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

Percentage Accurate: 42.4% → 88.3%

Time: 10.4s

Alternatives: 6

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.8 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(im, re\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.8e+73)
   (sqrt (* 0.5 (- (hypot im re) re)))
   (* (* 0.5 im) (pow re -0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.8e+73) {
		tmp = sqrt((0.5 * (hypot(im, re) - re)));
	} else {
		tmp = (0.5 * im) * pow(re, -0.5);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.8e+73) {
		tmp = Math.sqrt((0.5 * (Math.hypot(im, re) - re)));
	} else {
		tmp = (0.5 * im) * Math.pow(re, -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.8e+73:
		tmp = math.sqrt((0.5 * (math.hypot(im, re) - re)))
	else:
		tmp = (0.5 * im) * math.pow(re, -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.8e+73)
		tmp = sqrt(Float64(0.5 * Float64(hypot(im, re) - re)));
	else
		tmp = Float64(Float64(0.5 * im) * (re ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.8e+73)
		tmp = sqrt((0.5 * (hypot(im, re) - re)));
	else
		tmp = (0.5 * im) * (re ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.8e+73], N[Sqrt[N[(0.5 * N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.80000000000000008e73

   1. Initial program 48.2%

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

      1. pow1 48.2%

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

   4. Applied egg-rr 90.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 90.3%

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

      2. *-commutative 90.3%

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

      3. associate-*r* 90.3%

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

      4. metadata-eval 90.3%

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

      5. hypot-undefine 48.2%

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

      6. unpow2 48.2%

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

      7. unpow2 48.2%

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

      8. +-commutative 48.2%

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

      9. unpow2 48.2%

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

      10. unpow2 48.2%

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

      11. hypot-undefine 90.3%

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

   6. Simplified 90.3%

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

   if 2.80000000000000008e73 < re

   1. Initial program 8.8%

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

   3. Taylor expanded in re around inf 81.7%

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

      1. associate-*l* 81.7%

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

      2. *-commutative 81.7%

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

      3. associate-*r* 82.1%

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

   5. Simplified 82.1%

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

      1. pow1 82.1%

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

      2. add-log-exp 13.6%

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

      3. associate-*l* 13.6%

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

      4. exp-prod 13.6%

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

      5. sqrt-unprod 13.6%

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

      6. metadata-eval 13.6%

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

      7. metadata-eval 13.6%

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

      8. pow1 13.6%

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

      9. add-log-exp 82.6%

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

      10. inv-pow 82.6%

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

      11. sqrt-pow1 82.8%

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

      12. metadata-eval 82.8%

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

   7. Applied egg-rr 82.8%

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

      1. unpow1 82.8%

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

      2. associate-*r* 82.8%

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

      3. *-commutative 82.8%

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

   9. Simplified 82.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.8 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(im, re\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.9e+90)
   (sqrt (- re))
   (if (<= re 3.9e+70) (sqrt (* 0.5 (- im re))) (* (* 0.5 im) (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.9e+90) {
		tmp = sqrt(-re);
	} else if (re <= 3.9e+70) {
		tmp = sqrt((0.5 * (im - re)));
	} 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) :: tmp
    if (re <= (-3.9d+90)) then
        tmp = sqrt(-re)
    else if (re <= 3.9d+70) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = (0.5d0 * im) * (re ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.9e+90) {
		tmp = Math.sqrt(-re);
	} else if (re <= 3.9e+70) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = (0.5 * im) * Math.pow(re, -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.9e+90:
		tmp = math.sqrt(-re)
	elif re <= 3.9e+70:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = (0.5 * im) * math.pow(re, -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.9e+90)
		tmp = sqrt(Float64(-re));
	elseif (re <= 3.9e+70)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(0.5 * im) * (re ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.9e+90)
		tmp = sqrt(-re);
	elseif (re <= 3.9e+70)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = (0.5 * im) * (re ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.9e+90], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 3.9e+70], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.9000000000000002e90

