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

Percentage Accurate: 41.4% → 90.1%

Time: 6.0s

Alternatives: 5

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: 41.4% 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: 90.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.9%

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

      1. sqr-neg 4.9%

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

      2. sqr-neg 4.9%

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

      3. hypot-def 4.9%

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

   3. Simplified 4.9%

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

   4. Taylor expanded in re around inf 57.4%

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

      1. add-log-exp 8.5%

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

      2. *-un-lft-identity 8.5%

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

      3. log-prod 8.5%

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

      4. metadata-eval 8.5%

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

      5. add-log-exp 57.4%

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

      6. sqrt-div 62.4%

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

      7. unpow2 62.4%

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

      8. sqrt-prod 99.3%

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

      9. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 99.8%

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

   8. Simplified 99.8%

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

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

   1. Initial program 47.1%

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

      1. hypot-udef 89.5%

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

      2. add-sqr-sqrt 88.9%

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

      3. sqrt-unprod 89.5%

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

      4. *-commutative 89.5%

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

      5. *-commutative 89.5%

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

      6. swap-sqr 89.5%

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

      7. add-sqr-sqrt 89.5%

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

      8. metadata-eval 89.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \color{blue}{0.25}} \]

   3. Applied egg-rr 89.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}} \]
   4. Step-by-step derivation

      1. *-commutative 89.5%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

      2. associate-*r* 89.5%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      3. metadata-eval 89.5%

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

   5. Simplified 89.5%

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

4. Final simplification 91.3%

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

Alternative 2: 72.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -5.2 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -3.9 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -400000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-147} \lor \neg \left(re \leq 1.95 \cdot 10^{+28}\right) \land re \leq 1.5 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (* 0.5 (sqrt (* 2.0 im)))))
   (if (<= re -5.2e+71)
     t_0
     (if (<= re -3.9e+33)
       t_1
       (if (<= re -400000000000.0)
         t_0
         (if (or (<= re 1.45e-147)
                 (and (not (<= re 1.95e+28)) (<= re 1.5e+53)))
           t_1
           (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((2.0 * im));
	double tmp;
	if (re <= -5.2e+71) {
		tmp = t_0;
	} else if (re <= -3.9e+33) {
		tmp = t_1;
	} else if (re <= -400000000000.0) {
		tmp = t_0;
	} else if ((re <= 1.45e-147) || (!(re <= 1.95e+28) && (re <= 1.5e+53))) {
		tmp = t_1;
	} 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((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((2.0d0 * im))
    if (re <= (-5.2d+71)) then
        tmp = t_0
    else if (re <= (-3.9d+33)) then
        tmp = t_1
    else if (re <= (-400000000000.0d0)) then
        tmp = t_0
    else if ((re <= 1.45d-147) .or. (.not. (re <= 1.95d+28)) .and. (re <= 1.5d+53)) then
        tmp = t_1
    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((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((2.0 * im));
	double tmp;
	if (re <= -5.2e+71) {
		tmp = t_0;
	} else if (re <= -3.9e+33) {
		tmp = t_1;
	} else if (re <= -400000000000.0) {
		tmp = t_0;
	} else if ((re <= 1.45e-147) || (!(re <= 1.95e+28) && (re <= 1.5e+53))) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((2.0 * im))
	tmp = 0
	if re <= -5.2e+71:
		tmp = t_0
	elif re <= -3.9e+33:
		tmp = t_1
	elif re <= -400000000000.0:
		tmp = t_0
	elif (re <= 1.45e-147) or (not (re <= 1.95e+28) and (re <= 1.5e+53)):
		tmp = t_1
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * im)))
	tmp = 0.0
	if (re <= -5.2e+71)
		tmp = t_0;
	elseif (re <= -3.9e+33)
		tmp = t_1;
	elseif (re <= -400000000000.0)
		tmp = t_0;
	elseif ((re <= 1.45e-147) || (!(re <= 1.95e+28) && (re <= 1.5e+53)))
		tmp = t_1;
	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((re * -4.0));
	t_1 = 0.5 * sqrt((2.0 * im));
	tmp = 0.0;
	if (re <= -5.2e+71)
		tmp = t_0;
	elseif (re <= -3.9e+33)
		tmp = t_1;
	elseif (re <= -400000000000.0)
		tmp = t_0;
	elseif ((re <= 1.45e-147) || (~((re <= 1.95e+28)) && (re <= 1.5e+53)))
		tmp = t_1;
	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[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -5.2e+71], t$95$0, If[LessEqual[re, -3.9e+33], t$95$1, If[LessEqual[re, -400000000000.0], t$95$0, If[Or[LessEqual[re, 1.45e-147], And[N[Not[LessEqual[re, 1.95e+28]], $MachinePrecision], LessEqual[re, 1.5e+53]]], t$95$1, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.19999999999999983e71 or -3.9000000000000002e33 < re < -4e11

