math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 38.3 → 8.1

Time: 12.3s

Precision: binary64

Cost: 13708

\[im > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{if}\;re \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 7600000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{1}{\frac{1}{im} + \frac{re}{im \cdot im}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(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 (pow re -0.5))))
        (t_1 (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))
   (if (<= re 1.05e-78)
     t_1
     (if (<= re 4.4e-36)
       t_0
       (if (<= re 7600000000000.0)
         t_1
         (if (<= re 2.1e+39)
           (* 0.5 (/ 1.0 (/ (sqrt re) im)))
           (if (<= re 2.2e+113)
             (* 0.5 (sqrt (* 2.0 (/ 1.0 (+ (/ 1.0 im) (/ re (* im im)))))))
             t_0)))))))

C

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 * pow(re, -0.5));
	double t_1 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	double tmp;
	if (re <= 1.05e-78) {
		tmp = t_1;
	} else if (re <= 4.4e-36) {
		tmp = t_0;
	} else if (re <= 7600000000000.0) {
		tmp = t_1;
	} else if (re <= 2.1e+39) {
		tmp = 0.5 * (1.0 / (sqrt(re) / im));
	} else if (re <= 2.2e+113) {
		tmp = 0.5 * sqrt((2.0 * (1.0 / ((1.0 / im) + (re / (im * im))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

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.pow(re, -0.5));
	double t_1 = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	double tmp;
	if (re <= 1.05e-78) {
		tmp = t_1;
	} else if (re <= 4.4e-36) {
		tmp = t_0;
	} else if (re <= 7600000000000.0) {
		tmp = t_1;
	} else if (re <= 2.1e+39) {
		tmp = 0.5 * (1.0 / (Math.sqrt(re) / im));
	} else if (re <= 2.2e+113) {
		tmp = 0.5 * Math.sqrt((2.0 * (1.0 / ((1.0 / im) + (re / (im * im))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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.pow(re, -0.5))
	t_1 = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	tmp = 0
	if re <= 1.05e-78:
		tmp = t_1
	elif re <= 4.4e-36:
		tmp = t_0
	elif re <= 7600000000000.0:
		tmp = t_1
	elif re <= 2.1e+39:
		tmp = 0.5 * (1.0 / (math.sqrt(re) / im))
	elif re <= 2.2e+113:
		tmp = 0.5 * math.sqrt((2.0 * (1.0 / ((1.0 / im) + (re / (im * im))))))
	else:
		tmp = t_0
	return tmp

Julia

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 * (re ^ -0.5)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))))
	tmp = 0.0
	if (re <= 1.05e-78)
		tmp = t_1;
	elseif (re <= 4.4e-36)
		tmp = t_0;
	elseif (re <= 7600000000000.0)
		tmp = t_1;
	elseif (re <= 2.1e+39)
		tmp = Float64(0.5 * Float64(1.0 / Float64(sqrt(re) / im)));
	elseif (re <= 2.2e+113)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(1.0 / Float64(Float64(1.0 / im) + Float64(re / Float64(im * im)))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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 * (re ^ -0.5));
	t_1 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	tmp = 0.0;
	if (re <= 1.05e-78)
		tmp = t_1;
	elseif (re <= 4.4e-36)
		tmp = t_0;
	elseif (re <= 7600000000000.0)
		tmp = t_1;
	elseif (re <= 2.1e+39)
		tmp = 0.5 * (1.0 / (sqrt(re) / im));
	elseif (re <= 2.2e+113)
		tmp = 0.5 * sqrt((2.0 * (1.0 / ((1.0 / im) + (re / (im * im))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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[Power[re, -0.5], $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, 1.05e-78], t$95$1, If[LessEqual[re, 4.4e-36], t$95$0, If[LessEqual[re, 7600000000000.0], t$95$1, If[LessEqual[re, 2.1e+39], N[(0.5 * N[(1.0 / N[(N[Sqrt[re], $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.2e+113], N[(0.5 * N[Sqrt[N[(2.0 * N[(1.0 / N[(N[(1.0 / im), $MachinePrecision] + N[(re / N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < 1.05e-78 or 4.3999999999999999e-36 < re < 7.6e12