   1. Initial program 19.4%

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

      1. pow1 19.4%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 19.4%

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

      6. unpow2 19.4%

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

      7. unpow2 19.4%

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

      8. +-commutative 19.4%

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

      9. unpow2 19.4%

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

      10. unpow2 19.4%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 83.1%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 83.1%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 83.1%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

   if -3.9000000000000002e90 < re < 3.89999999999999975e70

   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. pow1 54.2%

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

   4. Applied egg-rr 87.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-*r* 87.9%

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

      4. metadata-eval 87.9%

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

      5. hypot-undefine 54.2%

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

      6. unpow2 54.2%

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

      7. unpow2 54.2%

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

      8. +-commutative 54.2%

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

      9. unpow2 54.2%

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

      10. unpow2 54.2%

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

      11. hypot-undefine 87.9%

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

   6. Simplified 87.9%

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

   7. Taylor expanded in re around 0 77.7%

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

      1. neg-mul-1 77.7%

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

      2. sub-neg 77.7%

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

   9. Simplified 77.7%

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

   if 3.89999999999999975e70 < re

   1. Initial program 12.1%

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

   3. Taylor expanded in re around inf 80.7%

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

      1. associate-*l* 80.7%

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

      2. *-commutative 80.7%

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

      3. associate-*r* 81.1%

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

   5. Simplified 81.1%

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

      1. pow1 81.1%

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

      2. add-log-exp 15.0%

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

      3. associate-*l* 15.0%

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

      4. exp-prod 15.0%

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

      5. sqrt-unprod 15.0%

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

      6. metadata-eval 15.0%

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

      7. metadata-eval 15.0%

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

      8. pow1 15.0%

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

      9. add-log-exp 81.6%

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

      10. inv-pow 81.6%

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

      11. sqrt-pow1 81.7%

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

      12. metadata-eval 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. unpow1 81.7%

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

      2. associate-*r* 81.7%

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

      3. *-commutative 81.7%

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

   9. Simplified 81.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot im\right) \cdot {re}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.42 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.42e+89)
   (sqrt (- re))
   (if (<= re 3.2e+70) (sqrt (* 0.5 (- im re))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.42e+89) {
		tmp = sqrt(-re);
	} else if (re <= 3.2e+70) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / 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 <= (-1.42d+89)) then
        tmp = sqrt(-re)
    else if (re <= 3.2d+70) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.42e+89) {
		tmp = Math.sqrt(-re);
	} else if (re <= 3.2e+70) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.42e+89:
		tmp = math.sqrt(-re)
	elif re <= 3.2e+70:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.42e+89)
		tmp = sqrt(Float64(-re));
	elseif (re <= 3.2e+70)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.42e+89)
		tmp = sqrt(-re);
	elseif (re <= 3.2e+70)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.42e+89], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 3.2e+70], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.42e89

   1. Initial program 19.4%

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

      1. pow1 19.4%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 19.4%

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

      6. unpow2 19.4%

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

      7. unpow2 19.4%

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

      8. +-commutative 19.4%

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

      9. unpow2 19.4%

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

      10. unpow2 19.4%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 83.1%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 83.1%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 83.1%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

