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

Percentage Accurate: 41.3% → 90.0%

Time: 18.6s

Alternatives: 7

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 90.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.2%

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

   3. Taylor expanded in re around inf 29.9%

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

      1. pow1/2 29.9%

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

      2. div-inv 30.0%

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

      3. unpow-prod-down 32.8%

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

      4. pow1/2 32.8%

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

      5. inv-pow 32.8%

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

      6. pow-pow 32.8%

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

      7. metadata-eval 32.8%

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

   5. Applied egg-rr 32.8%

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

      1. *-commutative 32.8%

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

      2. unpow2 32.8%

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

      3. rem-sqrt-square 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in re around 0 97.2%

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

      1. rem-square-sqrt 96.6%

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

      2. fabs-sqr 96.6%

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

      3. rem-square-sqrt 97.2%

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

      4. unpow1/2 97.2%

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

      5. rem-exp-log 91.2%

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

      6. exp-neg 91.2%

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

      7. exp-prod 91.2%

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

      8. distribute-lft-neg-out 91.2%

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

      9. distribute-rgt-neg-in 91.2%

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

      10. metadata-eval 91.2%

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

      11. exp-to-pow 97.2%

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

   10. Simplified 97.2%

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

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

   1. Initial program 48.5%

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

      1. sub-neg 48.5%

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

      2. sqr-neg 48.5%

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

      3. sub-neg 48.5%

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

      4. sqr-neg 48.5%

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

      5. hypot-define 90.6%

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

   3. Simplified 90.6%

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

4. Final simplification 91.5%

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

Alternative 2: 74.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ t_1 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_2 := 0.5 \cdot \sqrt{im \cdot 2}\\ t_3 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -2.1 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -6.5:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq -4.5 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-93}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq -5.5 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -1.28 \cdot 10^{-135}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq -1.32 \cdot 10^{-136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2 + re \cdot \left(\frac{re}{im} - 2\right)}\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+58}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+125}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ im (sqrt re))))
        (t_1 (* 0.5 (sqrt (* re -4.0))))
        (t_2 (* 0.5 (sqrt (* im 2.0))))
        (t_3 (* 0.5 (sqrt (* 2.0 (- im re))))))
   (if (<= re -2.1e+77)
     t_1
     (if (<= re -6.5)
       t_3
       (if (<= re -4.5e-61)
         t_1
         (if (<= re -4.2e-93)
           t_2
           (if (<= re -5.5e-99)
             t_1
             (if (<= re -1.28e-135)
               t_3
               (if (<= re -1.32e-136)
                 t_1
                 (if (<= re 9.5e-54)
                   (* 0.5 (sqrt (+ (* im 2.0) (* re (- (/ re im) 2.0)))))
                   (if (<= re 1.6e+25)
                     t_0
                     (if (<= re 1.95e+58)
                       t_2
                       (if (<= re 5.5e+72)
                         t_0
                         (if (<= re 2.3e+125)
                           t_2
                           (* 0.5 (* im (pow re -0.5)))))))))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im / sqrt(re));
	double t_1 = 0.5 * sqrt((re * -4.0));
	double t_2 = 0.5 * sqrt((im * 2.0));
	double t_3 = 0.5 * sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -2.1e+77) {
		tmp = t_1;
