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

Percentage Accurate: 41.0% → 90.0%

Time: 9.9s

Alternatives: 8

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: 41.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* im (sqrt (/ 0.25 re)))
   (sqrt (* 0.5 (- (hypot re im) re)))))

C

double code(double re, double im) {
	double tmp;
	if (sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = im * sqrt((0.25 / re));
	} else {
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = im * Math.sqrt((0.25 / re));
	} else {
		tmp = Math.sqrt((0.5 * (Math.hypot(re, im) - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re))) <= 0.0:
		tmp = im * math.sqrt((0.25 / re))
	else:
		tmp = math.sqrt((0.5 * (math.hypot(re, im) - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))) <= 0.0)
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	else
		tmp = sqrt(Float64(0.5 * Float64(hypot(re, im) - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0)
		tmp = im * sqrt((0.25 / re));
	else
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 15.7%

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

   2. Taylor expanded in im around 0 98.3%

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

      1. associate-*l* 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*l* 98.7%

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

   4. Simplified 98.7%

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

      1. associate-*r* 98.4%

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

      2. sqrt-unprod 99.7%

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

      3. metadata-eval 99.7%

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

      4. metadata-eval 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. sqrt-div 99.5%

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

      7. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*r* 99.5%

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

      2. un-div-inv 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-lft-identity 99.5%

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

      3. times-frac 99.5%

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

      4. /-rgt-identity 99.5%

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

   10. Simplified 99.5%

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

       1. expm1-log1p-u 98.2%

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

       2. expm1-udef 57.1%

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

       3. add-sqr-sqrt 57.1%

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

       4. sqrt-unprod 57.1%

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

       5. frac-times 57.1%

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

       6. metadata-eval 57.1%

          \[\leadsto im \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}}\right)} - 1\right) \]

       7. add-sqr-sqrt 57.1%

          \[\leadsto im \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{\color{blue}{re}}}\right)} - 1\right) \]

   12. Applied egg-rr 57.1%

       \[\leadsto im \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{re}}\right)} - 1\right)} \]
   13. Step-by-step derivation

       1. expm1-def 98.3%

          \[\leadsto im \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{0.25}{re}}\right)\right)} \]

       2. expm1-log1p 99.7%

          \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.25}{re}}} \]

   14. Simplified 99.7%

       \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.25}{re}}} \]

