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

Percentage Accurate: 40.5% → 89.7%

Time: 12.3s

Alternatives: 6

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.7% 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 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 7.6%

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

      1. sub-neg 7.6%

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

      2. flip-+ 7.6%

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

      3. pow2 7.6%

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

      4. hypot-define 7.6%

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

      5. hypot-define 7.6%

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

   4. Applied egg-rr 7.6%

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

   5. Taylor expanded in re around inf 41.7%

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

      1. associate-*r/ 41.7%

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

   7. Simplified 41.7%

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

      1. expm1-log1p-u 41.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \frac{0.5 \cdot {im}^{2}}{re}}\right)\right)} \]

      2. expm1-undefine 16.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \frac{0.5 \cdot {im}^{2}}{re}}\right)} - 1\right)} \]

      3. associate-*r/ 16.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{2 \cdot \left(0.5 \cdot {im}^{2}\right)}{re}}}\right)} - 1\right) \]

      4. sqrt-div 16.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{2 \cdot \left(0.5 \cdot {im}^{2}\right)}}{\sqrt{re}}}\right)} - 1\right) \]

      5. associate-*r* 16.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot {im}^{2}}}}{\sqrt{re}}\right)} - 1\right) \]

      6. metadata-eval 16.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{1} \cdot {im}^{2}}}{\sqrt{re}}\right)} - 1\right) \]

      7. *-un-lft-identity 16.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{{im}^{2}}}}{\sqrt{re}}\right)} - 1\right) \]

      8. sqrt-pow1 16.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{im}^{\left(\frac{2}{2}\right)}}}{\sqrt{re}}\right)} - 1\right) \]

      9. metadata-eval 16.7%

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

      10. pow1 16.7%

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

   9. Applied egg-rr 16.7%

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

       1. log1p-undefine 16.7%

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

       2. rem-exp-log 16.9%

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

       3. +-commutative 16.9%

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

       4. associate--l+ 99.6%

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

       5. metadata-eval 99.6%

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

       6. metadata-eval 99.6%

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

       7. div0 99.6%

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

       8. sub-neg 99.6%

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

       9. div-sub 99.6%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{im - 0}{\sqrt{re}}} \]

       10. --rgt-identity 99.6%

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

   11. Simplified 99.6%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 42.8%

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

      1. sub-neg 42.8%

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

      2. sqr-neg 42.8%

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

      3. sub-neg 42.8%

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

      4. sqr-neg 42.8%

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

      5. hypot-define 87.2%

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

   3. Simplified 87.2%

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

4. Final simplification 88.7%

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

Alternative 2: 75.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-65} \lor \neg \left(re \leq 850000\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(re + im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1e+36)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 2.1e-97)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (or (<= re 1.35e-65) (not (<= re 850000.0)))
       (* 0.5 (/ im (sqrt re)))
       (* 0.5 (sqrt (* 2.0 (- (+ re im) re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e+36) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.1e-97) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if ((re <= 1.35e-65) || !(re <= 850000.0)) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * ((re + im) - 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 <= (-1d+36)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 2.1d-97) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if ((re <= 1.35d-65) .or. (.not. (re <= 850000.0d0))) then
        tmp = 0.5d0 * (im / sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * ((re + im) - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1e+36) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.1e-97) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if ((re <= 1.35e-65) || !(re <= 850000.0)) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * ((re + im) - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1e+36:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 2.1e-97:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif (re <= 1.35e-65) or not (re <= 850000.0):
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * ((re + im) - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e+36)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 2.1e-97)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif ((re <= 1.35e-65) || !(re <= 850000.0))
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(re + im) - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1e+36)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 2.1e-97)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif ((re <= 1.35e-65) || ~((re <= 850000.0)))
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * ((re + im) - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e+36], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e-97], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.35e-65], N[Not[LessEqual[re, 850000.0]], $MachinePrecision]], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(re + im), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.00000000000000004e36

