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

Percentage Accurate: 40.6% → 89.8%

Time: 6.4s

Alternatives: 6

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: 40.6% 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: 89.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (/ 0.5 (/ (sqrt re) im))
   (* 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 / (sqrt(re) / im);
	} 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 / (Math.sqrt(re) / im);
	} 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 / (math.sqrt(re) / im)
	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(sqrt(re) / im));
	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 / (sqrt(re) / im);
	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[(N[Sqrt[re], $MachinePrecision] / im), $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 7.1%

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

   2. Taylor expanded in im around 0 94.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   3. Step-by-step derivation

      1. associate-*l* 94.4%

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

      2. *-commutative 94.4%

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

      3. associate-*l* 94.4%

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

   4. Simplified 94.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-log-exp 9.5%

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

      2. *-un-lft-identity 9.5%

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

      3. log-prod 9.5%

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

      4. metadata-eval 9.5%

         \[\leadsto \color{blue}{0} + \log \left(e^{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}\right) \]

      5. add-log-exp 94.4%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      6. associate-*r* 94.4%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      7. *-commutative 94.4%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\sqrt{0.5} \cdot im\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      8. associate-*l* 94.2%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      9. associate-*r* 94.2%

         \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \]

      10. sqrt-div 94.2%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}} \]

      11. metadata-eval 94.2%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}} \]

      12. un-div-inv 94.3%

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

   6. Applied egg-rr 95.1%

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

      1. +-lft-identity 95.1%

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

      2. associate-/l* 95.3%

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

   8. Simplified 95.3%

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

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

   1. Initial program 52.0%

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

      1. hypot-def 89.1%

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

   3. Simplified 89.1%

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

4. Final simplification 90.1%

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

Alternative 2: 75.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-59} \lor \neg \left(re \leq 4.1 \cdot 10^{+98}\right) \land re \leq 2.3 \cdot 10^{+129}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.7e-16)
   (* 0.5 (sqrt (* re -4.0)))
   (if (or (<= re 2.8e-59) (and (not (<= re 4.1e+98)) (<= re 2.3e+129)))
     (* 0.5 (sqrt (* im 2.0)))
     (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.7e-16) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if ((re <= 2.8e-59) || (!(re <= 4.1e+98) && (re <= 2.3e+129))) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = im * sqrt((0.25 / 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 <= (-2.7d-16)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if ((re <= 2.8d-59) .or. (.not. (re <= 4.1d+98)) .and. (re <= 2.3d+129)) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = im * sqrt((0.25d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.7e-16) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if ((re <= 2.8e-59) || (!(re <= 4.1e+98) && (re <= 2.3e+129))) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = im * Math.sqrt((0.25 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.7e-16:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif (re <= 2.8e-59) or (not (re <= 4.1e+98) and (re <= 2.3e+129)):
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = im * math.sqrt((0.25 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.7e-16)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif ((re <= 2.8e-59) || (!(re <= 4.1e+98) && (re <= 2.3e+129)))
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.7e-16)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif ((re <= 2.8e-59) || (~((re <= 4.1e+98)) && (re <= 2.3e+129)))
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = im * sqrt((0.25 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.7e-16], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 2.8e-59], And[N[Not[LessEqual[re, 4.1e+98]], $MachinePrecision], LessEqual[re, 2.3e+129]]], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.69999999999999999e-16

   1. Initial program 49.9%

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

      1. 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 83.9%

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

      1. *-commutative 83.9%

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

   6. Simplified 83.9%

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

   if -2.69999999999999999e-16 < re < 2.79999999999999981e-59 or 4.1e98 < re < 2.2999999999999999e129

   1. Initial program 61.6%

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

      1. hypot-def 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in re around 0 86.9%

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

      1. *-commutative 86.9%

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

   6. Simplified 86.9%

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

   if 2.79999999999999981e-59 < re < 4.1e98 or 2.2999999999999999e129 < re

   1. Initial program 15.4%

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

   2. Taylor expanded in im around 0 73.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   3. Step-by-step derivation

      1. associate-*l* 73.6%

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

      2. *-commutative 73.6%

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

      3. associate-*l* 73.7%

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

   4. Simplified 73.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-log-exp 7.1%

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

      2. *-un-lft-identity 7.1%

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

      3. log-prod 7.1%

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

      4. metadata-eval 7.1%

         \[\leadsto \color{blue}{0} + \log \left(e^{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}\right) \]