   if -1.42e89 < re < 3.2000000000000002e70

   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. pow1 54.2%

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

   4. Applied egg-rr 87.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-*r* 87.9%

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

      4. metadata-eval 87.9%

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

      5. hypot-undefine 54.2%

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

      6. unpow2 54.2%

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

      7. unpow2 54.2%

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

      8. +-commutative 54.2%

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

      9. unpow2 54.2%

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

      10. unpow2 54.2%

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

      11. hypot-undefine 87.9%

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

   6. Simplified 87.9%

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

   7. Taylor expanded in re around 0 77.7%

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

      1. neg-mul-1 77.7%

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

      2. sub-neg 77.7%

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

   9. Simplified 77.7%

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

   if 3.2000000000000002e70 < re

   1. Initial program 12.1%

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

   3. Taylor expanded in re around inf 80.7%

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

      1. associate-*l* 80.7%

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

      2. *-commutative 80.7%

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

      3. associate-*r* 81.1%

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

   5. Simplified 81.1%

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

      1. pow1 81.1%

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

      2. add-log-exp 15.0%

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

      3. associate-*l* 15.0%

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

      4. exp-prod 15.0%

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

      5. sqrt-unprod 15.0%

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

      6. metadata-eval 15.0%

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

      7. metadata-eval 15.0%

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

      8. pow1 15.0%

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

      9. add-log-exp 81.6%

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

      10. inv-pow 81.6%

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

      11. sqrt-pow1 81.7%

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

      12. metadata-eval 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. unpow1 81.7%

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

      2. associate-*r* 81.7%

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

      3. *-commutative 81.7%

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

   9. Simplified 81.7%

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

       1. *-commutative 81.7%

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

       2. associate-*r* 81.7%

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

       3. rem-cube-cbrt 80.6%

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

       4. rem-cube-cbrt 81.7%

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

       5. add-sqr-sqrt 81.4%

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

       6. sqrt-unprod 58.1%

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

       7. swap-sqr 43.6%

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

       8. unpow2 43.6%

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

       9. pow-prod-up 43.6%

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

       10. metadata-eval 43.6%

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

       11. inv-pow 43.6%

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

       12. div-inv 43.6%

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

       13. sqrt-div 59.7%

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

       14. sqrt-pow1 81.6%

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

       15. metadata-eval 81.6%

           \[\leadsto 0.5 \cdot \frac{{im}^{\color{blue}{1}}}{\sqrt{re}} \]

       16. pow1 81.6%

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

       17. associate-*r/ 81.7%

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

   11. Applied egg-rr 81.7%

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

       1. *-commutative 81.7%

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

       2. associate-/l* 81.7%

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

   13. Applied egg-rr 81.7%

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

Alternative 4: 63.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.02 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.02e+96) (sqrt (- re)) (sqrt (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.02e+96) {
		tmp = sqrt(-re);
	} else {
		tmp = sqrt((0.5 * 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 <= (-1.02d+96)) then
        tmp = sqrt(-re)
    else
        tmp = sqrt((0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.02e+96) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.02e+96:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.02e+96)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.02e+96)
		tmp = sqrt(-re);
	else
		tmp = sqrt((0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.02e+96], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(0.5 * im), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.02000000000000001e96

   1. Initial program 19.4%

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

      1. pow1 19.4%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 19.4%

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

      6. unpow2 19.4%

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

      7. unpow2 19.4%

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

      8. +-commutative 19.4%

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

      9. unpow2 19.4%

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

      10. unpow2 19.4%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 83.1%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 83.1%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 83.1%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

   if -1.02000000000000001e96 < re

   1. Initial program 43.6%

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

      1. pow1 43.6%

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

   4. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-*r* 74.4%

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

      4. metadata-eval 74.4%

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

      5. hypot-undefine 43.6%

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

      6. unpow2 43.6%

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

      7. unpow2 43.6%

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

      8. +-commutative 43.6%

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

      9. unpow2 43.6%

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

      10. unpow2 43.6%

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

      11. hypot-undefine 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in im around inf 63.3%

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

      1. *-commutative 63.3%

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

   9. Simplified 63.3%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.02 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-311) (sqrt (- re)) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-311) {
		tmp = sqrt(-re);
	} else {
		tmp = 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 <= (-2d-311)) then
        tmp = sqrt(-re)
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2e-311) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-311:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-311)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2e-311)
		tmp = sqrt(-re);
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-311], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.9999999999999e-311