	} else if (re <= -6.5) {
		tmp = t_3;
	} else if (re <= -4.5e-61) {
		tmp = t_1;
	} else if (re <= -4.2e-93) {
		tmp = t_2;
	} else if (re <= -5.5e-99) {
		tmp = t_1;
	} else if (re <= -1.28e-135) {
		tmp = t_3;
	} else if (re <= -1.32e-136) {
		tmp = t_1;
	} else if (re <= 9.5e-54) {
		tmp = 0.5 * sqrt(((im * 2.0) + (re * ((re / im) - 2.0))));
	} else if (re <= 1.6e+25) {
		tmp = t_0;
	} else if (re <= 1.95e+58) {
		tmp = t_2;
	} else if (re <= 5.5e+72) {
		tmp = t_0;
	} else if (re <= 2.3e+125) {
		tmp = t_2;
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.5d0 * (im / sqrt(re))
    t_1 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_2 = 0.5d0 * sqrt((im * 2.0d0))
    t_3 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    if (re <= (-2.1d+77)) then
        tmp = t_1
    else if (re <= (-6.5d0)) then
        tmp = t_3
    else if (re <= (-4.5d-61)) then
        tmp = t_1
    else if (re <= (-4.2d-93)) then
        tmp = t_2
    else if (re <= (-5.5d-99)) then
        tmp = t_1
    else if (re <= (-1.28d-135)) then
        tmp = t_3
    else if (re <= (-1.32d-136)) then
        tmp = t_1
    else if (re <= 9.5d-54) then
        tmp = 0.5d0 * sqrt(((im * 2.0d0) + (re * ((re / im) - 2.0d0))))
    else if (re <= 1.6d+25) then
        tmp = t_0
    else if (re <= 1.95d+58) then
        tmp = t_2
    else if (re <= 5.5d+72) then
        tmp = t_0
    else if (re <= 2.3d+125) then
        tmp = t_2
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im / Math.sqrt(re));
	double t_1 = 0.5 * Math.sqrt((re * -4.0));
	double t_2 = 0.5 * Math.sqrt((im * 2.0));
	double t_3 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -2.1e+77) {
		tmp = t_1;
	} else if (re <= -6.5) {
		tmp = t_3;
	} else if (re <= -4.5e-61) {
		tmp = t_1;
	} else if (re <= -4.2e-93) {
		tmp = t_2;
	} else if (re <= -5.5e-99) {
		tmp = t_1;
	} else if (re <= -1.28e-135) {
		tmp = t_3;
	} else if (re <= -1.32e-136) {
		tmp = t_1;
	} else if (re <= 9.5e-54) {
		tmp = 0.5 * Math.sqrt(((im * 2.0) + (re * ((re / im) - 2.0))));
	} else if (re <= 1.6e+25) {
		tmp = t_0;
	} else if (re <= 1.95e+58) {
		tmp = t_2;
	} else if (re <= 5.5e+72) {
		tmp = t_0;
	} else if (re <= 2.3e+125) {
		tmp = t_2;
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im / math.sqrt(re))
	t_1 = 0.5 * math.sqrt((re * -4.0))
	t_2 = 0.5 * math.sqrt((im * 2.0))
	t_3 = 0.5 * math.sqrt((2.0 * (im - re)))
	tmp = 0
	if re <= -2.1e+77:
		tmp = t_1
	elif re <= -6.5:
		tmp = t_3
	elif re <= -4.5e-61:
		tmp = t_1
	elif re <= -4.2e-93:
		tmp = t_2
	elif re <= -5.5e-99:
		tmp = t_1
	elif re <= -1.28e-135:
		tmp = t_3
	elif re <= -1.32e-136:
		tmp = t_1
	elif re <= 9.5e-54:
		tmp = 0.5 * math.sqrt(((im * 2.0) + (re * ((re / im) - 2.0))))
	elif re <= 1.6e+25:
		tmp = t_0
	elif re <= 1.95e+58:
		tmp = t_2
	elif re <= 5.5e+72:
		tmp = t_0
	elif re <= 2.3e+125:
		tmp = t_2
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im / sqrt(re)))
	t_1 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_2 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	t_3 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	tmp = 0.0
	if (re <= -2.1e+77)
		tmp = t_1;
	elseif (re <= -6.5)
		tmp = t_3;
	elseif (re <= -4.5e-61)
		tmp = t_1;
	elseif (re <= -4.2e-93)
		tmp = t_2;
	elseif (re <= -5.5e-99)
		tmp = t_1;
	elseif (re <= -1.28e-135)
		tmp = t_3;
	elseif (re <= -1.32e-136)
		tmp = t_1;
	elseif (re <= 9.5e-54)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * 2.0) + Float64(re * Float64(Float64(re / im) - 2.0)))));
	elseif (re <= 1.6e+25)
		tmp = t_0;
	elseif (re <= 1.95e+58)
		tmp = t_2;
	elseif (re <= 5.5e+72)
		tmp = t_0;
	elseif (re <= 2.3e+125)
		tmp = t_2;
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im / sqrt(re));
	t_1 = 0.5 * sqrt((re * -4.0));
	t_2 = 0.5 * sqrt((im * 2.0));
	t_3 = 0.5 * sqrt((2.0 * (im - re)));
	tmp = 0.0;
	if (re <= -2.1e+77)
		tmp = t_1;
	elseif (re <= -6.5)
		tmp = t_3;
	elseif (re <= -4.5e-61)
		tmp = t_1;
	elseif (re <= -4.2e-93)
		tmp = t_2;
	elseif (re <= -5.5e-99)
		tmp = t_1;
	elseif (re <= -1.28e-135)
		tmp = t_3;
	elseif (re <= -1.32e-136)
		tmp = t_1;
	elseif (re <= 9.5e-54)
		tmp = 0.5 * sqrt(((im * 2.0) + (re * ((re / im) - 2.0))));
	elseif (re <= 1.6e+25)
		tmp = t_0;
	elseif (re <= 1.95e+58)
		tmp = t_2;
	elseif (re <= 5.5e+72)
		tmp = t_0;
	elseif (re <= 2.3e+125)
		tmp = t_2;
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.1e+77], t$95$1, If[LessEqual[re, -6.5], t$95$3, If[LessEqual[re, -4.5e-61], t$95$1, If[LessEqual[re, -4.2e-93], t$95$2, If[LessEqual[re, -5.5e-99], t$95$1, If[LessEqual[re, -1.28e-135], t$95$3, If[LessEqual[re, -1.32e-136], t$95$1, If[LessEqual[re, 9.5e-54], N[(0.5 * N[Sqrt[N[(N[(im * 2.0), $MachinePrecision] + N[(re * N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.6e+25], t$95$0, If[LessEqual[re, 1.95e+58], t$95$2, If[LessEqual[re, 5.5e+72], t$95$0, If[LessEqual[re, 2.3e+125], t$95$2, N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if re < -2.0999999999999999e77 or -6.5 < re < -4.5e-61 or -4.2000000000000002e-93 < re < -5.49999999999999991e-99 or -1.27999999999999997e-135 < re < -1.3200000000000001e-136

