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

Percentage Accurate: 41.3% → 90.1%

Time: 8.4s

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: 41.3% 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

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* 0.5 (/ im (sqrt re)))
   (* 0.5 (sqrt (* 2.0 (- (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 = 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((2.0 * (Math.sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} 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((2.0 * (math.sqrt(((re * re) + (im * im))) - re))) <= 0.0:
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (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 = 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((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0)
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (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[(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 (sqrt.f64 (*.f64 2 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 10.8%

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

   2. Taylor expanded in re around inf 57.1%

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

      1. unpow2 57.1%

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

   4. Simplified 57.1%

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

   5. Taylor expanded in im around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. rem-exp-log 91.1%

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

      3. unpow1/2 91.1%

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

      4. exp-to-pow 91.0%

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

      5. *-commutative 91.0%

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

      6. log-pow 91.0%

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

      7. exp-to-pow 91.0%

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

      8. log-rec 91.0%

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

      9. distribute-lft-neg-out 91.0%

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

      10. exp-neg 91.0%

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

      11. log-rec 91.0%

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

      12. exp-to-pow 91.0%

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

      13. unpow1/2 91.0%

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

      14. exp-sum 90.6%

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

      15. sub-neg 90.6%

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

      16. log-div 91.3%

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

      17. rem-exp-log 99.7%

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

   7. Simplified 99.7%

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

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

   1. Initial program 43.6%

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

      1. sqr-neg 43.6%

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

      2. sqr-neg 43.6%

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

      3. hypot-def 87.3%

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

   3. Simplified 87.3%

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

4. Final simplification 89.2%

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

Alternative 2: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-47} \lor \neg \left(re \leq 1.6 \cdot 10^{+92}\right) \land re \leq 3.8 \cdot 10^{+138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.15e+72)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (or (<= re 1.35e-47) (and (not (<= re 1.6e+92)) (<= re 3.8e+138)))
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.15e+72) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 1.35e-47) || (!(re <= 1.6e+92) && (re <= 3.8e+138))) {
		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 <= (-1.15d+72)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if ((re <= 1.35d-47) .or. (.not. (re <= 1.6d+92)) .and. (re <= 3.8d+138)) 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 <= -1.15e+72) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 1.35e-47) || (!(re <= 1.6e+92) && (re <= 3.8e+138))) {
		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 <= -1.15e+72:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif (re <= 1.35e-47) or (not (re <= 1.6e+92) and (re <= 3.8e+138)):
		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 <= -1.15e+72)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif ((re <= 1.35e-47) || (!(re <= 1.6e+92) && (re <= 3.8e+138)))
		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 <= -1.15e+72)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif ((re <= 1.35e-47) || (~((re <= 1.6e+92)) && (re <= 3.8e+138)))
		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, -1.15e+72], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.35e-47], And[N[Not[LessEqual[re, 1.6e+92]], $MachinePrecision], LessEqual[re, 3.8e+138]]], 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 < -1.15e72

   1. Initial program 30.0%

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

   2. Taylor expanded in re around -inf 91.0%

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

      1. *-commutative 91.0%

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

   4. Simplified 91.0%

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

   if -1.15e72 < re < 1.3499999999999999e-47 or 1.60000000000000013e92 < re < 3.80000000000000012e138

   1. Initial program 55.4%

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

   2. Taylor expanded in re around 0 77.9%

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

   if 1.3499999999999999e-47 < re < 1.60000000000000013e92 or 3.80000000000000012e138 < re

   1. Initial program 11.8%

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

   2. Taylor expanded in re around inf 50.3%

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

      1. unpow2 50.3%

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

   4. Simplified 50.3%

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

   5. Taylor expanded in im around 0 81.2%

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

      1. *-commutative 81.2%

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

      2. rem-exp-log 75.8%

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

      3. unpow1/2 75.8%

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

      4. exp-to-pow 75.0%

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

      5. *-commutative 75.0%

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

      6. log-pow 75.0%

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

      7. exp-to-pow 75.0%

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

      8. log-rec 75.0%

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

      9. distribute-lft-neg-out 75.0%

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

      10. exp-neg 75.0%

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

      11. log-rec 75.0%

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

      12. exp-to-pow 75.0%

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

      13. unpow1/2 75.0%

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

      14. exp-sum 74.6%

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

      15. sub-neg 74.6%

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

      16. log-div 75.1%

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

      17. rem-exp-log 81.2%

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

   7. Simplified 81.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-47} \lor \neg \left(re \leq 1.6 \cdot 10^{+92}\right) \land re \leq 3.8 \cdot 10^{+138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -7.2 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+138}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- im re))))))
   (if (<= re -7.2e+68)
     (* 0.5 (sqrt (* 2.0 (* re -2.0))))
     (if (<= re 3.2e-47)
       t_0
       (if (<= re 2.4e+97)
         (* 0.5 (/ im (sqrt re)))
         (if (<= re 3.8e+138) t_0 (* 0.5 (* im (sqrt (/ 1.0 re))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -7.2e+68) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.2e-47) {
		tmp = t_0;
	} else if (re <= 2.4e+97) {
		tmp = 0.5 * (im / sqrt(re));
	} else if (re <= 3.8e+138) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im * sqrt((1.0 / 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((2.0d0 * (im - re)))
    if (re <= (-7.2d+68)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 3.2d-47) then
        tmp = t_0
    else if (re <= 2.4d+97) then
        tmp = 0.5d0 * (im / sqrt(re))
    else if (re <= 3.8d+138) then
        tmp = t_0
    else
        tmp = 0.5d0 * (im * sqrt((1.0d0 / 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 tmp;
	if (re <= -7.2e+68) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.2e-47) {
		tmp = t_0;
	} else if (re <= 2.4e+97) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else if (re <= 3.8e+138) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (im - re)))
	tmp = 0
	if re <= -7.2e+68:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 3.2e-47:
		tmp = t_0
	elif re <= 2.4e+97:
		tmp = 0.5 * (im / math.sqrt(re))
	elif re <= 3.8e+138:
		tmp = t_0
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	tmp = 0.0
	if (re <= -7.2e+68)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 3.2e-47)
		tmp = t_0;
	elseif (re <= 2.4e+97)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	elseif (re <= 3.8e+138)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (im - re)));
	tmp = 0.0;
	if (re <= -7.2e+68)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 3.2e-47)
		tmp = t_0;
	elseif (re <= 2.4e+97)
		tmp = 0.5 * (im / sqrt(re));
	elseif (re <= 3.8e+138)
		tmp = t_0;
	else
		tmp = 0.5 * (im * sqrt((1.0 / 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]}, If[LessEqual[re, -7.2e+68], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.2e-47], t$95$0, If[LessEqual[re, 2.4e+97], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.8e+138], t$95$0, N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.1999999999999998e68

