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

Percentage Accurate: 41.4% → 87.4%

Time: 9.6s

Alternatives: 5

Speedup: 2.1×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.4% 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: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.05 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.05e-55)
   (sqrt (* 0.5 (- (hypot re im) re)))
   (* im (/ 0.5 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.05e-55) {
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.05e-55) {
		tmp = Math.sqrt((0.5 * (Math.hypot(re, im) - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.05e-55:
		tmp = math.sqrt((0.5 * (math.hypot(re, im) - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.05e-55)
		tmp = sqrt(Float64(0.5 * Float64(hypot(re, 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 <= 3.05e-55)
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.05e-55], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.0500000000000001e-55

   1. Initial program 51.6%

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

      1. hypot-udef 94.8%

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

      2. add-sqr-sqrt 94.0%

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

      3. sqrt-unprod 94.8%

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

      4. *-commutative 94.8%

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

      5. *-commutative 94.8%

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

      6. swap-sqr 94.8%

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

      7. add-sqr-sqrt 94.8%

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

      8. metadata-eval 94.8%

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

   4. Applied egg-rr 94.8%

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

      1. *-commutative 94.8%

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

      2. associate-*r* 95.3%

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

      3. metadata-eval 95.3%

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

   6. Simplified 95.3%

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

   if 3.0500000000000001e-55 < re

   1. Initial program 10.2%

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

   3. Taylor expanded in im around 0 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-*l* 79.6%

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

      3. *-commutative 79.6%

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

   5. Simplified 79.6%

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

      1. expm1-log1p-u 79.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot 0.5\right)\right)\right)} \]

      2. expm1-udef 27.3%

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

      3. sqrt-unprod 27.3%

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

      4. metadata-eval 27.3%

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

      5. metadata-eval 27.3%

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

      6. *-rgt-identity 27.3%

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

      7. *-commutative 27.3%

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

      8. sqrt-div 27.3%

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

      9. metadata-eval 27.3%

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

      10. un-div-inv 27.3%

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

   7. Applied egg-rr 27.3%

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

      1. expm1-def 79.8%

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

      2. expm1-log1p 80.7%

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

      3. associate-*r/ 80.6%

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

   9. Simplified 80.6%

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

       1. associate-*r/ 80.7%

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

       2. *-commutative 80.7%

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

   11. Applied egg-rr 80.7%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.05 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+54}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.5e+54)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.6e-50) (sqrt (* 0.5 (- im re))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.5e+54) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.6e-50) {
		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 <= (-1.5d+54)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.6d-50) 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 <= -1.5e+54) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.6e-50) {
		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 <= -1.5e+54:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.6e-50:
		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 <= -1.5e+54)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.6e-50)
		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 <= -1.5e+54)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.6e-50)
		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, -1.5e+54], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.6e-50], 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 < -1.4999999999999999e54

   1. Initial program 32.0%

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

      1. sub-neg 32.0%

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

      2. sqr-neg 32.0%

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

      3. sub-neg 32.0%

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

      4. sqr-neg 32.0%

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

      5. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around -inf 82.2%

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

      1. *-commutative 82.2%

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

   7. Simplified 82.2%

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

   if -1.4999999999999999e54 < re < 2.6000000000000001e-50

   1. Initial program 58.7%

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

      1. hypot-udef 92.9%

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

      2. add-sqr-sqrt 92.2%

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

      3. sqrt-unprod 92.9%

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

      4. *-commutative 92.9%

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

      5. *-commutative 92.9%

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

      6. swap-sqr 92.9%

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

      7. add-sqr-sqrt 92.9%

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

      8. metadata-eval 92.9%

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

   4. Applied egg-rr 92.9%

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

      1. *-commutative 92.9%

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

      2. associate-*r* 93.6%

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

      3. metadata-eval 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in re around 0 79.6%

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

      1. neg-mul-1 79.6%

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

      2. unsub-neg 79.6%

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

   9. Simplified 79.6%

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

   if 2.6000000000000001e-50 < re

   1. Initial program 10.2%

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

   3. Taylor expanded in im around 0 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-*l* 79.6%

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

      3. *-commutative 79.6%

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

   5. Simplified 79.6%

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

      1. expm1-log1p-u 79.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot 0.5\right)\right)\right)} \]