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

   1. Initial program 44.0%

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

      1. hypot-udef 90.0%

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

      2. add-sqr-sqrt 89.3%

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

      3. sqrt-unprod 90.0%

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

      4. *-commutative 90.0%

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

      5. *-commutative 90.0%

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

      6. swap-sqr 90.0%

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

      7. add-sqr-sqrt 90.0%

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

      8. metadata-eval 90.0%

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

   3. Applied egg-rr 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*r* 90.4%

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

      3. metadata-eval 90.4%

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

   5. Simplified 90.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]

Alternative 2: 74.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -2 + \left(2 \cdot im + re \cdot \frac{re}{im}\right)}\\ \mathbf{if}\;re \leq -1.05 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3.05 \cdot 10^{+23}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (+ (* re -2.0) (+ (* 2.0 im) (* re (/ re im))))))))
   (if (<= re -1.05e-72)
     (* 0.5 (sqrt (* re -4.0)))
     (if (<= re 1.7e-9)
       t_0
       (if (<= re 3.05e+23)
         (* im (sqrt (/ 0.25 re)))
         (if (<= re 3.6e+69) t_0 (* im (/ 0.5 (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt(((re * -2.0) + ((2.0 * im) + (re * (re / im)))));
	double tmp;
	if (re <= -1.05e-72) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.7e-9) {
		tmp = t_0;
	} else if (re <= 3.05e+23) {
		tmp = im * sqrt((0.25 / re));
	} else if (re <= 3.6e+69) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt(((re * (-2.0d0)) + ((2.0d0 * im) + (re * (re / im)))))
    if (re <= (-1.05d-72)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.7d-9) then
        tmp = t_0
    else if (re <= 3.05d+23) then
        tmp = im * sqrt((0.25d0 / re))
    else if (re <= 3.6d+69) then
        tmp = t_0
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt(((re * -2.0) + ((2.0 * im) + (re * (re / im)))));
	double tmp;
	if (re <= -1.05e-72) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.7e-9) {
		tmp = t_0;
	} else if (re <= 3.05e+23) {
		tmp = im * Math.sqrt((0.25 / re));
	} else if (re <= 3.6e+69) {
		tmp = t_0;
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt(((re * -2.0) + ((2.0 * im) + (re * (re / im)))))
	tmp = 0
	if re <= -1.05e-72:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.7e-9:
		tmp = t_0
	elif re <= 3.05e+23:
		tmp = im * math.sqrt((0.25 / re))
	elif re <= 3.6e+69:
		tmp = t_0
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(Float64(re * -2.0) + Float64(Float64(2.0 * im) + Float64(re * Float64(re / im))))))
	tmp = 0.0
	if (re <= -1.05e-72)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.7e-9)
		tmp = t_0;
	elseif (re <= 3.05e+23)
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	elseif (re <= 3.6e+69)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt(((re * -2.0) + ((2.0 * im) + (re * (re / im)))));
	tmp = 0.0;
	if (re <= -1.05e-72)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.7e-9)
		tmp = t_0;
	elseif (re <= 3.05e+23)
		tmp = im * sqrt((0.25 / re));
	elseif (re <= 3.6e+69)
		tmp = t_0;
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(N[(re * -2.0), $MachinePrecision] + N[(N[(2.0 * im), $MachinePrecision] + N[(re * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.05e-72], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.7e-9], t$95$0, If[LessEqual[re, 3.05e+23], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.6e+69], t$95$0, N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.05e-72

   1. Initial program 47.6%

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

      1. sqr-neg 47.6%

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

      2. sqr-neg 47.6%

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

      3. 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. Taylor expanded in re around -inf 74.9%

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

      1. *-commutative 74.9%

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

   6. Simplified 74.9%

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

   if -1.05e-72 < re < 1.6999999999999999e-9 or 3.0499999999999999e23 < re < 3.6000000000000003e69

   1. Initial program 53.1%

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

      1. sqr-neg 53.1%

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

      2. sqr-neg 53.1%

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

      3. hypot-def 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in re around 0 82.5%

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

      1. unpow2 82.5%

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

      2. *-un-lft-identity 82.5%

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

      3. times-frac 83.0%

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

      4. /-rgt-identity 83.0%

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

   6. Applied egg-rr 83.0%

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

   if 1.6999999999999999e-9 < re < 3.0499999999999999e23

   1. Initial program 6.4%

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

   2. Taylor expanded in im around 0 85.0%

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

      1. associate-*l* 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-*l* 86.0%

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

   4. Simplified 86.0%

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

      1. associate-*r* 85.4%

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

      2. sqrt-unprod 86.4%

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

      3. metadata-eval 86.4%

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

      4. metadata-eval 86.4%

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

      5. *-un-lft-identity 86.4%

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

      6. sqrt-div 86.0%

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

      7. metadata-eval 86.0%

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

   6. Applied egg-rr 86.0%

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

      1. associate-*r* 86.0%

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

      2. un-div-inv 86.0%

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

   8. Applied egg-rr 86.0%

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

      1. *-commutative 86.0%

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

      2. *-lft-identity 86.0%

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

      3. times-frac 86.0%

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

      4. /-rgt-identity 86.0%

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

   10. Simplified 86.0%

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

       1. expm1-log1p-u 86.0%

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

       2. expm1-udef 71.4%

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

       3. add-sqr-sqrt 71.4%

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

       4. sqrt-unprod 71.4%

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

       5. frac-times 71.4%

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

       6. metadata-eval 71.4%

          \[\leadsto im \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}}\right)} - 1\right) \]