   1. Initial program 49.8%

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

      1. pow1 49.8%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 49.8%

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

      6. unpow2 49.8%

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

      7. unpow2 49.8%

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

      8. +-commutative 49.8%

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

      9. unpow2 49.8%

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

      10. unpow2 49.8%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 45.8%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 45.8%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 45.8%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

   if -1.9999999999999e-311 < re

   1. Initial program 31.9%

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

      1. pow1 31.9%

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

   4. Applied egg-rr 60.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 60.1%

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

      2. *-commutative 60.1%

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

      3. associate-*r* 60.1%

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

      4. metadata-eval 60.1%

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

      5. hypot-undefine 31.9%

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

      6. unpow2 31.9%

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

      7. unpow2 31.9%

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

      8. +-commutative 31.9%

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

      9. unpow2 31.9%

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

      10. unpow2 31.9%

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

      11. hypot-undefine 60.1%

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

   6. Simplified 60.1%

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

   7. Taylor expanded in re around -inf 0.0%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 0.0%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 0.0%

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

       1. expm1-log1p-u 0.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-re}\right)\right)} \]

       2. expm1-undefine 0.0%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 4.0%

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

       5. sqr-neg 4.0%

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

       6. sqrt-unprod 4.6%

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

       7. add-sqr-sqrt 4.6%

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

   11. Applied egg-rr 4.6%

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

       1. sub-neg 4.6%

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

       2. metadata-eval 4.6%

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

       3. +-commutative 4.6%

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

       4. log1p-undefine 4.6%

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

       5. rem-exp-log 4.6%

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

       6. associate-+r+ 5.8%

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

       7. metadata-eval 5.8%

          \[\leadsto \color{blue}{0} + \sqrt{re} \]

       8. metadata-eval 5.8%

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

       9. remove-double-neg 5.8%

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

       10. distribute-neg-in 5.8%

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

       11. +-lft-identity 5.8%

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

       12. remove-double-neg 5.8%

           \[\leadsto \color{blue}{\sqrt{re}} \]

   13. Simplified 5.8%

       \[\leadsto \color{blue}{\sqrt{re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 2.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt re))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(re)
end function

Java

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

Python

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

Julia

function code(re, im)
	return sqrt(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 40.0%

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

   1. pow1 40.0%

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

4. Applied egg-rr 78.2%

   \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
5. Step-by-step derivation

   1. unpow1 78.2%

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

   2. *-commutative 78.2%

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

   3. associate-*r* 78.2%

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

   4. metadata-eval 78.2%

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

   5. hypot-undefine 40.0%

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

   6. unpow2 40.0%

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

   7. unpow2 40.0%

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

   8. +-commutative 40.0%

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

   9. unpow2 40.0%

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

   10. unpow2 40.0%

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

   11. hypot-undefine 78.2%

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

6. Simplified 78.2%

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

7. Taylor expanded in re around -inf 20.8%

   \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
8. Step-by-step derivation

   1. neg-mul-1 20.8%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

9. Simplified 20.8%

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

    1. expm1-log1p-u 19.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-re}\right)\right)} \]

    2. expm1-undefine 14.2%

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

    3. add-sqr-sqrt 14.2%

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

    4. sqrt-unprod 7.7%

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

    5. sqr-neg 7.7%

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

    6. sqrt-unprod 2.5%

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

    7. add-sqr-sqrt 2.5%

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

11. Applied egg-rr 2.5%

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

    1. sub-neg 2.5%

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

    2. metadata-eval 2.5%

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

    3. +-commutative 2.5%

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

    4. log1p-undefine 2.5%

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

    5. rem-exp-log 2.5%

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

    6. associate-+r+ 3.2%

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

    7. metadata-eval 3.2%

       \[\leadsto \color{blue}{0} + \sqrt{re} \]

    8. metadata-eval 3.2%

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

    9. remove-double-neg 3.2%

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

    10. distribute-neg-in 3.2%

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

    11. +-lft-identity 3.2%

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

    12. remove-double-neg 3.2%

        \[\leadsto \color{blue}{\sqrt{re}} \]

13. Simplified 3.2%

    \[\leadsto \color{blue}{\sqrt{re}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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)))))