      5. add-log-exp 73.7%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      6. associate-*r* 73.6%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      7. *-commutative 73.6%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\sqrt{0.5} \cdot im\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      8. associate-*l* 73.6%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      9. associate-*r* 73.6%

         \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \]

      10. sqrt-div 73.5%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}} \]

      11. metadata-eval 73.5%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}} \]

      12. un-div-inv 73.4%

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

   6. Applied egg-rr 74.2%

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

      1. +-lft-identity 74.2%

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

      2. associate-*l/ 74.2%

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

   8. Simplified 74.2%

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

      1. add-sqr-sqrt 73.9%

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

      2. sqrt-unprod 74.2%

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

      3. frac-times 74.1%

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

      4. metadata-eval 74.1%

         \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}} \cdot im \]

      5. add-sqr-sqrt 74.3%

         \[\leadsto \sqrt{\frac{0.25}{\color{blue}{re}}} \cdot im \]

   10. Applied egg-rr 74.3%

       \[\leadsto \color{blue}{\sqrt{\frac{0.25}{re}}} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-59} \lor \neg \left(re \leq 4.1 \cdot 10^{+98}\right) \land re \leq 2.3 \cdot 10^{+129}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]

Alternative 3: 74.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq -1.08 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 4 \cdot 10^{+98}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re -1.08e-16)
     (* 0.5 (sqrt (* re -4.0)))
     (if (<= re 2.8e-59)
       t_0
       (if (<= re 4e+98)
         (/ 0.5 (/ (sqrt re) im))
         (if (<= re 2.3e+129) t_0 (* im (sqrt (/ 0.25 re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((im * 2.0));
	double tmp;
	if (re <= -1.08e-16) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.8e-59) {
		tmp = t_0;
	} else if (re <= 4e+98) {
		tmp = 0.5 / (sqrt(re) / im);
	} else if (re <= 2.3e+129) {
		tmp = t_0;
	} else {
		tmp = im * sqrt((0.25 / 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) :: tmp
    t_0 = 0.5d0 * sqrt((im * 2.0d0))
    if (re <= (-1.08d-16)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.8d-59) then
        tmp = t_0
    else if (re <= 4d+98) then
        tmp = 0.5d0 / (sqrt(re) / im)
    else if (re <= 2.3d+129) then
        tmp = t_0
    else
        tmp = im * sqrt((0.25d0 / re))
    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 <= -1.08e-16) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.8e-59) {
		tmp = t_0;
	} else if (re <= 4e+98) {
		tmp = 0.5 / (Math.sqrt(re) / im);
	} else if (re <= 2.3e+129) {
		tmp = t_0;
	} else {
		tmp = im * Math.sqrt((0.25 / re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((im * 2.0))
	tmp = 0
	if re <= -1.08e-16:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.8e-59:
		tmp = t_0
	elif re <= 4e+98:
		tmp = 0.5 / (math.sqrt(re) / im)
	elif re <= 2.3e+129:
		tmp = t_0
	else:
		tmp = im * math.sqrt((0.25 / re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	tmp = 0.0
	if (re <= -1.08e-16)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.8e-59)
		tmp = t_0;
	elseif (re <= 4e+98)
		tmp = Float64(0.5 / Float64(sqrt(re) / im));
	elseif (re <= 2.3e+129)
		tmp = t_0;
	else
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((im * 2.0));
	tmp = 0.0;
	if (re <= -1.08e-16)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.8e-59)
		tmp = t_0;
	elseif (re <= 4e+98)
		tmp = 0.5 / (sqrt(re) / im);
	elseif (re <= 2.3e+129)
		tmp = t_0;
	else
		tmp = im * sqrt((0.25 / re));
	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, -1.08e-16], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.8e-59], t$95$0, If[LessEqual[re, 4e+98], N[(0.5 / N[(N[Sqrt[re], $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.3e+129], t$95$0, N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.08e-16

   1. Initial program 49.9%

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

      1. 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 83.9%

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

      1. *-commutative 83.9%

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

   6. Simplified 83.9%

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

   if -1.08e-16 < re < 2.79999999999999981e-59 or 3.99999999999999999e98 < re < 2.2999999999999999e129

   1. Initial program 61.6%

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

      1. hypot-def 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in re around 0 86.9%

