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

Percentage Accurate: 40.9% → 90.0%

Time: 7.8s

Alternatives: 7

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
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]

Initial Program: 40.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
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]

Alternative 1: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* 0.5 (* im (pow re -0.5)))
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))

C

double code(double re, double im) {
	double tmp;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 14.1%

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

   2. Taylor expanded in re around inf 43.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 43.4%

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

   4. Simplified 43.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im \cdot im}{re}\right)}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 43.2%

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

      2. add-sqr-sqrt 43.1%

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

      3. associate-*r/ 43.1%

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

      4. sqrt-div 43.1%

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

      5. add-sqr-sqrt 43.0%

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

      6. swap-sqr 43.0%

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

      7. sqrt-unprod 43.0%

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

      8. add-sqr-sqrt 43.0%

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

      9. un-div-inv 43.1%

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

      10. metadata-eval 43.1%

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

      11. sqrt-div 43.1%

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

      12. associate-*r/ 43.1%

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

      13. sqrt-div 43.0%

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

   6. Applied egg-rr 95.5%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 47.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def 90.0%

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

   3. Simplified 90.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]

Alternative 2: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -9.5 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -4.8 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -4.9 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -5.8 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im - re} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{-36} \lor \neg \left(re \leq 1.05 \cdot 10^{+26}\right) \land re \leq 1.45 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- im re)))))
        (t_1 (* 0.5 (sqrt (* 2.0 (* re -2.0))))))
   (if (<= re -9.5e+62)
     t_1
     (if (<= re -4.8e-21)
       t_0
       (if (<= re -4.9e-39)
         t_1
         (if (<= re -5.8e-53)
           (* 0.5 (* (sqrt (- im re)) (sqrt 2.0)))
           (if (or (<= re 4.8e-36)
                   (and (not (<= re 1.05e+26)) (<= re 1.45e+40)))
             t_0
             (* 0.5 (/ im (sqrt re))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (im - re)));
	double t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -9.5e+62) {
		tmp = t_1;
	} else if (re <= -4.8e-21) {
		tmp = t_0;
	} else if (re <= -4.9e-39) {
		tmp = t_1;
	} else if (re <= -5.8e-53) {
		tmp = 0.5 * (sqrt((im - re)) * sqrt(2.0));
	} else if ((re <= 4.8e-36) || (!(re <= 1.05e+26) && (re <= 1.45e+40))) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    t_1 = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    if (re <= (-9.5d+62)) then
        tmp = t_1
    else if (re <= (-4.8d-21)) then
        tmp = t_0
    else if (re <= (-4.9d-39)) then
        tmp = t_1
    else if (re <= (-5.8d-53)) then
        tmp = 0.5d0 * (sqrt((im - re)) * sqrt(2.0d0))
    else if ((re <= 4.8d-36) .or. (.not. (re <= 1.05d+26)) .and. (re <= 1.45d+40)) then
        tmp = t_0
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double t_1 = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -9.5e+62) {
		tmp = t_1;
	} else if (re <= -4.8e-21) {
		tmp = t_0;
	} else if (re <= -4.9e-39) {
		tmp = t_1;
	} else if (re <= -5.8e-53) {
		tmp = 0.5 * (Math.sqrt((im - re)) * Math.sqrt(2.0));
	} else if ((re <= 4.8e-36) || (!(re <= 1.05e+26) && (re <= 1.45e+40))) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (im - re)))
	t_1 = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	tmp = 0
	if re <= -9.5e+62:
		tmp = t_1
	elif re <= -4.8e-21:
		tmp = t_0
	elif re <= -4.9e-39:
		tmp = t_1
	elif re <= -5.8e-53:
		tmp = 0.5 * (math.sqrt((im - re)) * math.sqrt(2.0))
	elif (re <= 4.8e-36) or (not (re <= 1.05e+26) and (re <= 1.45e+40)):
		tmp = t_0
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))))
	tmp = 0.0
	if (re <= -9.5e+62)
		tmp = t_1;
	elseif (re <= -4.8e-21)
		tmp = t_0;
	elseif (re <= -4.9e-39)
		tmp = t_1;
	elseif (re <= -5.8e-53)
		tmp = Float64(0.5 * Float64(sqrt(Float64(im - re)) * sqrt(2.0)));
	elseif ((re <= 4.8e-36) || (!(re <= 1.05e+26) && (re <= 1.45e+40)))
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (im - re)));
	t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	tmp = 0.0;
	if (re <= -9.5e+62)
		tmp = t_1;
	elseif (re <= -4.8e-21)
		tmp = t_0;
	elseif (re <= -4.9e-39)
		tmp = t_1;
	elseif (re <= -5.8e-53)
		tmp = 0.5 * (sqrt((im - re)) * sqrt(2.0));
	elseif ((re <= 4.8e-36) || (~((re <= 1.05e+26)) && (re <= 1.45e+40)))
		tmp = t_0;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -9.5e+62], t$95$1, If[LessEqual[re, -4.8e-21], t$95$0, If[LessEqual[re, -4.9e-39], t$95$1, If[LessEqual[re, -5.8e-53], N[(0.5 * N[(N[Sqrt[N[(im - re), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 4.8e-36], And[N[Not[LessEqual[re, 1.05e+26]], $MachinePrecision], LessEqual[re, 1.45e+40]]], t$95$0, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -9.5000000000000003e62 or -4.7999999999999999e-21 < re < -4.89999999999999974e-39