   1. Initial program 32.0

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

   2. Simplified 3.7

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

      [Start] 32.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 32.0	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 32.0	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 32.0	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 32.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 32.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 3.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   if 1.05e-78 < re < 4.3999999999999999e-36 or 2.2000000000000001e113 < re

   1. Initial program 57.3

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

   2. Simplified 37.4

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

      [Start] 57.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 57.3	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 57.3	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 57.3	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 57.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 57.3	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 37.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Taylor expanded in re around inf 34.8

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

   4. Simplified 34.8

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

      [Start] 34.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      unpow2 [=>] 34.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]

   5. Applied egg-rr 44.3

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]

   6. Simplified 14.7

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

      [Start] 44.3	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right) \]
      expm1-def [=>] 15.1	\[ 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]
      expm1-log1p [=>] 14.7	\[ 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   7. Applied egg-rr 14.7

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

   if 7.6e12 < re < 2.0999999999999999e39

   1. Initial program 44.3

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

   2. Simplified 30.8

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

      [Start] 44.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 44.3	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 44.3	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 44.3	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 44.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 44.3	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 30.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Taylor expanded in re around inf 41.4

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

   4. Simplified 41.4

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

      [Start] 41.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      unpow2 [=>] 41.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]

   5. Applied egg-rr 29.0

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

   if 2.0999999999999999e39 < re < 2.2000000000000001e113

   1. Initial program 50.9

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

   2. Simplified 34.4

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

      [Start] 50.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 50.9	\[ 0.5 \cdot \sqrt{\color{blue}{\left(2 \cdot 1\right)} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      metadata-eval [<=] 50.9	\[ 0.5 \cdot \sqrt{\left(2 \cdot \color{blue}{\left(--1\right)}\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
      associate-*r* [<=] 50.9	\[ 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(--1\right) \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}} \]
      metadata-eval [=>] 50.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)} \]
      *-lft-identity [=>] 50.9	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right)}} \]
      hypot-def [=>] 34.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Applied egg-rr 34.4

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

   4. Taylor expanded in re around 0 31.8

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

   5. Simplified 31.8

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

      [Start] 31.8	\[ 0.5 \cdot \sqrt{2 \cdot \frac{1}{\frac{re}{{im}^{2}} + \frac{1}{im}}} \]
      +-commutative [=>] 31.8	\[ 0.5 \cdot \sqrt{2 \cdot \frac{1}{\color{blue}{\frac{1}{im} + \frac{re}{{im}^{2}}}}} \]
      unpow2 [=>] 31.8	\[ 0.5 \cdot \sqrt{2 \cdot \frac{1}{\frac{1}{im} + \frac{re}{\color{blue}{im \cdot im}}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;re \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 7600000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{1}{\frac{1}{im} + \frac{re}{im \cdot im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	16.5
Cost	8152

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \frac{1}{\frac{1}{im} + \frac{re}{im \cdot im}}}\\ \mathbf{if}\;re \leq -3.8 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 6500000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	16.8
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{if}\;re \leq -3.8 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 8000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 9.2 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 7.8 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	16.7
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{if}\;re \leq -3.8 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 32500000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	15.9
Cost	7576

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ t_1 := 0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{if}\;re \leq -7.5 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 17000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	23.5
Cost	7444

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ t_1 := 0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{if}\;re \leq 7.6 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 16000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	23.5
Cost	7382

\[\begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{-78} \lor \neg \left(re \leq 1.1 \cdot 10^{-36}\right) \land \left(re \leq 27000000000 \lor \neg \left(re \leq 10^{+40}\right) \land re \leq 6 \cdot 10^{+93}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 7
Error	30.1
Cost	6720

\[0.5 \cdot \sqrt{2 \cdot im} \]

Reproduce

herbie shell --seed 2023056 
(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)))))