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

      1. *-commutative 86.9%

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

   6. Simplified 86.9%

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

   if 2.79999999999999981e-59 < re < 3.99999999999999999e98

   1. Initial program 24.8%

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

   2. Taylor expanded in im around 0 62.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   3. Step-by-step derivation

      1. associate-*l* 62.8%

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

      2. *-commutative 62.8%

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

      3. associate-*l* 62.8%

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

   4. Simplified 62.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-log-exp 8.7%

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

      2. *-un-lft-identity 8.7%

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

      3. log-prod 8.7%

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

      4. metadata-eval 8.7%

         \[\leadsto \color{blue}{0} + \log \left(e^{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}\right) \]

      5. add-log-exp 62.8%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      6. associate-*r* 62.8%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      7. *-commutative 62.8%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\sqrt{0.5} \cdot im\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      8. associate-*l* 62.7%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      9. associate-*r* 62.7%

         \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \]

      10. sqrt-div 62.6%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}} \]

      11. metadata-eval 62.6%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}} \]

      12. un-div-inv 62.6%

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

   6. Applied egg-rr 63.2%

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

      1. +-lft-identity 63.2%

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

      2. associate-/l* 63.4%

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

   8. Simplified 63.4%

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

   if 2.2999999999999999e129 < re

   1. Initial program 3.5%

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

   2. Taylor expanded in im around 0 87.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   3. Step-by-step derivation

      1. associate-*l* 87.3%

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

      2. *-commutative 87.3%

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

      3. associate-*l* 87.5%

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

   4. Simplified 87.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-log-exp 5.0%

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

      2. *-un-lft-identity 5.0%

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

      3. log-prod 5.0%

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

      4. metadata-eval 5.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}\right) \]

      5. add-log-exp 87.5%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      6. associate-*r* 87.3%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      7. *-commutative 87.3%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\sqrt{0.5} \cdot im\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      8. associate-*l* 87.3%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      9. associate-*r* 87.3%

         \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \]

      10. sqrt-div 87.2%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}} \]

      11. metadata-eval 87.2%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}} \]

      12. un-div-inv 87.1%

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

   6. Applied egg-rr 88.0%

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

      1. +-lft-identity 88.0%

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

      2. associate-*l/ 88.1%

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

   8. Simplified 88.1%

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

      1. add-sqr-sqrt 87.6%

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

      2. sqrt-unprod 88.1%

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

      3. frac-times 87.9%

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

      4. metadata-eval 87.9%

         \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}} \cdot im \]

      5. add-sqr-sqrt 88.3%

         \[\leadsto \sqrt{\frac{0.25}{\color{blue}{re}}} \cdot im \]

   10. Applied egg-rr 88.3%

       \[\leadsto \color{blue}{\sqrt{\frac{0.25}{re}}} \cdot im \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.08 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{+98}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+129}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]

Alternative 4: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.22 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+97}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+129}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.6e-18)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.22e-61)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 1.4e+97)
       (/ 0.5 (/ (sqrt re) im))
       (if (<= re 2.3e+129)
         (* 0.5 (sqrt (* im 2.0)))
         (* im (sqrt (/ 0.25 re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.6e-18) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.22e-61) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 1.4e+97) {
		tmp = 0.5 / (sqrt(re) / im);
	} else if (re <= 2.3e+129) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = im * sqrt((0.25 / 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 <= (-4.6d-18)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.22d-61) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 1.4d+97) then
        tmp = 0.5d0 / (sqrt(re) / im)
    else if (re <= 2.3d+129) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = im * sqrt((0.25d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.6e-18) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.22e-61) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 1.4e+97) {
		tmp = 0.5 / (Math.sqrt(re) / im);
	} else if (re <= 2.3e+129) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = im * Math.sqrt((0.25 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.6e-18:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.22e-61:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 1.4e+97:
		tmp = 0.5 / (math.sqrt(re) / im)
	elif re <= 2.3e+129:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = im * math.sqrt((0.25 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.6e-18)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.22e-61)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 1.4e+97)
		tmp = Float64(0.5 / Float64(sqrt(re) / im));
	elseif (re <= 2.3e+129)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.6e-18)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.22e-61)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 1.4e+97)
		tmp = 0.5 / (sqrt(re) / im);
	elseif (re <= 2.3e+129)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = im * sqrt((0.25 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.6e-18], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.22e-61], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e+97], N[(0.5 / N[(N[Sqrt[re], $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.3e+129], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -4.6000000000000002e-18