      2. expm1-udef 27.3%

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

      3. sqrt-unprod 27.3%

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

      4. metadata-eval 27.3%

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

      5. metadata-eval 27.3%

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

      6. *-rgt-identity 27.3%

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

      7. *-commutative 27.3%

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

      8. sqrt-div 27.3%

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

      9. metadata-eval 27.3%

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

      10. un-div-inv 27.3%

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

   7. Applied egg-rr 27.3%

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

      1. expm1-def 79.8%

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

      2. expm1-log1p 80.7%

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

      3. associate-*r/ 80.6%

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

   9. Simplified 80.6%

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

       1. associate-*r/ 80.7%

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

       2. *-commutative 80.7%

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

   11. Applied egg-rr 80.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+54}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.9e-162) (sqrt (* 0.5 (- im re))) (sqrt (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.9e-162) {
		tmp = sqrt((0.5 * (im - 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 <= 1.9d-162) then
        tmp = sqrt((0.5d0 * (im - 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 <= 1.9e-162) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = Math.sqrt((0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.9e-162:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = math.sqrt((0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.9e-162)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = sqrt(Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.9e-162)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = sqrt((0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.9e-162], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * im), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.90000000000000002e-162

   1. Initial program 52.5%

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

      1. hypot-udef 97.3%

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

      2. add-sqr-sqrt 96.5%

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

      3. sqrt-unprod 97.3%

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

      4. *-commutative 97.3%

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

      5. *-commutative 97.3%

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

      6. swap-sqr 97.3%

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

      7. add-sqr-sqrt 97.3%

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

      8. metadata-eval 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. *-commutative 97.3%

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

      2. associate-*r* 97.9%

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

      3. metadata-eval 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in re around 0 66.8%

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

      1. neg-mul-1 66.8%

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

      2. unsub-neg 66.8%

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

   9. Simplified 66.8%

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

   if 1.90000000000000002e-162 < re

   1. Initial program 16.4%

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

      1. hypot-udef 40.5%

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

      2. add-sqr-sqrt 40.3%

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

      3. sqrt-unprod 40.5%

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

      4. *-commutative 40.5%

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

      5. *-commutative 40.5%

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

      6. swap-sqr 40.5%

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

      7. add-sqr-sqrt 40.5%

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

      8. metadata-eval 40.5%

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

   4. Applied egg-rr 40.5%

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

      1. *-commutative 40.5%

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

      2. associate-*r* 40.5%

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

      3. metadata-eval 40.5%

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

   6. Simplified 40.5%

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

   7. Taylor expanded in re around 0 33.5%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3e-103) (* 0.5 (sqrt (* re -4.0))) (sqrt (* 0.5 (- im re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3e-103) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = sqrt((0.5 * (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 (im <= 3d-103) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = sqrt((0.5d0 * (im - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3e-103) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = Math.sqrt((0.5 * (im - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3e-103:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = math.sqrt((0.5 * (im - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3e-103)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3e-103)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = sqrt((0.5 * (im - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3e-103], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3e-103

   1. Initial program 31.1%

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

      1. sub-neg 31.1%

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

      2. sqr-neg 31.1%

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

      3. sub-neg 31.1%

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

      4. sqr-neg 31.1%

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

      5. hypot-def 64.2%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in re around -inf 44.1%

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

      1. *-commutative 44.1%

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

   7. Simplified 44.1%

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

   if 3e-103 < im

   1. Initial program 40.5%

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

      1. hypot-udef 78.1%

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

      2. add-sqr-sqrt 77.5%

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

      3. sqrt-unprod 78.1%

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

      4. *-commutative 78.1%

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

      5. *-commutative 78.1%

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

      6. swap-sqr 78.1%

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

      7. add-sqr-sqrt 78.1%

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

      8. metadata-eval 78.1%

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

   4. Applied egg-rr 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r* 78.6%

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

      3. metadata-eval 78.6%

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

   6. Simplified 78.6%

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

   7. Taylor expanded in re around 0 67.7%

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

      1. neg-mul-1 67.7%

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

      2. unsub-neg 67.7%

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

   9. Simplified 67.7%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.5%

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

   1. hypot-udef 73.7%

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

   2. add-sqr-sqrt 73.2%

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

   3. sqrt-unprod 73.7%

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

   4. *-commutative 73.7%

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

   5. *-commutative 73.7%

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

   6. swap-sqr 73.7%

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

   7. add-sqr-sqrt 73.7%

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

   8. metadata-eval 73.7%

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

4. Applied egg-rr 73.7%

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

   1. *-commutative 73.7%

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

   2. associate-*r* 74.1%

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

   3. metadata-eval 74.1%

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

6. Simplified 74.1%

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

7. Taylor expanded in re around 0 49.6%

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

8. Final simplification 49.6%

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

Reproduce

herbie shell --seed 2024024 
(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)))))