   1. Initial program 41.9%

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

   3. Taylor expanded in re around -inf 84.9%

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

      1. *-commutative 84.9%

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

   5. Simplified 84.9%

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

   if -1.00000000000000004e36 < re < 2.1000000000000001e-97

   1. Initial program 60.8%

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

   3. Taylor expanded in re around 0 86.6%

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

   if 2.1000000000000001e-97 < re < 1.3499999999999999e-65 or 8.5e5 < re

   1. Initial program 8.1%

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

      1. sub-neg 8.1%

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

      2. flip-+ 6.1%

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

      3. pow2 6.1%

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

      4. hypot-define 6.1%

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

      5. hypot-define 6.4%

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

   4. Applied egg-rr 6.4%

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

   5. Taylor expanded in re around inf 47.0%

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

      1. associate-*r/ 47.0%

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

   7. Simplified 47.0%

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

      1. expm1-log1p-u 46.5%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \frac{0.5 \cdot {im}^{2}}{re}}\right)\right)} \]

      2. expm1-undefine 29.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \frac{0.5 \cdot {im}^{2}}{re}}\right)} - 1\right)} \]

      3. associate-*r/ 29.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{2 \cdot \left(0.5 \cdot {im}^{2}\right)}{re}}}\right)} - 1\right) \]

      4. sqrt-div 29.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{2 \cdot \left(0.5 \cdot {im}^{2}\right)}}{\sqrt{re}}}\right)} - 1\right) \]

      5. associate-*r* 29.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot {im}^{2}}}}{\sqrt{re}}\right)} - 1\right) \]

      6. metadata-eval 29.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{1} \cdot {im}^{2}}}{\sqrt{re}}\right)} - 1\right) \]

      7. *-un-lft-identity 29.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{{im}^{2}}}}{\sqrt{re}}\right)} - 1\right) \]

      8. sqrt-pow1 38.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{im}^{\left(\frac{2}{2}\right)}}}{\sqrt{re}}\right)} - 1\right) \]

      9. metadata-eval 38.6%

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

      10. pow1 38.6%

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

   9. Applied egg-rr 38.6%

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

       1. log1p-undefine 38.6%

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

       2. rem-exp-log 39.8%

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

       3. +-commutative 39.8%

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

       4. associate--l+ 81.7%

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

       5. metadata-eval 81.7%

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

       6. metadata-eval 81.7%

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

       7. div0 81.7%

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

       8. sub-neg 81.7%

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

       9. div-sub 81.7%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{im - 0}{\sqrt{re}}} \]

       10. --rgt-identity 81.7%

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

   11. Simplified 81.7%

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

   if 1.3499999999999999e-65 < re < 8.5e5

   1. Initial program 24.1%

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

      1. sub-neg 24.1%

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

      2. flip-+ 22.2%

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

      3. pow2 22.2%

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

      4. hypot-define 22.2%

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

      5. hypot-define 24.1%

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

   4. Applied egg-rr 24.1%

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

      1. unpow2 24.1%

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

      2. flip-+ 67.2%

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

      3. sub-neg 67.2%

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

   6. Applied egg-rr 66.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.35 \cdot 10^{-65} \lor \neg \left(re \leq 850000\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(re + im\right) - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.65 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{-78} \lor \neg \left(re \leq 1.35 \cdot 10^{-65}\right) \land re \leq 57:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.65e+34)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (or (<= re 1.55e-78) (and (not (<= re 1.35e-65)) (<= re 57.0)))
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.65e+34) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 1.55e-78) || (!(re <= 1.35e-65) && (re <= 57.0))) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= (-1.65d+34)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if ((re <= 1.55d-78) .or. (.not. (re <= 1.35d-65)) .and. (re <= 57.0d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= -1.65e+34) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 1.55e-78) || (!(re <= 1.35e-65) && (re <= 57.0))) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.65e+34:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif (re <= 1.55e-78) or (not (re <= 1.35e-65) and (re <= 57.0)):
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.65e+34)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif ((re <= 1.55e-78) || (!(re <= 1.35e-65) && (re <= 57.0)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= -1.65e+34)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif ((re <= 1.55e-78) || (~((re <= 1.35e-65)) && (re <= 57.0)))
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.65e+34], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.55e-78], And[N[Not[LessEqual[re, 1.35e-65]], $MachinePrecision], LessEqual[re, 57.0]]], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.64999999999999994e34