   1. Initial program 38.5%

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

   2. Taylor expanded in re around -inf 84.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}} \]
   3. Step-by-step derivation

      1. *-commutative 84.4%

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

   4. Simplified 84.4%

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

   if -9.5000000000000003e62 < re < -4.7999999999999999e-21 or -5.7999999999999996e-53 < re < 4.8e-36 or 1.05e26 < re < 1.45000000000000009e40

   1. Initial program 58.1%

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

   2. Taylor expanded in re around 0 88.6%

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

   if -4.89999999999999974e-39 < re < -5.7999999999999996e-53

   1. Initial program 51.7%

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

   2. Taylor expanded in re around 0 99.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)} \]
   3. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. sqrt-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 4.8e-36 < re < 1.05e26 or 1.45000000000000009e40 < re

   1. Initial program 15.6%

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

   2. Taylor expanded in re around inf 53.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 53.8%

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

   4. Simplified 53.8%

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

   6. Applied egg-rr 25.3%

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

      1. expm1-def 83.3%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 83.4%

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

   8. Simplified 83.4%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.5 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -4.8 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -4.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -5.8 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im - re} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{-36} \lor \neg \left(re \leq 1.05 \cdot 10^{+26}\right) \land re \leq 1.45 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 76.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-34} \lor \neg \left(re \leq 6 \cdot 10^{+25}\right) \land re \leq 1.5 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e+58)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (or (<= re 4e-34) (and (not (<= re 6e+25)) (<= re 1.5e+40)))
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+58) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 4e-34) || (!(re <= 6e+25) && (re <= 1.5e+40))) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.1d+58)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if ((re <= 4d-34) .or. (.not. (re <= 6d+25)) .and. (re <= 1.5d+40)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+58) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 4e-34) || (!(re <= 6e+25) && (re <= 1.5e+40))) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e+58:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif (re <= 4e-34) or (not (re <= 6e+25) and (re <= 1.5e+40)):
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e+58)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif ((re <= 4e-34) || (!(re <= 6e+25) && (re <= 1.5e+40)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1e+58)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif ((re <= 4e-34) || (~((re <= 6e+25)) && (re <= 1.5e+40)))
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e+58], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 4e-34], And[N[Not[LessEqual[re, 6e+25]], $MachinePrecision], LessEqual[re, 1.5e+40]]], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.0999999999999999e58

   1. Initial program 32.1%

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

   2. Taylor expanded in re around -inf 82.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}} \]
   3. Step-by-step derivation

      1. *-commutative 82.8%

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

   4. Simplified 82.8%

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

   if -3.0999999999999999e58 < re < 3.99999999999999971e-34 or 6.00000000000000011e25 < re < 1.5000000000000001e40

