Average Error: 38.8 → 8.3
Time: 4.1s
Precision: binary64
\[im > 0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
↓
\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot \sqrt{0.5 \cdot \frac{2}{re}}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{if}\;re \leq 8.7 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 6 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))
↓
(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im (sqrt (* 0.5 (/ 2.0 re))))))
        (t_1 (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))
   (if (<= re 8.7e-104)
     t_1
     (if (<= re 6e-67) t_0 (if (<= re 1e+14) t_1 t_0)))))
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 t_0 = 0.5 * (im * sqrt((0.5 * (2.0 / re))));
	double t_1 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	double tmp;
	if (re <= 8.7e-104) {
		tmp = t_1;
	} else if (re <= 6e-67) {
		tmp = t_0;
	} else if (re <= 1e+14) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 t_0 = 0.5 * (im * Math.sqrt((0.5 * (2.0 / re))));
	double t_1 = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	double tmp;
	if (re <= 8.7e-104) {
		tmp = t_1;
	} else if (re <= 6e-67) {
		tmp = t_0;
	} else if (re <= 1e+14) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))
↓
def code(re, im):
	t_0 = 0.5 * (im * math.sqrt((0.5 * (2.0 / re))))
	t_1 = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	tmp = 0
	if re <= 8.7e-104:
		tmp = t_1
	elif re <= 6e-67:
		tmp = t_0
	elif re <= 1e+14:
		tmp = t_1
	else:
		tmp = t_0
	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)
	t_0 = Float64(0.5 * Float64(im * sqrt(Float64(0.5 * Float64(2.0 / re)))))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))))
	tmp = 0.0
	if (re <= 8.7e-104)
		tmp = t_1;
	elseif (re <= 6e-67)
		tmp = t_0;
	elseif (re <= 1e+14)
		tmp = t_1;
	else
		tmp = t_0;
	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)
	t_0 = 0.5 * (im * sqrt((0.5 * (2.0 / re))));
	t_1 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	tmp = 0.0;
	if (re <= 8.7e-104)
		tmp = t_1;
	elseif (re <= 6e-67)
		tmp = t_0;
	elseif (re <= 1e+14)
		tmp = t_1;
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(0.5 * N[(im * N[Sqrt[N[(0.5 * N[(2.0 / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 8.7e-104], t$95$1, If[LessEqual[re, 6e-67], t$95$0, If[LessEqual[re, 1e+14], t$95$1, t$95$0]]]]]

0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
t_0 := 0.5 \cdot \left(im \cdot \sqrt{0.5 \cdot \frac{2}{re}}\right)\\
t_1 := 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\
\mathbf{if}\;re \leq 8.7 \cdot 10^{-104}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;re \leq 6 \cdot 10^{-67}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;re \leq 10^{+14}:\\
\;\;\;\;t_1\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if re < 8.70000000000000034e-104 or 6.00000000000000065e-67 < re < 1e14

    1. Initial program 32.6

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

    2. Simplified5.0

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

    if 8.70000000000000034e-104 < re < 6.00000000000000065e-67 or 1e14 < re

    1. Initial program 55.6

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

    2. Simplified37.0

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

    3. Applied egg-rr37.0

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

    4. Taylor expanded in im around 0 17.4

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

    5. Simplified17.4

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

    6. Applied egg-rr17.2

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \sqrt{\frac{2}{re}} \cdot \left(\sqrt{0.5} \cdot im\right)\right)} \]

    7. Applied egg-rr17.0

      \[\leadsto 0.5 \cdot \left(0 + \color{blue}{{\left(im \cdot \sqrt{\frac{2}{re} \cdot 0.5}\right)}^{1}}\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8.7 \cdot 10^{-104}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{0.5 \cdot \frac{2}{re}}\right)\\ \mathbf{elif}\;re \leq 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{0.5 \cdot \frac{2}{re}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(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)))))