       7. add-sqr-sqrt 71.4%

          \[\leadsto im \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{\color{blue}{re}}}\right)} - 1\right) \]

   12. Applied egg-rr 71.4%

       \[\leadsto im \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{re}}\right)} - 1\right)} \]
   13. Step-by-step derivation

       1. expm1-def 86.2%

          \[\leadsto im \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{0.25}{re}}\right)\right)} \]

       2. expm1-log1p 86.4%

          \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.25}{re}}} \]

   14. Simplified 86.4%

       \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.25}{re}}} \]

   if 3.6000000000000003e69 < re

   1. Initial program 10.3%

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

   2. Taylor expanded in im around 0 77.0%

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

      1. associate-*l* 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-*l* 77.2%

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

   4. Simplified 77.2%

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

      1. associate-*r* 77.0%

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

      2. sqrt-unprod 77.8%

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

      3. metadata-eval 77.8%

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

      4. metadata-eval 77.8%

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

      5. *-un-lft-identity 77.8%

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

      6. sqrt-div 77.9%

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

      7. metadata-eval 77.9%

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

   6. Applied egg-rr 77.9%

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

      1. associate-*r* 77.9%

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

      2. un-div-inv 77.9%

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

   8. Applied egg-rr 77.9%

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

      1. *-commutative 77.9%

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

      2. *-lft-identity 77.9%

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

      3. times-frac 77.9%

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

      4. /-rgt-identity 77.9%

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

   10. Simplified 77.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -2 + \left(2 \cdot im + re \cdot \frac{re}{im}\right)}\\ \mathbf{elif}\;re \leq 3.05 \cdot 10^{+23}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -2 + \left(2 \cdot im + re \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{-9} \lor \neg \left(re \leq 1.42 \cdot 10^{+23}\right) \land re \leq 3.6 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.7e-57)
   (* 0.5 (sqrt (* re -4.0)))
   (if (or (<= re 1.7e-9) (and (not (<= re 1.42e+23)) (<= re 3.6e+66)))
     (sqrt (* 0.5 (- im re)))
     (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.7e-57) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if ((re <= 1.7e-9) || (!(re <= 1.42e+23) && (re <= 3.6e+66))) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * sqrt((0.25 / 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 <= (-1.7d-57)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if ((re <= 1.7d-9) .or. (.not. (re <= 1.42d+23)) .and. (re <= 3.6d+66)) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * sqrt((0.25d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.7e-57) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if ((re <= 1.7e-9) || (!(re <= 1.42e+23) && (re <= 3.6e+66))) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * Math.sqrt((0.25 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.7e-57:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif (re <= 1.7e-9) or (not (re <= 1.42e+23) and (re <= 3.6e+66)):
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * math.sqrt((0.25 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.7e-57)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif ((re <= 1.7e-9) || (!(re <= 1.42e+23) && (re <= 3.6e+66)))
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.7e-57)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif ((re <= 1.7e-9) || (~((re <= 1.42e+23)) && (re <= 3.6e+66)))
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * sqrt((0.25 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.7e-57], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.7e-9], And[N[Not[LessEqual[re, 1.42e+23]], $MachinePrecision], LessEqual[re, 3.6e+66]]], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.70000000000000008e-57

   1. Initial program 46.4%

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

      1. sqr-neg 46.4%

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

      2. sqr-neg 46.4%

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

      3. 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. Taylor expanded in re around -inf 75.3%

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

      1. *-commutative 75.3%

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

   6. Simplified 75.3%

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

   if -1.70000000000000008e-57 < re < 1.6999999999999999e-9 or 1.42000000000000004e23 < re < 3.6e66