   1. Initial program 30.0%

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

   2. Taylor expanded in re around -inf 91.0%

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

      1. *-commutative 91.0%

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

   4. Simplified 91.0%

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

   if -7.1999999999999998e68 < re < 3.1999999999999999e-47 or 2.4e97 < re < 3.80000000000000012e138

   1. Initial program 55.4%

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

   2. Taylor expanded in re around 0 77.9%

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

   if 3.1999999999999999e-47 < re < 2.4e97

   1. Initial program 21.9%

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

   2. Taylor expanded in re around inf 43.7%

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

      1. unpow2 43.7%

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

   4. Simplified 43.7%

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

   5. Taylor expanded in im around 0 71.2%

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

      1. *-commutative 71.2%

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

      2. rem-exp-log 65.3%

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

      3. unpow1/2 65.3%

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

      4. exp-to-pow 65.3%

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

      5. *-commutative 65.3%

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

      6. log-pow 65.3%

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

      7. exp-to-pow 65.3%

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

      8. log-rec 65.3%

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

      9. distribute-lft-neg-out 65.3%

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

      10. exp-neg 65.3%

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

      11. log-rec 65.3%

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

      12. exp-to-pow 65.3%

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

      13. unpow1/2 65.3%

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

      14. exp-sum 65.1%

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

      15. sub-neg 65.1%

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

      16. log-div 65.7%

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

      17. rem-exp-log 71.3%

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

   7. Simplified 71.3%

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

   if 3.80000000000000012e138 < re

   1. Initial program 2.5%

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

      1. *-commutative 2.5%

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

      2. hypot-udef 31.6%

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

      3. *-commutative 31.6%

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

      4. add-cbrt-cube 24.2%

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

      5. add-sqr-sqrt 24.3%

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

      6. pow1 24.3%

         \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      7. pow1/2 24.3%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}}} \]

      8. pow-prod-up 24.3%

         \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\left(1 + 0.5\right)}}} \]

      9. metadata-eval 24.3%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\color{blue}{1.5}}} \]

   3. Applied egg-rr 24.3%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}}} \]

   4. Taylor expanded in im around 0 90.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{-47}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]

Alternative 4: 52.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e-310)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.5 * sqrt((2.0 * (re * -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 <= (-5d-310)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-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 <= -5e-310) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-310:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -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 <= -5e-310)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-310], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 53.4%

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

   2. Taylor expanded in re around -inf 52.8%

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

      1. *-commutative 52.8%

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

   4. Simplified 52.8%

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

   if -4.999999999999985e-310 < re

   1. Initial program 26.8%

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

   2. Taylor expanded in re around inf 32.1%

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

      1. unpow2 32.1%

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

   4. Simplified 32.1%

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

   5. Taylor expanded in im around 0 55.4%

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

      1. *-commutative 55.4%

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

      2. rem-exp-log 51.6%

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

      3. unpow1/2 51.6%

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

      4. exp-to-pow 51.2%

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

      5. *-commutative 51.2%

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

      6. log-pow 51.2%

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

      7. exp-to-pow 51.2%

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

      8. log-rec 51.2%

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

      9. distribute-lft-neg-out 51.2%

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

      10. exp-neg 51.2%

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

      11. log-rec 51.2%

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

      12. exp-to-pow 51.2%

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

      13. unpow1/2 51.2%

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

      14. exp-sum 51.0%

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

      15. sub-neg 51.0%

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

      16. log-div 51.3%

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

      17. rem-exp-log 55.4%

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

   7. Simplified 55.4%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 26.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

2. Taylor expanded in re around inf 18.5%

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

   1. unpow2 18.5%

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

4. Simplified 18.5%

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

5. Taylor expanded in im around 0 30.9%

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

   1. *-commutative 30.9%

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

   2. rem-exp-log 28.8%

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

   3. unpow1/2 28.8%

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

   4. exp-to-pow 28.6%

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

   5. *-commutative 28.6%

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

   6. log-pow 28.6%

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

   7. exp-to-pow 28.6%

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

   8. log-rec 28.6%

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

   9. distribute-lft-neg-out 28.6%

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

   10. exp-neg 28.6%

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

   11. log-rec 28.6%

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

   12. exp-to-pow 28.6%

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

   13. unpow1/2 28.6%

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

   14. exp-sum 28.5%

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

   15. sub-neg 28.5%

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

   16. log-div 28.7%

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

   17. rem-exp-log 31.0%

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

7. Simplified 31.0%

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

8. Final simplification 31.0%

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

Reproduce

herbie shell --seed 2023274 
(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)))))