   1. Initial program 41.9%

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

   3. Taylor expanded in re around -inf 84.9%

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

      1. *-commutative 84.9%

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

   5. Simplified 84.9%

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

   if -1.64999999999999994e34 < re < 1.55000000000000009e-78 or 1.3499999999999999e-65 < re < 57

   1. Initial program 54.2%

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

      1. sub-neg 54.2%

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

      2. flip-+ 44.5%

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

      3. pow2 44.5%

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

      4. hypot-define 44.4%

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

      5. hypot-define 45.5%

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

   4. Applied egg-rr 45.5%

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

   5. Taylor expanded in re around 0 81.1%

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

   if 1.55000000000000009e-78 < re < 1.3499999999999999e-65 or 57 < re

   1. Initial program 8.2%

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

      1. sub-neg 8.2%

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

      2. flip-+ 6.2%

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

      3. pow2 6.2%

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

      4. hypot-define 6.2%

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

      5. hypot-define 6.5%

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

   4. Applied egg-rr 6.5%

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

   5. Taylor expanded in re around inf 48.2%

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

      1. associate-*r/ 48.2%

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

   7. Simplified 48.2%

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

      1. expm1-log1p-u 47.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \frac{0.5 \cdot {im}^{2}}{re}}\right)\right)} \]

      2. expm1-undefine 30.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \frac{0.5 \cdot {im}^{2}}{re}}\right)} - 1\right)} \]

      3. associate-*r/ 30.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{2 \cdot \left(0.5 \cdot {im}^{2}\right)}{re}}}\right)} - 1\right) \]

      4. sqrt-div 30.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{2 \cdot \left(0.5 \cdot {im}^{2}\right)}}{\sqrt{re}}}\right)} - 1\right) \]

      5. associate-*r* 30.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot {im}^{2}}}}{\sqrt{re}}\right)} - 1\right) \]

      6. metadata-eval 30.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{1} \cdot {im}^{2}}}{\sqrt{re}}\right)} - 1\right) \]

      7. *-un-lft-identity 30.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{{im}^{2}}}}{\sqrt{re}}\right)} - 1\right) \]

      8. sqrt-pow1 39.5%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{im}^{\left(\frac{2}{2}\right)}}}{\sqrt{re}}\right)} - 1\right) \]

      9. metadata-eval 39.5%

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

      10. pow1 39.5%

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

   9. Applied egg-rr 39.5%

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

       1. log1p-undefine 39.5%

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

       2. rem-exp-log 40.8%

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

       3. +-commutative 40.8%

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

       4. associate--l+ 82.5%

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

       5. metadata-eval 82.5%

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

       6. metadata-eval 82.5%

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

       7. div0 82.5%

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

       8. sub-neg 82.5%

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

       9. div-sub 82.5%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{im - 0}{\sqrt{re}}} \]

       10. --rgt-identity 82.5%

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

   11. Simplified 82.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.65 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{-78} \lor \neg \left(re \leq 1.35 \cdot 10^{-65}\right) \land re \leq 57:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.75 \cdot 10^{-65} \lor \neg \left(re \leq 0.051\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.05e+36)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 3.8e-97)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (or (<= re 2.75e-65) (not (<= re 0.051)))
       (* 0.5 (/ im (sqrt re)))
       (* 0.5 (sqrt (* 2.0 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.05e+36) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.8e-97) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if ((re <= 2.75e-65) || !(re <= 0.051)) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.05d+36)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 3.8d-97) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if ((re <= 2.75d-65) .or. (.not. (re <= 0.051d0))) then
        tmp = 0.5d0 * (im / sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.05e+36) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.8e-97) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if ((re <= 2.75e-65) || !(re <= 0.051)) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.05e+36:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 3.8e-97:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif (re <= 2.75e-65) or not (re <= 0.051):
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.05e+36)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 3.8e-97)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif ((re <= 2.75e-65) || !(re <= 0.051))
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.05e+36)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 3.8e-97)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif ((re <= 2.75e-65) || ~((re <= 0.051)))
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.05e+36], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.8e-97], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 2.75e-65], N[Not[LessEqual[re, 0.051]], $MachinePrecision]], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.05000000000000002e36