   1. Initial program 53.9%

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

      1. hypot-udef 91.5%

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

      2. add-sqr-sqrt 90.7%

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

      3. sqrt-unprod 91.5%

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

      4. *-commutative 91.5%

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

      5. *-commutative 91.5%

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

      6. swap-sqr 91.5%

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

      7. add-sqr-sqrt 91.5%

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

      8. metadata-eval 91.5%

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

   3. Applied egg-rr 91.5%

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

      1. *-commutative 91.5%

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

      2. associate-*r* 91.5%

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

      3. metadata-eval 91.5%

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

   5. Simplified 91.5%

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

   6. Taylor expanded in re around 0 81.7%

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

      1. neg-mul-1 81.7%

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

      2. unsub-neg 81.7%

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

   8. Simplified 81.7%

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

   if 1.6999999999999999e-9 < re < 1.42000000000000004e23 or 3.6e66 < re

   1. Initial program 9.8%

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

   2. Taylor expanded in im around 0 78.0%

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

      1. associate-*l* 78.1%

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

      2. *-commutative 78.1%

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

      3. associate-*l* 78.2%

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

   4. Simplified 78.2%

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

      1. associate-*r* 78.1%

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

      2. sqrt-unprod 78.9%

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

      3. metadata-eval 78.9%

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

      4. metadata-eval 78.9%

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

      5. *-un-lft-identity 78.9%

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

      6. sqrt-div 78.9%

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

      7. metadata-eval 78.9%

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

   6. Applied egg-rr 78.9%

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

      1. associate-*r* 78.9%

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

      2. un-div-inv 78.9%

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

   8. Applied egg-rr 78.9%

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

      1. *-commutative 78.9%

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

      2. *-lft-identity 78.9%

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

      3. times-frac 78.9%

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

      4. /-rgt-identity 78.9%

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

   10. Simplified 78.9%

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

       1. expm1-log1p-u 78.9%

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

       2. expm1-udef 27.1%

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

       3. add-sqr-sqrt 27.1%

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

       4. sqrt-unprod 27.1%

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

       5. frac-times 27.1%

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

       6. metadata-eval 27.1%

          \[\leadsto im \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}}\right)} - 1\right) \]

       7. add-sqr-sqrt 27.1%

          \[\leadsto im \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{\color{blue}{re}}}\right)} - 1\right) \]

   12. Applied egg-rr 27.1%

       \[\leadsto im \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{re}}\right)} - 1\right)} \]
   13. Step-by-step derivation

       1. expm1-def 78.9%

          \[\leadsto im \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{0.25}{re}}\right)\right)} \]

       2. expm1-log1p 78.9%

          \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.25}{re}}} \]

   14. Simplified 78.9%

       \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.25}{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{-9} \lor \neg \left(re \leq 1.42 \cdot 10^{+23}\right) \land re \leq 3.6 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]

Alternative 4: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -7.5 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.36 \cdot 10^{+25}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{elif}\;re \leq 2.95 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (* 0.5 (- im re)))))
   (if (<= re -7.5e-64)
     (* 0.5 (sqrt (* re -4.0)))
     (if (<= re 5.6e-9)
       t_0
       (if (<= re 1.36e+25)
         (* im (sqrt (/ 0.25 re)))
         (if (<= re 2.95e+66) t_0 (* im (/ 0.5 (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -7.5e-64) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5.6e-9) {
		tmp = t_0;
	} else if (re <= 1.36e+25) {
		tmp = im * sqrt((0.25 / re));
	} else if (re <= 2.95e+66) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.5d0 * (im - re)))
    if (re <= (-7.5d-64)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5.6d-9) then
        tmp = t_0
    else if (re <= 1.36d+25) then
        tmp = im * sqrt((0.25d0 / re))
    else if (re <= 2.95d+66) then
        tmp = t_0
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -7.5e-64) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5.6e-9) {
		tmp = t_0;
	} else if (re <= 1.36e+25) {
		tmp = im * Math.sqrt((0.25 / re));
	} else if (re <= 2.95e+66) {
		tmp = t_0;
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt((0.5 * (im - re)))
	tmp = 0
	if re <= -7.5e-64:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5.6e-9:
		tmp = t_0
	elif re <= 1.36e+25:
		tmp = im * math.sqrt((0.25 / re))
	elif re <= 2.95e+66:
		tmp = t_0
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(0.5 * Float64(im - re)))
	tmp = 0.0
	if (re <= -7.5e-64)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5.6e-9)
		tmp = t_0;
	elseif (re <= 1.36e+25)
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	elseif (re <= 2.95e+66)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt((0.5 * (im - re)));
	tmp = 0.0;
	if (re <= -7.5e-64)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5.6e-9)
		tmp = t_0;
	elseif (re <= 1.36e+25)
		tmp = im * sqrt((0.25 / re));
	elseif (re <= 2.95e+66)
		tmp = t_0;
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[re, -7.5e-64], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.6e-9], t$95$0, If[LessEqual[re, 1.36e+25], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.95e+66], t$95$0, N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.49999999999999949e-64