   1. Initial program 59.5%

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

   2. Taylor expanded in re around 0 86.4%

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

   if 3.99999999999999971e-34 < re < 6.00000000000000011e25 or 1.5000000000000001e40 < re

   1. Initial program 15.6%

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

   2. Taylor expanded in re around inf 53.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 53.8%

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

   4. Simplified 53.8%

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

   6. Applied egg-rr 25.3%

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

      1. expm1-def 83.3%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 83.4%

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

   8. Simplified 83.4%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-34} \lor \neg \left(re \leq 6 \cdot 10^{+25}\right) \land re \leq 1.5 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq -1.05 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re -1.05e-37)
     (* 0.5 (sqrt (* 2.0 (* re -2.0))))
     (if (<= re 2.1e-10)
       t_0
       (if (<= re 1.12e+26)
         (* 0.5 (/ im (sqrt re)))
         (if (<= re 7e+41) t_0 (* 0.5 (* im (pow re -0.5)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((im * 2.0));
	double tmp;
	if (re <= -1.05e-37) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.1e-10) {
		tmp = t_0;
	} else if (re <= 1.12e+26) {
		tmp = 0.5 * (im / sqrt(re));
	} else if (re <= 7e+41) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((im * 2.0d0))
    if (re <= (-1.05d-37)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 2.1d-10) then
        tmp = t_0
    else if (re <= 1.12d+26) then
        tmp = 0.5d0 * (im / sqrt(re))
    else if (re <= 7d+41) then
        tmp = t_0
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((im * 2.0));
	double tmp;
	if (re <= -1.05e-37) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.1e-10) {
		tmp = t_0;
	} else if (re <= 1.12e+26) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else if (re <= 7e+41) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((im * 2.0))
	tmp = 0
	if re <= -1.05e-37:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 2.1e-10:
		tmp = t_0
	elif re <= 1.12e+26:
		tmp = 0.5 * (im / math.sqrt(re))
	elif re <= 7e+41:
		tmp = t_0
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	tmp = 0.0
	if (re <= -1.05e-37)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 2.1e-10)
		tmp = t_0;
	elseif (re <= 1.12e+26)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	elseif (re <= 7e+41)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((im * 2.0));
	tmp = 0.0;
	if (re <= -1.05e-37)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 2.1e-10)
		tmp = t_0;
	elseif (re <= 1.12e+26)
		tmp = 0.5 * (im / sqrt(re));
	elseif (re <= 7e+41)
		tmp = t_0;
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.05e-37], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e-10], t$95$0, If[LessEqual[re, 1.12e+26], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7e+41], t$95$0, N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.05e-37

   1. Initial program 44.0%

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

   2. Taylor expanded in re around -inf 75.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}} \]
   3. Step-by-step derivation

      1. *-commutative 75.8%

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

   4. Simplified 75.8%

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

   if -1.05e-37 < re < 2.1e-10 or 1.1200000000000001e26 < re < 6.9999999999999998e41

   1. Initial program 55.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-udef 90.9%

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

      2. add-cbrt-cube 36.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\sqrt[3]{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}} \]

      3. pow1/3 34.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.3333333333333333}}} \]

      4. pow-to-exp 34.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{e^{\log \left(\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.3333333333333333}}} \]

      5. pow3 34.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot e^{\log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right) - re\right)}^{3}\right)} \cdot 0.3333333333333333}} \]

      6. log-pow 83.4%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot e^{\color{blue}{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)} \cdot 0.3333333333333333}} \]

   3. Applied egg-rr 83.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{e^{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.3333333333333333}}} \]

   4. Taylor expanded in re around 0 87.8%

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

   if 2.1e-10 < re < 1.1200000000000001e26

   1. Initial program 20.6%

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

   2. Taylor expanded in re around inf 45.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 45.4%

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

   4. Simplified 45.4%

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

   6. Applied egg-rr 12.0%

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

      1. expm1-def 84.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 84.4%

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

   8. Simplified 84.4%

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

   if 6.9999999999999998e41 < re

   1. Initial program 13.8%

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

   2. Taylor expanded in re around inf 59.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 59.4%

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

   4. Simplified 59.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im \cdot im}{re}\right)}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 59.2%