   1. Initial program 42.5%

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

   3. Taylor expanded in re around -inf 83.1%

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

      1. *-commutative 83.1%

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

   5. Simplified 83.1%

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

   if -2.0999999999999999e77 < re < -6.5 or -5.49999999999999991e-99 < re < -1.27999999999999997e-135

   1. Initial program 61.7%

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

   3. Taylor expanded in re around 0 74.5%

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

   if -4.5e-61 < re < -4.2000000000000002e-93 or 1.6e25 < re < 1.95000000000000005e58 or 5.5e72 < re < 2.30000000000000013e125

   1. Initial program 53.4%

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

   3. Taylor expanded in re around 0 86.3%

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

      1. *-commutative 86.3%

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

   5. Simplified 86.3%

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

   if -1.3200000000000001e-136 < re < 9.4999999999999994e-54

   1. Initial program 55.0%

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

   3. Taylor expanded in re around 0 82.4%

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

   if 9.4999999999999994e-54 < re < 1.6e25 or 1.95000000000000005e58 < re < 5.5e72

   1. Initial program 31.1%

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

   3. Taylor expanded in re around inf 27.4%

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

      1. pow1/2 27.4%

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

      2. div-inv 27.4%

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

      3. unpow-prod-down 31.9%

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

      4. pow1/2 31.9%

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

      5. inv-pow 31.9%

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

      6. pow-pow 31.9%

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

      7. metadata-eval 31.9%

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

   5. Applied egg-rr 31.9%

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

      1. *-commutative 31.9%

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

      2. unpow2 31.9%

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

      3. rem-sqrt-square 73.3%

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

   7. Simplified 73.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{-0.5} \cdot \left|im\right|\right)} \]
   8. Step-by-step derivation

      1. *-commutative 73.3%

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

      2. metadata-eval 73.3%

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

      3. pow-flip 73.3%

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

      4. pow1/2 73.3%

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

      5. div-inv 73.4%

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

      6. add-sqr-sqrt 72.9%

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

      7. fabs-sqr 72.9%

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

      8. add-sqr-sqrt 73.4%

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

   9. Applied egg-rr 73.4%

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

   if 2.30000000000000013e125 < re

   1. Initial program 3.0%

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

   3. Taylor expanded in re around inf 54.9%

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

      1. pow1/2 54.9%

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

      2. div-inv 54.9%

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

      3. unpow-prod-down 68.8%

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

      4. pow1/2 68.8%

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

      5. inv-pow 68.8%

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

      6. pow-pow 68.8%

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

      7. metadata-eval 68.8%

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

   5. Applied egg-rr 68.8%

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

      1. *-commutative 68.8%

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

      2. unpow2 68.8%

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

      3. rem-sqrt-square 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in re around 0 87.4%