   1. Initial program 46.4%

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

      1. sqr-neg 46.4%

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

      2. sqr-neg 46.4%

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

      3. 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. Taylor expanded in re around -inf 75.3%

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

      1. *-commutative 75.3%

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

   6. Simplified 75.3%

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

   if -7.49999999999999949e-64 < re < 5.59999999999999969e-9 or 1.36e25 < re < 2.94999999999999994e66

   1. Initial program 53.9%

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

      1. hypot-udef 91.5%

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

      2. add-sqr-sqrt 90.7%

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

      3. sqrt-unprod 91.5%

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

      4. *-commutative 91.5%

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

      5. *-commutative 91.5%

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

      6. swap-sqr 91.5%

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

      7. add-sqr-sqrt 91.5%

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

      8. metadata-eval 91.5%

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

   3. Applied egg-rr 91.5%

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

      1. *-commutative 91.5%

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

      2. associate-*r* 91.5%

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

      3. metadata-eval 91.5%

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

   5. Simplified 91.5%

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

   6. Taylor expanded in re around 0 81.7%

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

      1. neg-mul-1 81.7%

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

      2. unsub-neg 81.7%

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

   8. Simplified 81.7%

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

   if 5.59999999999999969e-9 < re < 1.36e25

   1. Initial program 6.4%

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

   2. Taylor expanded in im around 0 85.0%

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

      1. associate-*l* 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-*l* 86.0%

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

   4. Simplified 86.0%

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

      1. associate-*r* 85.4%

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

      2. sqrt-unprod 86.4%

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

      3. metadata-eval 86.4%

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

      4. metadata-eval 86.4%

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

      5. *-un-lft-identity 86.4%

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

      6. sqrt-div 86.0%

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

      7. metadata-eval 86.0%

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

   6. Applied egg-rr 86.0%

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

      1. associate-*r* 86.0%

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

      2. un-div-inv 86.0%

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

   8. Applied egg-rr 86.0%

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

      1. *-commutative 86.0%

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

      2. *-lft-identity 86.0%

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

      3. times-frac 86.0%

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

      4. /-rgt-identity 86.0%

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

   10. Simplified 86.0%

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

       1. expm1-log1p-u 86.0%

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

       2. expm1-udef 71.4%

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

       3. add-sqr-sqrt 71.4%

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

       4. sqrt-unprod 71.4%

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

       5. frac-times 71.4%

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

       6. metadata-eval 71.4%

          \[\leadsto im \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}}\right)} - 1\right) \]

       7. add-sqr-sqrt 71.4%

          \[\leadsto im \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{\color{blue}{re}}}\right)} - 1\right) \]

   12. Applied egg-rr 71.4%

       \[\leadsto im \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{re}}\right)} - 1\right)} \]
   13. Step-by-step derivation

       1. expm1-def 86.2%

          \[\leadsto im \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{0.25}{re}}\right)\right)} \]

       2. expm1-log1p 86.4%

          \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.25}{re}}} \]