   1. Initial program 41.9%

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

   3. Taylor expanded in re around -inf 84.9%

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

      1. *-commutative 84.9%

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

   5. Simplified 84.9%

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

   if -1.05000000000000002e36 < re < 3.8000000000000001e-97

   1. Initial program 60.8%

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

   3. Taylor expanded in re around 0 86.6%

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

   if 3.8000000000000001e-97 < re < 2.7499999999999999e-65 or 0.0509999999999999967 < re

   1. Initial program 8.1%

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

      1. sub-neg 8.1%

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

      2. flip-+ 6.1%

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

      3. pow2 6.1%

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

      4. hypot-define 6.1%

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

      5. hypot-define 6.4%

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

   4. Applied egg-rr 6.4%

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

   5. Taylor expanded in re around inf 47.0%

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

      1. associate-*r/ 47.0%

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

   7. Simplified 47.0%

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

      1. expm1-log1p-u 46.5%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \frac{0.5 \cdot {im}^{2}}{re}}\right)\right)} \]

      2. expm1-undefine 29.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \frac{0.5 \cdot {im}^{2}}{re}}\right)} - 1\right)} \]

      3. associate-*r/ 29.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{2 \cdot \left(0.5 \cdot {im}^{2}\right)}{re}}}\right)} - 1\right) \]

      4. sqrt-div 29.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{2 \cdot \left(0.5 \cdot {im}^{2}\right)}}{\sqrt{re}}}\right)} - 1\right) \]

      5. associate-*r* 29.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot {im}^{2}}}}{\sqrt{re}}\right)} - 1\right) \]

      6. metadata-eval 29.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{1} \cdot {im}^{2}}}{\sqrt{re}}\right)} - 1\right) \]

      7. *-un-lft-identity 29.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{{im}^{2}}}}{\sqrt{re}}\right)} - 1\right) \]

      8. sqrt-pow1 38.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{im}^{\left(\frac{2}{2}\right)}}}{\sqrt{re}}\right)} - 1\right) \]

      9. metadata-eval 38.6%

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

      10. pow1 38.6%

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

   9. Applied egg-rr 38.6%

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

       1. log1p-undefine 38.6%

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

       2. rem-exp-log 39.8%

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

       3. +-commutative 39.8%

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

       4. associate--l+ 81.7%

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

       5. metadata-eval 81.7%

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

       6. metadata-eval 81.7%

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

       7. div0 81.7%

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

       8. sub-neg 81.7%

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

       9. div-sub 81.7%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{im - 0}{\sqrt{re}}} \]