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

      1. rem-square-sqrt 87.0%

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

      2. fabs-sqr 87.0%

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

      3. rem-square-sqrt 87.4%

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

      4. unpow1/2 87.4%

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

      5. rem-exp-log 82.0%

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

      6. exp-neg 82.0%

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

      7. exp-prod 82.0%

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

      8. distribute-lft-neg-out 82.0%

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

      9. distribute-rgt-neg-in 82.0%

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

      10. metadata-eval 82.0%

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

      11. exp-to-pow 87.4%

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

   10. Simplified 87.4%

       \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{-0.5} \cdot im\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -6.5:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -4.5 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq -5.5 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.28 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -1.32 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2 + re \cdot \left(\frac{re}{im} - 2\right)}\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+125}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{im \cdot 2}\\ t_2 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ t_3 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -150000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq -5.8 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -1.35 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -5.5 \cdot 10^{-99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -8.5 \cdot 10^{-133}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq -1.32 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-52}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;re \leq 4 \cdot 10^{+24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{+72}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0))))
        (t_1 (* 0.5 (sqrt (* im 2.0))))
        (t_2 (* 0.5 (/ im (sqrt re))))
        (t_3 (* 0.5 (sqrt (* 2.0 (- im re))))))
   (if (<= re -1.15e+77)
     t_0
     (if (<= re -150000.0)
       t_3
       (if (<= re -5.8e-61)
         t_0
         (if (<= re -1.35e-92)
           t_1
           (if (<= re -5.5e-99)
             t_0
             (if (<= re -8.5e-133)
               t_3
               (if (<= re -1.32e-136)
                 t_0
                 (if (<= re 1.2e-52)
                   t_3
                   (if (<= re 4e+24)
                     t_2
                     (if (<= re 1.02e+58)
                       t_1
                       (if (<= re 2.9e+72)
                         t_2
                         (if (<= re 2.3e+125)
                           t_1
                           (* 0.5 (* im (pow re -0.5)))))))))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((im * 2.0));
	double t_2 = 0.5 * (im / sqrt(re));
	double t_3 = 0.5 * sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -1.15e+77) {
		tmp = t_0;
	} else if (re <= -150000.0) {
		tmp = t_3;
	} else if (re <= -5.8e-61) {
		tmp = t_0;
	} else if (re <= -1.35e-92) {
		tmp = t_1;
	} else if (re <= -5.5e-99) {
		tmp = t_0;
	} else if (re <= -8.5e-133) {
		tmp = t_3;
	} else if (re <= -1.32e-136) {
		tmp = t_0;
	} else if (re <= 1.2e-52) {
		tmp = t_3;
	} else if (re <= 4e+24) {
		tmp = t_2;
	} else if (re <= 1.02e+58) {
		tmp = t_1;
	} else if (re <= 2.9e+72) {
		tmp = t_2;
	} else if (re <= 2.3e+125) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((im * 2.0d0))
    t_2 = 0.5d0 * (im / sqrt(re))
    t_3 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    if (re <= (-1.15d+77)) then
        tmp = t_0
    else if (re <= (-150000.0d0)) then
        tmp = t_3
    else if (re <= (-5.8d-61)) then
        tmp = t_0
    else if (re <= (-1.35d-92)) then
        tmp = t_1
    else if (re <= (-5.5d-99)) then
        tmp = t_0
    else if (re <= (-8.5d-133)) then
        tmp = t_3
    else if (re <= (-1.32d-136)) then
        tmp = t_0
    else if (re <= 1.2d-52) then
        tmp = t_3
    else if (re <= 4d+24) then
        tmp = t_2
    else if (re <= 1.02d+58) then
        tmp = t_1
    else if (re <= 2.9d+72) then
        tmp = t_2
    else if (re <= 2.3d+125) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((im * 2.0));
	double t_2 = 0.5 * (im / Math.sqrt(re));
	double t_3 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -1.15e+77) {
		tmp = t_0;
	} else if (re <= -150000.0) {
		tmp = t_3;
	} else if (re <= -5.8e-61) {
		tmp = t_0;
	} else if (re <= -1.35e-92) {
		tmp = t_1;
	} else if (re <= -5.5e-99) {
		tmp = t_0;
	} else if (re <= -8.5e-133) {
		tmp = t_3;
	} else if (re <= -1.32e-136) {
		tmp = t_0;
	} else if (re <= 1.2e-52) {
		tmp = t_3;
	} else if (re <= 4e+24) {
		tmp = t_2;
	} else if (re <= 1.02e+58) {
		tmp = t_1;
	} else if (re <= 2.9e+72) {
		tmp = t_2;
	} else if (re <= 2.3e+125) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((im * 2.0))
	t_2 = 0.5 * (im / math.sqrt(re))
	t_3 = 0.5 * math.sqrt((2.0 * (im - re)))
	tmp = 0
	if re <= -1.15e+77:
		tmp = t_0
	elif re <= -150000.0:
		tmp = t_3
	elif re <= -5.8e-61:
		tmp = t_0
	elif re <= -1.35e-92:
		tmp = t_1
	elif re <= -5.5e-99:
		tmp = t_0
	elif re <= -8.5e-133:
		tmp = t_3
	elif re <= -1.32e-136:
		tmp = t_0
	elif re <= 1.2e-52:
		tmp = t_3
	elif re <= 4e+24:
		tmp = t_2
	elif re <= 1.02e+58:
		tmp = t_1
	elif re <= 2.9e+72:
		tmp = t_2
	elif re <= 2.3e+125:
		tmp = t_1
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	t_2 = Float64(0.5 * Float64(im / sqrt(re)))
	t_3 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	tmp = 0.0
	if (re <= -1.15e+77)
		tmp = t_0;
	elseif (re <= -150000.0)
		tmp = t_3;
	elseif (re <= -5.8e-61)
		tmp = t_0;
	elseif (re <= -1.35e-92)
		tmp = t_1;
	elseif (re <= -5.5e-99)
		tmp = t_0;
	elseif (re <= -8.5e-133)
		tmp = t_3;
	elseif (re <= -1.32e-136)
		tmp = t_0;
	elseif (re <= 1.2e-52)
		tmp = t_3;
	elseif (re <= 4e+24)
		tmp = t_2;
	elseif (re <= 1.02e+58)
		tmp = t_1;
	elseif (re <= 2.9e+72)
		tmp = t_2;
	elseif (re <= 2.3e+125)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * sqrt((im * 2.0));
	t_2 = 0.5 * (im / sqrt(re));
	t_3 = 0.5 * sqrt((2.0 * (im - re)));
	tmp = 0.0;
	if (re <= -1.15e+77)
		tmp = t_0;
	elseif (re <= -150000.0)
		tmp = t_3;
	elseif (re <= -5.8e-61)
		tmp = t_0;
	elseif (re <= -1.35e-92)
		tmp = t_1;
	elseif (re <= -5.5e-99)
		tmp = t_0;
	elseif (re <= -8.5e-133)
		tmp = t_3;
	elseif (re <= -1.32e-136)
		tmp = t_0;
	elseif (re <= 1.2e-52)
		tmp = t_3;
	elseif (re <= 4e+24)
		tmp = t_2;
	elseif (re <= 1.02e+58)
		tmp = t_1;
	elseif (re <= 2.9e+72)
		tmp = t_2;
	elseif (re <= 2.3e+125)
		tmp = t_1;
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.15e+77], t$95$0, If[LessEqual[re, -150000.0], t$95$3, If[LessEqual[re, -5.8e-61], t$95$0, If[LessEqual[re, -1.35e-92], t$95$1, If[LessEqual[re, -5.5e-99], t$95$0, If[LessEqual[re, -8.5e-133], t$95$3, If[LessEqual[re, -1.32e-136], t$95$0, If[LessEqual[re, 1.2e-52], t$95$3, If[LessEqual[re, 4e+24], t$95$2, If[LessEqual[re, 1.02e+58], t$95$1, If[LessEqual[re, 2.9e+72], t$95$2, If[LessEqual[re, 2.3e+125], t$95$1, N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -1.14999999999999997e77 or -1.5e5 < re < -5.7999999999999999e-61 or -1.34999999999999998e-92 < re < -5.49999999999999991e-99 or -8.49999999999999957e-133 < re < -1.3200000000000001e-136

