[re, 7.962544944086704e-55], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(im * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[(N[Power[2.0, -0.5], $MachinePrecision] * N[Power[re, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq 7.962544944086704 \cdot 10^{-55}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{0.5}}{{2}^{-0.5} \cdot {re}^{0.5}}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if re < 7.962544944086704e-55

    1. Initial program 31.5

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

    2. Simplified3.5

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

    if 7.962544944086704e-55 < re

    1. Initial program 55.0

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

    2. Simplified35.6

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

    3. Taylor expanded in im around 0 18.8

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

    4. Simplified18.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{2}\right)\right)} \]

    5. Applied egg-rr18.7

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im \cdot \sqrt{0.5}}{\frac{1}{\frac{\sqrt{2}}{\sqrt{re}}}}} \]

    6. Applied egg-rr18.6

      \[\leadsto 0.5 \cdot \frac{im \cdot \sqrt{0.5}}{\color{blue}{{\left(\frac{2}{re}\right)}^{-0.5}}} \]

    7. Taylor expanded in re around 0 21.1

      \[\leadsto 0.5 \cdot \frac{im \cdot \sqrt{0.5}}{\color{blue}{e^{-0.5 \cdot \left(\log 2 + -1 \cdot \log re\right)}}} \]

    8. Simplified18.5

      \[\leadsto 0.5 \cdot \frac{im \cdot \sqrt{0.5}}{\color{blue}{{2}^{-0.5} \cdot {re}^{0.5}}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.962544944086704 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{0.5}}{{2}^{-0.5} \cdot {re}^{0.5}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022203 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))