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

      2. add-sqr-sqrt 59.1%

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

      3. associate-*r/ 59.1%

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

      4. sqrt-div 59.1%

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

      5. add-sqr-sqrt 58.9%

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

      6. swap-sqr 59.0%

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

      7. sqrt-unprod 59.0%

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

      8. add-sqr-sqrt 59.0%

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

      9. un-div-inv 59.0%

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

      10. metadata-eval 59.0%

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

      11. sqrt-div 59.1%

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

      12. associate-*r/ 59.1%

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

      13. sqrt-div 59.0%

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

   6. Applied egg-rr 86.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{-0.5} \cdot im\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 5: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq 1.55 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 5.9 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re 1.55e-12)
     t_0
     (if (<= re 4.8e+26)
       (* 0.5 (/ im (sqrt re)))
       (if (<= re 5.9e+41) t_0 (* 0.5 (* im (pow re -0.5))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((im * 2.0));
	double tmp;
	if (re <= 1.55e-12) {
		tmp = t_0;
	} else if (re <= 4.8e+26) {
		tmp = 0.5 * (im / sqrt(re));
	} else if (re <= 5.9e+41) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((im * 2.0d0))
    if (re <= 1.55d-12) then
        tmp = t_0
    else if (re <= 4.8d+26) then
        tmp = 0.5d0 * (im / sqrt(re))
    else if (re <= 5.9d+41) then
        tmp = t_0
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((im * 2.0));
	double tmp;
	if (re <= 1.55e-12) {
		tmp = t_0;
	} else if (re <= 4.8e+26) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else if (re <= 5.9e+41) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((im * 2.0))
	tmp = 0
	if re <= 1.55e-12:
		tmp = t_0
	elif re <= 4.8e+26:
		tmp = 0.5 * (im / math.sqrt(re))
	elif re <= 5.9e+41:
		tmp = t_0
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	tmp = 0.0
	if (re <= 1.55e-12)
		tmp = t_0;
	elseif (re <= 4.8e+26)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	elseif (re <= 5.9e+41)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((im * 2.0));
	tmp = 0.0;
	if (re <= 1.55e-12)
		tmp = t_0;
	elseif (re <= 4.8e+26)
		tmp = 0.5 * (im / sqrt(re));
	elseif (re <= 5.9e+41)
		tmp = t_0;
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 1.55e-12], t$95$0, If[LessEqual[re, 4.8e+26], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.9e+41], t$95$0, N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.5500000000000001e-12 or 4.80000000000000009e26 < re < 5.9000000000000001e41

   1. Initial program 51.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-udef 94.0%

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

      2. add-cbrt-cube 33.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\sqrt[3]{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}} \]

      3. pow1/3 31.8%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.3333333333333333}}} \]

      4. pow-to-exp 31.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{e^{\log \left(\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.3333333333333333}}} \]

      5. pow3 31.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot e^{\log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right) - re\right)}^{3}\right)} \cdot 0.3333333333333333}} \]

      6. log-pow 85.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot e^{\color{blue}{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)} \cdot 0.3333333333333333}} \]

   3. Applied egg-rr 85.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{e^{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.3333333333333333}}} \]

   4. Taylor expanded in re around 0 68.2%

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

   if 1.5500000000000001e-12 < re < 4.80000000000000009e26

   1. Initial program 20.6%

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

   2. Taylor expanded in re around inf 45.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 45.4%

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

   4. Simplified 45.4%

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

   6. Applied egg-rr 12.0%

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

      1. expm1-def 84.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 84.4%

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

   8. Simplified 84.4%

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

   if 5.9000000000000001e41 < re

   1. Initial program 13.8%

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

   2. Taylor expanded in re around inf 59.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 59.4%

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

   4. Simplified 59.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im \cdot im}{re}\right)}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 59.2%