   1. Initial program 42.5%

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

   3. Taylor expanded in re around -inf 83.1%

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

      1. *-commutative 83.1%

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

   5. Simplified 83.1%

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

   if -1.14999999999999997e77 < re < -1.5e5 or -5.49999999999999991e-99 < re < -8.49999999999999957e-133 or -1.3200000000000001e-136 < re < 1.2000000000000001e-52

   1. Initial program 57.0%

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

   3. Taylor expanded in re around 0 79.8%

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

   if -5.7999999999999999e-61 < re < -1.34999999999999998e-92 or 3.9999999999999999e24 < re < 1.02000000000000005e58 or 2.90000000000000017e72 < re < 2.30000000000000013e125

   1. Initial program 53.4%

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

   3. Taylor expanded in re around 0 86.3%

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

      1. *-commutative 86.3%

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

   5. Simplified 86.3%

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

   if 1.2000000000000001e-52 < re < 3.9999999999999999e24 or 1.02000000000000005e58 < re < 2.90000000000000017e72

   1. Initial program 27.7%

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

   3. Taylor expanded in re around inf 28.4%

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

      1. pow1/2 28.4%

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

      2. div-inv 28.4%

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

      3. unpow-prod-down 33.1%

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

      4. pow1/2 33.1%

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

      5. inv-pow 33.1%

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

      6. pow-pow 33.1%

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

      7. metadata-eval 33.1%

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

   5. Applied egg-rr 33.1%

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

      1. *-commutative 33.1%

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

      2. unpow2 33.1%

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

      3. rem-sqrt-square 76.5%

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

   7. Simplified 76.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{-0.5} \cdot \left|im\right|\right)} \]
   8. Step-by-step derivation

      1. *-commutative 76.5%

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

      2. metadata-eval 76.5%

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

      3. pow-flip 76.5%

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

      4. pow1/2 76.5%

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

      5. div-inv 76.6%

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

      6. add-sqr-sqrt 76.1%

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

      7. fabs-sqr 76.1%

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

      8. add-sqr-sqrt 76.6%

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

   9. Applied egg-rr 76.6%

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

   if 2.30000000000000013e125 < re

   1. Initial program 3.0%

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

   3. Taylor expanded in re around inf 54.9%

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

      1. pow1/2 54.9%

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

      2. div-inv 54.9%

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

      3. unpow-prod-down 68.8%

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

      4. pow1/2 68.8%

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

      5. inv-pow 68.8%

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

      6. pow-pow 68.8%

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

      7. metadata-eval 68.8%

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

   5. Applied egg-rr 68.8%

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

      1. *-commutative 68.8%

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

      2. unpow2 68.8%

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

      3. rem-sqrt-square 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in re around 0 87.4%