   14. Simplified 86.4%

       \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.25}{re}}} \]

   if 2.94999999999999994e66 < re

   1. Initial program 10.3%

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

   2. Taylor expanded in im around 0 77.0%

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

      1. associate-*l* 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-*l* 77.2%

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

   4. Simplified 77.2%

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

      1. associate-*r* 77.0%

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

      2. sqrt-unprod 77.8%

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

      3. metadata-eval 77.8%

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

      4. metadata-eval 77.8%

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

      5. *-un-lft-identity 77.8%

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

      6. sqrt-div 77.9%

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

      7. metadata-eval 77.9%

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

   6. Applied egg-rr 77.9%

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

      1. associate-*r* 77.9%

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

      2. un-div-inv 77.9%

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

   8. Applied egg-rr 77.9%

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

      1. *-commutative 77.9%

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

      2. *-lft-identity 77.9%

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

      3. times-frac 77.9%

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

      4. /-rgt-identity 77.9%

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

   10. Simplified 77.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.36 \cdot 10^{+25}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{elif}\;re \leq 2.95 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e-72) (* 0.5 (sqrt (* re -4.0))) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e-72) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = sqrt((im * 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 <= (-1.8d-72)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e-72) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e-72:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e-72)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e-72)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e-72], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.8e-72

   1. Initial program 47.6%

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

      1. sqr-neg 47.6%

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

      2. sqr-neg 47.6%

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

      3. 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. Taylor expanded in re around -inf 74.9%

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

      1. *-commutative 74.9%

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

   6. Simplified 74.9%

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

   if -1.8e-72 < re

   1. Initial program 37.7%

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

      1. hypot-udef 73.0%

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

      2. add-sqr-sqrt 72.5%

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

      3. sqrt-unprod 73.0%

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

      4. *-commutative 73.0%

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

      5. *-commutative 73.0%

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

      6. swap-sqr 73.0%

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

      7. add-sqr-sqrt 73.0%

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

      8. metadata-eval 73.0%

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

   3. Applied egg-rr 73.0%

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

      1. *-commutative 73.0%

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

      2. associate-*r* 73.6%

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

      3. metadata-eval 73.6%

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

   5. Simplified 73.6%

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

   6. Taylor expanded in re around 0 61.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]

Alternative 6: 55.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

   1. hypot-udef 83.0%

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

   2. add-sqr-sqrt 82.4%

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

   3. sqrt-unprod 83.0%

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

   4. *-commutative 83.0%

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

   5. *-commutative 83.0%

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

   6. swap-sqr 83.0%

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

   7. add-sqr-sqrt 83.0%

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

   8. metadata-eval 83.0%

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

3. Applied egg-rr 83.0%

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

   1. *-commutative 83.0%

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

   2. associate-*r* 83.4%

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

   3. metadata-eval 83.4%

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

5. Simplified 83.4%

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

6. Taylor expanded in re around 0 53.6%

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

   1. neg-mul-1 53.6%

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

   2. unsub-neg 53.6%

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

8. Simplified 53.6%

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

9. Final simplification 53.6%

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

Alternative 7: 53.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

   1. hypot-udef 83.0%

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

   2. add-sqr-sqrt 82.4%

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

   3. sqrt-unprod 83.0%

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

   4. *-commutative 83.0%

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

   5. *-commutative 83.0%

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

   6. swap-sqr 83.0%

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

   7. add-sqr-sqrt 83.0%

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

   8. metadata-eval 83.0%

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

3. Applied egg-rr 83.0%

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

   1. *-commutative 83.0%

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

   2. associate-*r* 83.4%

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

   3. metadata-eval 83.4%

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

5. Simplified 83.4%

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

6. Taylor expanded in re around 0 50.0%

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

7. Final simplification 50.0%

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

Alternative 8: 6.6% accurate, 71.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