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

      2. add-sqr-sqrt 59.1%

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

      3. associate-*r/ 59.1%

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

      4. sqrt-div 59.1%

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

      5. add-sqr-sqrt 58.9%

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

      6. swap-sqr 59.0%

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

      7. sqrt-unprod 59.0%

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

      8. add-sqr-sqrt 59.0%

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

      9. un-div-inv 59.0%

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

      10. metadata-eval 59.0%

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

      11. sqrt-div 59.1%

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

      12. associate-*r/ 59.1%

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

      13. sqrt-div 59.0%

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

   6. Applied egg-rr 86.6%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.55 \cdot 10^{-12}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 5.9 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 6: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.6 \cdot 10^{-11} \lor \neg \left(re \leq 1.2 \cdot 10^{+26}\right) \land re \leq 1.08 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 2.6e-11) (and (not (<= re 1.2e+26)) (<= re 1.08e+40)))
   (* 0.5 (sqrt (* im 2.0)))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 2.6e-11) || (!(re <= 1.2e+26) && (re <= 1.08e+40))) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= 2.6d-11) .or. (.not. (re <= 1.2d+26)) .and. (re <= 1.08d+40)) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 2.6e-11) || (!(re <= 1.2e+26) && (re <= 1.08e+40))) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 2.6e-11) or (not (re <= 1.2e+26) and (re <= 1.08e+40)):
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 2.6e-11) || (!(re <= 1.2e+26) && (re <= 1.08e+40)))
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 2.6e-11) || (~((re <= 1.2e+26)) && (re <= 1.08e+40)))
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 2.6e-11], And[N[Not[LessEqual[re, 1.2e+26]], $MachinePrecision], LessEqual[re, 1.08e+40]]], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.6000000000000001e-11 or 1.20000000000000002e26 < re < 1.08000000000000001e40

   1. Initial program 51.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-udef 94.5%

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

      2. add-cbrt-cube 34.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\sqrt[3]{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}} \]

      3. pow1/3 32.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.3333333333333333}}} \]

      4. pow-to-exp 32.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{e^{\log \left(\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.3333333333333333}}} \]

      5. pow3 32.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot e^{\log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right) - re\right)}^{3}\right)} \cdot 0.3333333333333333}} \]

      6. log-pow 86.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot e^{\color{blue}{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)} \cdot 0.3333333333333333}} \]

   3. Applied egg-rr 86.3%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{e^{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.3333333333333333}}} \]

   4. Taylor expanded in re around 0 68.4%

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

   if 2.6000000000000001e-11 < re < 1.20000000000000002e26 or 1.08000000000000001e40 < re

   1. Initial program 14.8%

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

   2. Taylor expanded in re around inf 55.3%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 55.3%

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

   4. Simplified 55.3%

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

   6. Applied egg-rr 26.5%

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

      1. expm1-def 85.1%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 85.2%

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

   8. Simplified 85.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.6 \cdot 10^{-11} \lor \neg \left(re \leq 1.2 \cdot 10^{+26}\right) \land re \leq 1.08 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 7: 52.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{im \cdot 2} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (sqrt (* im 2.0))))

C

double code(double re, double im) {
	return 0.5 * sqrt((im * 2.0));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((im * 2.0d0))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((im * 2.0));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((im * 2.0))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(im * 2.0)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((im * 2.0));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.2%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
2. Step-by-step derivation

   1. hypot-udef 79.0%

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

   2. add-cbrt-cube 31.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\sqrt[3]{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}} \]

   3. pow1/3 29.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.3333333333333333}}} \]

   4. pow-to-exp 29.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{e^{\log \left(\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.3333333333333333}}} \]

   5. pow3 29.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot e^{\log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right) - re\right)}^{3}\right)} \cdot 0.3333333333333333}} \]

   6. log-pow 72.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot e^{\color{blue}{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)} \cdot 0.3333333333333333}} \]

3. Applied egg-rr 72.6%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{e^{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.3333333333333333}}} \]

4. Taylor expanded in re around 0 55.7%

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

5. Final simplification 55.7%

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

Reproduce

herbie shell --seed 2023217 
(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)))))