   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. sqr-neg 42.2%

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

      2. sqr-neg 42.2%

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

      3. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around -inf 79.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   5. Step-by-step derivation

      1. *-commutative 79.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   6. Simplified 79.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -5.19999999999999983e71 < re < -3.9000000000000002e33 or -4e11 < re < 1.4500000000000001e-147 or 1.9499999999999999e28 < re < 1.49999999999999999e53

   1. Initial program 61.3%

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

      1. sqr-neg 61.3%

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

      2. sqr-neg 61.3%

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

      3. hypot-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in re around 0 79.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 79.2%

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

   6. Simplified 79.2%

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

   if 1.4500000000000001e-147 < re < 1.9499999999999999e28 or 1.49999999999999999e53 < re

   1. Initial program 16.6%

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

      1. sqr-neg 16.6%

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

      2. sqr-neg 16.6%

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

      3. hypot-def 36.3%

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

   3. Simplified 36.3%

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

   4. Taylor expanded in re around inf 48.1%

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

      1. add-log-exp 13.7%

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

      2. *-un-lft-identity 13.7%

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

      3. log-prod 13.7%

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

      4. metadata-eval 13.7%

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

      5. add-log-exp 48.1%

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

      6. sqrt-div 51.3%

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

      7. unpow2 51.3%

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

      8. sqrt-prod 74.1%

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

      9. add-sqr-sqrt 74.4%

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

   6. Applied egg-rr 74.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 74.4%

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

   8. Simplified 74.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -3.9 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq -400000000000:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-147} \lor \neg \left(re \leq 1.95 \cdot 10^{+28}\right) \land re \leq 1.5 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 73.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -5.3 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -7.5 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -350000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 6.1 \cdot 10^{+28} \lor \neg \left(re \leq 1.5 \cdot 10^{+53}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0))))
        (t_1 (* 0.5 (sqrt (* 2.0 (- im re))))))
   (if (<= re -5.3e+71)
     t_0
     (if (<= re -7.5e+31)
       t_1
       (if (<= re -350000000000.0)
         t_0
         (if (<= re 2.05e-151)
           t_1
           (if (or (<= re 6.1e+28) (not (<= re 1.5e+53)))
             (* 0.5 (/ im (sqrt re)))
             (* 0.5 (sqrt (* 2.0 im))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -5.3e+71) {
		tmp = t_0;
	} else if (re <= -7.5e+31) {
		tmp = t_1;
	} else if (re <= -350000000000.0) {
		tmp = t_0;
	} else if (re <= 2.05e-151) {
		tmp = t_1;
	} else if ((re <= 6.1e+28) || !(re <= 1.5e+53)) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	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((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    if (re <= (-5.3d+71)) then
        tmp = t_0
    else if (re <= (-7.5d+31)) then
        tmp = t_1
    else if (re <= (-350000000000.0d0)) then
        tmp = t_0
    else if (re <= 2.05d-151) then
        tmp = t_1
    else if ((re <= 6.1d+28) .or. (.not. (re <= 1.5d+53))) then
        tmp = 0.5d0 * (im / sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -5.3e+71) {
		tmp = t_0;
	} else if (re <= -7.5e+31) {
		tmp = t_1;
	} else if (re <= -350000000000.0) {
		tmp = t_0;
	} else if (re <= 2.05e-151) {
		tmp = t_1;
	} else if ((re <= 6.1e+28) || !(re <= 1.5e+53)) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((2.0 * (im - re)))
	tmp = 0
	if re <= -5.3e+71:
		tmp = t_0
	elif re <= -7.5e+31:
		tmp = t_1
	elif re <= -350000000000.0:
		tmp = t_0
	elif re <= 2.05e-151:
		tmp = t_1
	elif (re <= 6.1e+28) or not (re <= 1.5e+53):
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	tmp = 0.0
	if (re <= -5.3e+71)
		tmp = t_0;
	elseif (re <= -7.5e+31)
		tmp = t_1;
	elseif (re <= -350000000000.0)
		tmp = t_0;
	elseif (re <= 2.05e-151)
		tmp = t_1;
	elseif ((re <= 6.1e+28) || !(re <= 1.5e+53))
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * sqrt((2.0 * (im - re)));
	tmp = 0.0;
	if (re <= -5.3e+71)
		tmp = t_0;
	elseif (re <= -7.5e+31)
		tmp = t_1;
	elseif (re <= -350000000000.0)
		tmp = t_0;
	elseif (re <= 2.05e-151)
		tmp = t_1;
	elseif ((re <= 6.1e+28) || ~((re <= 1.5e+53)))
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -5.3e+71], t$95$0, If[LessEqual[re, -7.5e+31], t$95$1, If[LessEqual[re, -350000000000.0], t$95$0, If[LessEqual[re, 2.05e-151], t$95$1, If[Or[LessEqual[re, 6.1e+28], N[Not[LessEqual[re, 1.5e+53]], $MachinePrecision]], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -5.2999999999999999e71 or -7.5e31 < re < -3.5e11