2. Taylor expanded in im around 0 22.7%

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

   1. associate-*l* 22.7%

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

   2. *-commutative 22.7%

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

   3. associate-*l* 22.8%

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

4. Simplified 22.8%

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

   1. add-sqr-sqrt 22.7%

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

   2. pow2 22.7%

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

   3. associate-*r* 22.7%

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

   4. sqrt-unprod 22.9%

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

   5. metadata-eval 22.9%

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

   6. metadata-eval 22.9%

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

   7. *-un-lft-identity 22.9%

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

   8. inv-pow 22.9%

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

   9. sqrt-pow1 22.9%

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

   10. metadata-eval 22.9%

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

6. Applied egg-rr 22.9%

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

   1. unpow2 22.9%

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

   2. add-sqr-sqrt 23.0%

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

   3. metadata-eval 23.0%

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

   4. sqrt-pow2 22.9%

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

   5. inv-pow 22.9%

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

   6. un-div-inv 22.9%

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

   7. *-un-lft-identity 22.9%

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

   8. add-sqr-sqrt 22.9%

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

   9. times-frac 22.8%

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

   10. metadata-eval 22.8%

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

   11. sqrt-div 22.9%

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

   12. inv-pow 22.9%

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

   13. sqrt-pow2 22.9%

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

   14. metadata-eval 22.9%

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

   15. sqrt-pow1 22.9%

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

   16. metadata-eval 22.9%

       \[\leadsto 0.5 \cdot \left({re}^{\color{blue}{-0.25}} \cdot \frac{im}{\sqrt{\sqrt{re}}}\right) \]

   17. add-exp-log 22.0%

       \[\leadsto 0.5 \cdot \left({re}^{-0.25} \cdot \frac{im}{\sqrt{\sqrt{\color{blue}{e^{\log re}}}}}\right) \]

   18. add-sqr-sqrt 17.2%

       \[\leadsto 0.5 \cdot \left({re}^{-0.25} \cdot \frac{im}{\sqrt{\sqrt{e^{\color{blue}{\sqrt{\log re} \cdot \sqrt{\log re}}}}}}\right) \]

   19. sqrt-unprod 18.8%

       \[\leadsto 0.5 \cdot \left({re}^{-0.25} \cdot \frac{im}{\sqrt{\sqrt{e^{\color{blue}{\sqrt{\log re \cdot \log re}}}}}}\right) \]

   20. sqr-neg 18.8%

       \[\leadsto 0.5 \cdot \left({re}^{-0.25} \cdot \frac{im}{\sqrt{\sqrt{e^{\sqrt{\color{blue}{\left(-\log re\right) \cdot \left(-\log re\right)}}}}}}\right) \]

   21. sqrt-unprod 1.4%

       \[\leadsto 0.5 \cdot \left({re}^{-0.25} \cdot \frac{im}{\sqrt{\sqrt{e^{\color{blue}{\sqrt{-\log re} \cdot \sqrt{-\log re}}}}}}\right) \]

   22. add-sqr-sqrt 3.1%

       \[\leadsto 0.5 \cdot \left({re}^{-0.25} \cdot \frac{im}{\sqrt{\sqrt{e^{\color{blue}{-\log re}}}}}\right) \]

8. Applied egg-rr 3.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{-0.25} \cdot \frac{im}{{re}^{-0.25}}\right)} \]
9. Step-by-step derivation

   1. *-commutative 3.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{im}{{re}^{-0.25}} \cdot {re}^{-0.25}\right)} \]

   2. associate-*l/ 3.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im \cdot {re}^{-0.25}}{{re}^{-0.25}}} \]

   3. associate-/l* 3.1%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\frac{{re}^{-0.25}}{{re}^{-0.25}}}} \]

   4. *-inverses 6.3%

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

   5. /-rgt-identity 6.3%

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

10. Simplified 6.3%

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

11. Final simplification 6.3%

    \[\leadsto im \cdot 0.5 \]

Reproduce

herbie shell --seed 2023320 
(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)))))