Average Error: 38.9 → 12.3
Time: 5.4s
Precision: binary64
\[im > 0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq 2.4641894740488898 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \left(im \cdot \frac{im}{re}\right)\right)}\\ \end{array} \]
(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re 2.4641894740488898e-42)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (sqrt (* 2.0 (* 0.5 (* im (/ im re))))))))
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 <= 2.4641894740488898e-42) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (0.5 * (im * (im / re)))));
	}
	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 <= 2.4641894740488898e-42) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (0.5 * (im * (im / re)))));
	}
	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 <= 2.4641894740488898e-42:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (0.5 * (im * (im / re)))))
	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 <= 2.4641894740488898e-42)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(0.5 * Float64(im * Float64(im / re))))));
	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 <= 2.4641894740488898e-42)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * sqrt((2.0 * (0.5 * (im * (im / re)))));
	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, 2.4641894740488898e-42], 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[Sqrt[N[(2.0 * N[(0.5 * N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]), $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 2.4641894740488898 \cdot 10^{-42}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \left(im \cdot \frac{im}{re}\right)\right)}\\


\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 < 2.46418947404888975e-42

    1. Initial program 32.0

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

    2. Simplified3.8

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

    if 2.46418947404888975e-42 < re

    1. Initial program 55.7

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

    2. Simplified36.9

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

    3. Taylor expanded in re around inf 36.5

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

    4. Simplified33.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \left(\frac{im}{re} \cdot im\right)\right)}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.4641894740488898 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \left(im \cdot \frac{im}{re}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022185 
(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)))))