   1. Initial program 49.9%

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

      1. 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 83.9%

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

      1. *-commutative 83.9%

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

   6. Simplified 83.9%

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

   if -4.6000000000000002e-18 < re < 1.22e-61

   1. Initial program 66.7%

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

   2. Taylor expanded in re around 0 89.2%

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

   if 1.22e-61 < re < 1.4e97

   1. Initial program 23.9%

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

   2. Taylor expanded in im around 0 62.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   3. Step-by-step derivation

      1. associate-*l* 62.3%

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

      2. *-commutative 62.3%

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

      3. associate-*l* 62.3%

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

   4. Simplified 62.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-log-exp 8.5%

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

      2. *-un-lft-identity 8.5%

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

      3. log-prod 8.5%

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

      4. metadata-eval 8.5%

         \[\leadsto \color{blue}{0} + \log \left(e^{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}\right) \]

      5. add-log-exp 62.3%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      6. associate-*r* 62.3%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      7. *-commutative 62.3%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\sqrt{0.5} \cdot im\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      8. associate-*l* 62.2%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      9. associate-*r* 62.2%

         \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \]

      10. sqrt-div 62.1%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}} \]

      11. metadata-eval 62.1%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}} \]

      12. un-div-inv 62.1%

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

   6. Applied egg-rr 62.8%

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

      1. +-lft-identity 62.8%

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

      2. associate-/l* 62.9%

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

   8. Simplified 62.9%

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

   if 1.4e97 < re < 2.2999999999999999e129

   1. Initial program 14.8%

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

      1. hypot-def 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in re around 0 79.4%

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

      1. *-commutative 79.4%

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

   6. Simplified 79.4%

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

   if 2.2999999999999999e129 < re

   1. Initial program 3.5%

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

   2. Taylor expanded in im around 0 87.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   3. Step-by-step derivation

      1. associate-*l* 87.3%

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

      2. *-commutative 87.3%

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

      3. associate-*l* 87.5%

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

   4. Simplified 87.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. add-log-exp 5.0%

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

      2. *-un-lft-identity 5.0%

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

      3. log-prod 5.0%

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

      4. metadata-eval 5.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}\right) \]

      5. add-log-exp 87.5%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      6. associate-*r* 87.3%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      7. *-commutative 87.3%

         \[\leadsto 0 + 0.5 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\sqrt{0.5} \cdot im\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      8. associate-*l* 87.3%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      9. associate-*r* 87.3%

         \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \]

      10. sqrt-div 87.2%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}} \]

      11. metadata-eval 87.2%

          \[\leadsto 0 + \left(0.5 \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}} \]

      12. un-div-inv 87.1%

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

   6. Applied egg-rr 88.0%

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

      1. +-lft-identity 88.0%

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

      2. associate-*l/ 88.1%

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

   8. Simplified 88.1%

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

      1. add-sqr-sqrt 87.6%

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

      2. sqrt-unprod 88.1%

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

      3. frac-times 87.9%

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

      4. metadata-eval 87.9%

         \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}} \cdot im \]

      5. add-sqr-sqrt 88.3%

         \[\leadsto \sqrt{\frac{0.25}{\color{blue}{re}}} \cdot im \]

   10. Applied egg-rr 88.3%

       \[\leadsto \color{blue}{\sqrt{\frac{0.25}{re}}} \cdot im \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.22 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+97}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+129}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]

Alternative 5: 63.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.8 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.8e-17) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.8e-17) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-7.8d-17)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.8e-17) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.8e-17:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.8e-17)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.8e-17)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.8e-17], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -7.79999999999999979e-17

   1. Initial program 49.9%

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

      1. 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 83.9%

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

      1. *-commutative 83.9%

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

   6. Simplified 83.9%

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

   if -7.79999999999999979e-17 < re

   1. Initial program 43.2%

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

      1. hypot-def 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in re around 0 64.4%

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

      1. *-commutative 64.4%

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

   6. Simplified 64.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.8 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6: 51.9% 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 44.8%

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

   1. hypot-def 76.8%

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

3. Simplified 76.8%

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

4. Taylor expanded in re around 0 54.0%

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

   1. *-commutative 54.0%

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

6. Simplified 54.0%

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

7. Final simplification 54.0%

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

Reproduce

herbie shell --seed 2023188 
(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)))))