       10. --rgt-identity 81.7%

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

   11. Simplified 81.7%

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

   if 2.7499999999999999e-65 < re < 0.0509999999999999967

   1. Initial program 24.1%

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

      1. sub-neg 24.1%

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

      2. flip-+ 22.2%

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

      3. pow2 22.2%

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

      4. hypot-define 22.2%

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

      5. hypot-define 24.1%

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

   4. Applied egg-rr 24.1%

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

   5. Taylor expanded in re around 0 66.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.75 \cdot 10^{-65} \lor \neg \left(re \leq 0.051\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.55 \cdot 10^{-78} \lor \neg \left(re \leq 1.35 \cdot 10^{-65}\right) \land re \leq 0.038:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 1.55e-78) (and (not (<= re 1.35e-65)) (<= re 0.038)))
   (* 0.5 (sqrt (* 2.0 im)))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 1.55e-78) || (!(re <= 1.35e-65) && (re <= 0.038))) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= 1.55d-78) .or. (.not. (re <= 1.35d-65)) .and. (re <= 0.038d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= 1.55e-78) || (!(re <= 1.35e-65) && (re <= 0.038))) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 1.55e-78) or (not (re <= 1.35e-65) and (re <= 0.038)):
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 1.55e-78) || (!(re <= 1.35e-65) && (re <= 0.038)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= 1.55e-78) || (~((re <= 1.35e-65)) && (re <= 0.038)))
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 1.55e-78], And[N[Not[LessEqual[re, 1.35e-65]], $MachinePrecision], LessEqual[re, 0.038]]], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.55000000000000009e-78 or 1.3499999999999999e-65 < re < 0.0379999999999999991

   1. Initial program 50.5%

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

      1. sub-neg 50.5%

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

      2. flip-+ 31.5%

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

      3. pow2 31.5%

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

      4. hypot-define 31.5%

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

      5. hypot-define 32.3%

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

   4. Applied egg-rr 32.3%

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

   5. Taylor expanded in re around 0 62.8%

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

   if 1.55000000000000009e-78 < re < 1.3499999999999999e-65 or 0.0379999999999999991 < re

   1. Initial program 8.2%

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

      1. sub-neg 8.2%

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

      2. flip-+ 6.2%

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

      3. pow2 6.2%

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

      4. hypot-define 6.2%

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

      5. hypot-define 6.5%

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

   4. Applied egg-rr 6.5%

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

   5. Taylor expanded in re around inf 48.2%

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

      1. associate-*r/ 48.2%

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

   7. Simplified 48.2%

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

      1. expm1-log1p-u 47.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \frac{0.5 \cdot {im}^{2}}{re}}\right)\right)} \]

      2. expm1-undefine 30.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \frac{0.5 \cdot {im}^{2}}{re}}\right)} - 1\right)} \]

      3. associate-*r/ 30.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{2 \cdot \left(0.5 \cdot {im}^{2}\right)}{re}}}\right)} - 1\right) \]

      4. sqrt-div 30.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{2 \cdot \left(0.5 \cdot {im}^{2}\right)}}{\sqrt{re}}}\right)} - 1\right) \]

      5. associate-*r* 30.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot {im}^{2}}}}{\sqrt{re}}\right)} - 1\right) \]

      6. metadata-eval 30.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{1} \cdot {im}^{2}}}{\sqrt{re}}\right)} - 1\right) \]

      7. *-un-lft-identity 30.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{{im}^{2}}}}{\sqrt{re}}\right)} - 1\right) \]

      8. sqrt-pow1 39.5%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{im}^{\left(\frac{2}{2}\right)}}}{\sqrt{re}}\right)} - 1\right) \]

      9. metadata-eval 39.5%

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

      10. pow1 39.5%

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

   9. Applied egg-rr 39.5%

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

       1. log1p-undefine 39.5%

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

       2. rem-exp-log 40.8%

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

       3. +-commutative 40.8%

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

       4. associate--l+ 82.5%

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

       5. metadata-eval 82.5%

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

       6. metadata-eval 82.5%

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

       7. div0 82.5%

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

       8. sub-neg 82.5%

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

       9. div-sub 82.5%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{im - 0}{\sqrt{re}}} \]

       10. --rgt-identity 82.5%

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

   11. Simplified 82.5%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.55 \cdot 10^{-78} \lor \neg \left(re \leq 1.35 \cdot 10^{-65}\right) \land re \leq 0.038:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. sub-neg 38.6%

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

   2. flip-+ 24.4%

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

   3. pow2 24.4%

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

   4. hypot-define 24.4%

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

   5. hypot-define 25.1%

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

4. Applied egg-rr 25.1%

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

5. Taylor expanded in re around 0 51.6%

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

6. Final simplification 51.6%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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)))))