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

      1. rem-square-sqrt 87.0%

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

      2. fabs-sqr 87.0%

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

      3. rem-square-sqrt 87.4%

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

      4. unpow1/2 87.4%

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

      5. rem-exp-log 82.0%

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

      6. exp-neg 82.0%

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

      7. exp-prod 82.0%

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

      8. distribute-lft-neg-out 82.0%

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

      9. distribute-rgt-neg-in 82.0%

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

      10. metadata-eval 82.0%

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

      11. exp-to-pow 87.4%

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

   10. Simplified 87.4%

       \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{-0.5} \cdot im\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -150000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -5.8 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.35 \cdot 10^{-92}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq -5.5 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -8.5 \cdot 10^{-133}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -1.32 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+125}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ t_2 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq -2.4 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -0.00022:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq -1.75 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-52}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+58}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{+99}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0))))
        (t_1 (* 0.5 (/ im (sqrt re))))
        (t_2 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re -2.4e+77)
     t_0
     (if (<= re -0.00022)
       t_2
       (if (<= re -1.75e-43)
         t_0
         (if (<= re 2.9e-52)
           t_2
           (if (<= re 1.45e+21)
             t_1
             (if (<= re 1.95e+58)
               t_2
               (if (<= re 5.6e+72)
                 t_1
                 (if (<= re 2.15e+99)
                   t_2
                   (* 0.5 (* im (pow re -0.5)))))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * (im / sqrt(re));
	double t_2 = 0.5 * sqrt((im * 2.0));
	double tmp;
	if (re <= -2.4e+77) {
		tmp = t_0;
	} else if (re <= -0.00022) {
		tmp = t_2;
	} else if (re <= -1.75e-43) {
		tmp = t_0;
	} else if (re <= 2.9e-52) {
		tmp = t_2;
	} else if (re <= 1.45e+21) {
		tmp = t_1;
	} else if (re <= 1.95e+58) {
		tmp = t_2;
	} else if (re <= 5.6e+72) {
		tmp = t_1;
	} else if (re <= 2.15e+99) {
		tmp = t_2;
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_1 = 0.5d0 * (im / sqrt(re))
    t_2 = 0.5d0 * sqrt((im * 2.0d0))
    if (re <= (-2.4d+77)) then
        tmp = t_0
    else if (re <= (-0.00022d0)) then
        tmp = t_2
    else if (re <= (-1.75d-43)) then
        tmp = t_0
    else if (re <= 2.9d-52) then
        tmp = t_2
    else if (re <= 1.45d+21) then
        tmp = t_1
    else if (re <= 1.95d+58) then
        tmp = t_2
    else if (re <= 5.6d+72) then
        tmp = t_1
    else if (re <= 2.15d+99) then
        tmp = t_2
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * (im / Math.sqrt(re));
	double t_2 = 0.5 * Math.sqrt((im * 2.0));
	double tmp;
	if (re <= -2.4e+77) {
		tmp = t_0;
	} else if (re <= -0.00022) {
		tmp = t_2;
	} else if (re <= -1.75e-43) {
		tmp = t_0;
	} else if (re <= 2.9e-52) {
		tmp = t_2;
	} else if (re <= 1.45e+21) {
		tmp = t_1;
	} else if (re <= 1.95e+58) {
		tmp = t_2;
	} else if (re <= 5.6e+72) {
		tmp = t_1;
	} else if (re <= 2.15e+99) {
		tmp = t_2;
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * (im / math.sqrt(re))
	t_2 = 0.5 * math.sqrt((im * 2.0))
	tmp = 0
	if re <= -2.4e+77:
		tmp = t_0
	elif re <= -0.00022:
		tmp = t_2
	elif re <= -1.75e-43:
		tmp = t_0
	elif re <= 2.9e-52:
		tmp = t_2
	elif re <= 1.45e+21:
		tmp = t_1
	elif re <= 1.95e+58:
		tmp = t_2
	elif re <= 5.6e+72:
		tmp = t_1
	elif re <= 2.15e+99:
		tmp = t_2
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * Float64(im / sqrt(re)))
	t_2 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	tmp = 0.0
	if (re <= -2.4e+77)
		tmp = t_0;
	elseif (re <= -0.00022)
		tmp = t_2;
	elseif (re <= -1.75e-43)
		tmp = t_0;
	elseif (re <= 2.9e-52)
		tmp = t_2;
	elseif (re <= 1.45e+21)
		tmp = t_1;
	elseif (re <= 1.95e+58)
		tmp = t_2;
	elseif (re <= 5.6e+72)
		tmp = t_1;
	elseif (re <= 2.15e+99)
		tmp = t_2;
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * (im / sqrt(re));
	t_2 = 0.5 * sqrt((im * 2.0));
	tmp = 0.0;
	if (re <= -2.4e+77)
		tmp = t_0;
	elseif (re <= -0.00022)
		tmp = t_2;
	elseif (re <= -1.75e-43)
		tmp = t_0;
	elseif (re <= 2.9e-52)
		tmp = t_2;
	elseif (re <= 1.45e+21)
		tmp = t_1;
	elseif (re <= 1.95e+58)
		tmp = t_2;
	elseif (re <= 5.6e+72)
		tmp = t_1;
	elseif (re <= 2.15e+99)
		tmp = t_2;
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.4e+77], t$95$0, If[LessEqual[re, -0.00022], t$95$2, If[LessEqual[re, -1.75e-43], t$95$0, If[LessEqual[re, 2.9e-52], t$95$2, If[LessEqual[re, 1.45e+21], t$95$1, If[LessEqual[re, 1.95e+58], t$95$2, If[LessEqual[re, 5.6e+72], t$95$1, If[LessEqual[re, 2.15e+99], t$95$2, N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.3999999999999999e77 or -2.20000000000000008e-4 < re < -1.74999999999999999e-43

   1. Initial program 32.6%

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

   3. Taylor expanded in re around -inf 83.1%

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

      1. *-commutative 83.1%

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

   5. Simplified 83.1%

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

   if -2.3999999999999999e77 < re < -2.20000000000000008e-4 or -1.74999999999999999e-43 < re < 2.9000000000000002e-52 or 1.45e21 < re < 1.95000000000000005e58 or 5.5999999999999998e72 < re < 2.1500000000000001e99

   1. Initial program 59.1%

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

   3. Taylor expanded in re around 0 75.9%

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

      1. *-commutative 75.9%

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

   5. Simplified 75.9%

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

   if 2.9000000000000002e-52 < re < 1.45e21 or 1.95000000000000005e58 < re < 5.5999999999999998e72