   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. sqr-neg 42.2%

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

      2. sqr-neg 42.2%

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

      3. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around -inf 79.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   5. Step-by-step derivation

      1. *-commutative 79.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   6. Simplified 79.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -5.2999999999999999e71 < re < -7.5e31 or -3.5e11 < re < 2.0500000000000001e-151

   1. Initial program 65.9%

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

   2. Taylor expanded in re around 0 82.6%

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

   if 2.0500000000000001e-151 < re < 6.1000000000000002e28 or 1.49999999999999999e53 < re

   1. Initial program 16.3%

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

      1. sqr-neg 16.3%

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

      2. sqr-neg 16.3%

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

      3. hypot-def 36.6%

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

   3. Simplified 36.6%

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

   4. Taylor expanded in re around inf 47.2%

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

      1. add-log-exp 13.5%

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

      2. *-un-lft-identity 13.5%

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

      3. log-prod 13.5%

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

      4. metadata-eval 13.5%

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

      5. add-log-exp 47.2%

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

      6. sqrt-div 50.4%

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

      7. unpow2 50.4%

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

      8. sqrt-prod 73.6%

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

      9. add-sqr-sqrt 74.0%

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

   6. Applied egg-rr 74.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 74.0%

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

   8. Simplified 74.0%

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

   if 6.1000000000000002e28 < re < 1.49999999999999999e53

   1. Initial program 18.1%

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

      1. sqr-neg 18.1%

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

      2. sqr-neg 18.1%

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

      3. hypot-def 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in re around 0 86.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 86.6%

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

   6. Simplified 86.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.3 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -7.5 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -350000000000:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{-151}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 6.1 \cdot 10^{+28} \lor \neg \left(re \leq 1.5 \cdot 10^{+53}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 4: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.8e-96) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.8e-96) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.8d-96) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.8e-96) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.8e-96:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.8e-96)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.8e-96)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.8e-96], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.80000000000000015e-96

   1. Initial program 37.0%

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

      1. sqr-neg 37.0%

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

      2. sqr-neg 37.0%

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

      3. hypot-def 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in re around -inf 43.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   5. Step-by-step derivation

      1. *-commutative 43.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   6. Simplified 43.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if 2.80000000000000015e-96 < im

   1. Initial program 41.1%

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

      1. sqr-neg 41.1%

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

      2. sqr-neg 41.1%

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

      3. hypot-def 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in re around 0 65.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 65.6%

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

   6. Simplified 65.6%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 5: 52.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

   1. sqr-neg 39.7%

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

   2. sqr-neg 39.7%

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

   3. hypot-def 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in re around 0 48.2%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
5. Step-by-step derivation

   1. *-commutative 48.2%

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

6. Simplified 48.2%

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

7. Final simplification 48.2%

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

Reproduce

herbie shell --seed 2023305 
(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)))))