   1. Initial program 27.7%

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

   3. Taylor expanded in re around inf 28.4%

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

      1. pow1/2 28.4%

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

      2. div-inv 28.4%

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

      3. unpow-prod-down 33.1%

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

      4. pow1/2 33.1%

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

      5. inv-pow 33.1%

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

      6. pow-pow 33.1%

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

      7. metadata-eval 33.1%

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

   5. Applied egg-rr 33.1%

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

      1. *-commutative 33.1%

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

      2. unpow2 33.1%

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

      3. rem-sqrt-square 76.5%

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

   7. Simplified 76.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{-0.5} \cdot \left|im\right|\right)} \]
   8. Step-by-step derivation

      1. *-commutative 76.5%

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

      2. metadata-eval 76.5%

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

      3. pow-flip 76.5%

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

      4. pow1/2 76.5%

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

      5. div-inv 76.6%

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

      6. add-sqr-sqrt 76.1%

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

      7. fabs-sqr 76.1%

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

      8. add-sqr-sqrt 76.6%

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

   9. Applied egg-rr 76.6%

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

   if 2.1500000000000001e99 < re

   1. Initial program 7.7%

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

   3. Taylor expanded in re around inf 52.6%

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

      1. pow1/2 52.6%

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

      2. div-inv 52.7%

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

      3. unpow-prod-down 65.2%

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

      4. pow1/2 65.2%

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

      5. inv-pow 65.2%

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

      6. pow-pow 65.2%

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

      7. metadata-eval 65.2%

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

   5. Applied egg-rr 65.2%

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

      1. *-commutative 65.2%

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

      2. unpow2 65.2%

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

      3. rem-sqrt-square 82.0%

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

   7. Simplified 82.0%

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

   8. Taylor expanded in re around 0 82.0%

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

      1. rem-square-sqrt 81.6%

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

      2. fabs-sqr 81.6%

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

      3. rem-square-sqrt 82.0%

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

      4. unpow1/2 82.0%

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

      5. rem-exp-log 77.0%

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

      6. exp-neg 77.0%

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

      7. exp-prod 77.0%

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

      8. distribute-lft-neg-out 77.0%

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

      9. distribute-rgt-neg-in 77.0%

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

      10. metadata-eval 77.0%

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

      11. exp-to-pow 82.0%

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

   10. Simplified 82.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -0.00022:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq -1.75 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq -4.1 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -1.6 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 3.15 \cdot 10^{-52} \lor \neg \left(re \leq 1.75 \cdot 10^{+22}\right) \land \left(re \leq 1.3 \cdot 10^{+58} \lor \neg \left(re \leq 3.4 \cdot 10^{+70}\right) \land re \leq 1.46 \cdot 10^{+99}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re -4.1e+77)
     t_0
     (if (<= re -1.15e-5)
       t_1
       (if (<= re -1.6e-43)
         t_0
         (if (or (<= re 3.15e-52)
                 (and (not (<= re 1.75e+22))
                      (or (<= re 1.3e+58)
                          (and (not (<= re 3.4e+70)) (<= re 1.46e+99)))))
           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((im * 2.0));
	double tmp;
	if (re <= -4.1e+77) {
		tmp = t_0;
	} else if (re <= -1.15e-5) {
		tmp = t_1;
	} else if (re <= -1.6e-43) {
		tmp = t_0;
	} else if ((re <= 3.15e-52) || (!(re <= 1.75e+22) && ((re <= 1.3e+58) || (!(re <= 3.4e+70) && (re <= 1.46e+99))))) {
		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((im * 2.0d0))
    if (re <= (-4.1d+77)) then
        tmp = t_0
    else if (re <= (-1.15d-5)) then
        tmp = t_1
    else if (re <= (-1.6d-43)) then
        tmp = t_0
    else if ((re <= 3.15d-52) .or. (.not. (re <= 1.75d+22)) .and. (re <= 1.3d+58) .or. (.not. (re <= 3.4d+70)) .and. (re <= 1.46d+99)) 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((im * 2.0));
	double tmp;
	if (re <= -4.1e+77) {
		tmp = t_0;
	} else if (re <= -1.15e-5) {
		tmp = t_1;
	} else if (re <= -1.6e-43) {
		tmp = t_0;
	} else if ((re <= 3.15e-52) || (!(re <= 1.75e+22) && ((re <= 1.3e+58) || (!(re <= 3.4e+70) && (re <= 1.46e+99))))) {
		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((im * 2.0))
	tmp = 0
	if re <= -4.1e+77:
		tmp = t_0
	elif re <= -1.15e-5:
		tmp = t_1
	elif re <= -1.6e-43:
		tmp = t_0
	elif (re <= 3.15e-52) or (not (re <= 1.75e+22) and ((re <= 1.3e+58) or (not (re <= 3.4e+70) and (re <= 1.46e+99)))):
		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(im * 2.0)))
	tmp = 0.0
	if (re <= -4.1e+77)
		tmp = t_0;
	elseif (re <= -1.15e-5)
		tmp = t_1;
	elseif (re <= -1.6e-43)
		tmp = t_0;
	elseif ((re <= 3.15e-52) || (!(re <= 1.75e+22) && ((re <= 1.3e+58) || (!(re <= 3.4e+70) && (re <= 1.46e+99)))))
		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((im * 2.0));
	tmp = 0.0;
	if (re <= -4.1e+77)
		tmp = t_0;
	elseif (re <= -1.15e-5)
		tmp = t_1;
	elseif (re <= -1.6e-43)
		tmp = t_0;
	elseif ((re <= 3.15e-52) || (~((re <= 1.75e+22)) && ((re <= 1.3e+58) || (~((re <= 3.4e+70)) && (re <= 1.46e+99)))))
		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[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4.1e+77], t$95$0, If[LessEqual[re, -1.15e-5], t$95$1, If[LessEqual[re, -1.6e-43], t$95$0, If[Or[LessEqual[re, 3.15e-52], And[N[Not[LessEqual[re, 1.75e+22]], $MachinePrecision], Or[LessEqual[re, 1.3e+58], And[N[Not[LessEqual[re, 3.4e+70]], $MachinePrecision], LessEqual[re, 1.46e+99]]]]], t$95$1, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.1000000000000001e77 or -1.15e-5 < re < -1.59999999999999992e-43

   1. Initial program 32.6%

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

   3. Taylor expanded in re around -inf 83.1%

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

      1. *-commutative 83.1%

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

   5. Simplified 83.1%

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

   if -4.1000000000000001e77 < re < -1.15e-5 or -1.59999999999999992e-43 < re < 3.1500000000000002e-52 or 1.75e22 < re < 1.29999999999999994e58 or 3.4000000000000001e70 < re < 1.4600000000000001e99

   1. Initial program 59.1%

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

   3. Taylor expanded in re around 0 75.9%

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

      1. *-commutative 75.9%

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

   5. Simplified 75.9%

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

   if 3.1500000000000002e-52 < re < 1.75e22 or 1.29999999999999994e58 < re < 3.4000000000000001e70 or 1.4600000000000001e99 < re

   1. Initial program 14.3%

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

   3. Taylor expanded in re around inf 44.7%

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

      1. pow1/2 44.7%

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

      2. div-inv 44.7%

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

      3. unpow-prod-down 54.7%

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

      4. pow1/2 54.7%

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

      5. inv-pow 54.7%

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

      6. pow-pow 54.7%

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

      7. metadata-eval 54.7%

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

   5. Applied egg-rr 54.7%

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

      1. *-commutative 54.7%

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

      2. unpow2 54.7%

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

      3. rem-sqrt-square 80.2%

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

   7. Simplified 80.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{-0.5} \cdot \left|im\right|\right)} \]
   8. Step-by-step derivation

      1. *-commutative 80.2%

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

      2. metadata-eval 80.2%

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

      3. pow-flip 80.2%

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

      4. pow1/2 80.2%

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

      5. div-inv 80.2%

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

      6. add-sqr-sqrt 79.8%

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

      7. fabs-sqr 79.8%

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

      8. add-sqr-sqrt 80.2%

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

   9. Applied egg-rr 80.2%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.1 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq -1.6 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.15 \cdot 10^{-52} \lor \neg \left(re \leq 1.75 \cdot 10^{+22}\right) \land \left(re \leq 1.3 \cdot 10^{+58} \lor \neg \left(re \leq 3.4 \cdot 10^{+70}\right) \land re \leq 1.46 \cdot 10^{+99}\right):\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{+77} \lor \neg \left(re \leq -0.85\right) \land re \leq -1.7 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -5.8e+77) (and (not (<= re -0.85)) (<= re -1.7e-43)))
   (* 0.5 (sqrt (* re -4.0)))
   (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -5.8e+77) || (!(re <= -0.85) && (re <= -1.7e-43))) {
		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 <= (-5.8d+77)) .or. (.not. (re <= (-0.85d0))) .and. (re <= (-1.7d-43))) 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 <= -5.8e+77) || (!(re <= -0.85) && (re <= -1.7e-43))) {
		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 <= -5.8e+77) or (not (re <= -0.85) and (re <= -1.7e-43)):
		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 <= -5.8e+77) || (!(re <= -0.85) && (re <= -1.7e-43)))
		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 <= -5.8e+77) || (~((re <= -0.85)) && (re <= -1.7e-43)))
		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[Or[LessEqual[re, -5.8e+77], And[N[Not[LessEqual[re, -0.85]], $MachinePrecision], LessEqual[re, -1.7e-43]]], 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 < -5.8000000000000003e77 or -0.849999999999999978 < re < -1.7e-43

   1. Initial program 32.6%

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

   3. Taylor expanded in re around -inf 83.1%

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

      1. *-commutative 83.1%

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

   5. Simplified 83.1%

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

   if -5.8000000000000003e77 < re < -0.849999999999999978 or -1.7e-43 < re

   1. Initial program 45.6%

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

   3. Taylor expanded in re around 0 60.8%

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

      1. *-commutative 60.8%

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

   5. Simplified 60.8%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{+77} \lor \neg \left(re \leq -0.85\right) \land re \leq -1.7 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.8%

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

3. Taylor expanded in re around 0 52.7%

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

   1. *-commutative 52.7%

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

5. Simplified 52.7%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))))