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

Percentage Accurate: 41.4% → 90.3%

Time: 9.7s

Alternatives: 6

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 90.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.4%

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

   3. Taylor expanded in re around inf 38.9%

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

      1. associate-*r* 38.9%

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

      2. metadata-eval 38.9%

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

      3. *-un-lft-identity 38.9%

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

      4. div-inv 39.0%

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

      5. sqrt-prod 45.5%

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

      6. unpow2 45.5%

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

      7. sqrt-prod 96.9%

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

      8. add-log-exp 10.1%

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

      9. add-sqr-sqrt 10.1%

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

      10. pow1 10.1%

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

      11. metadata-eval 10.1%

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

      12. metadata-eval 10.1%

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

      13. sqrt-unprod 10.1%

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

      14. exp-prod 10.1%

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

      15. associate-*l* 10.1%

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

      16. add-log-exp 96.5%

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

      17. *-commutative 96.5%

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

   5. Applied egg-rr 97.4%

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

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

   1. Initial program 43.3%

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

      1. sub-neg 43.3%

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

      2. sqr-neg 43.3%

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

      3. sub-neg 43.3%

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

      4. sqr-neg 43.3%

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

      5. hypot-define 88.6%

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

   3. Simplified 88.6%

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

4. Final simplification 89.9%

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

Alternative 2: 72.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq -5.4 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 9.6 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re -5.4e+60)
     (* 0.5 (sqrt (* 2.0 (* re -2.0))))
     (if (<= re 6e-164)
       t_0
       (if (<= re 5.4e-89)
         (* 0.5 (* im (pow re -0.5)))
         (if (<= re 9.6e-49) t_0 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((im * 2.0));
	double tmp;
	if (re <= -5.4e+60) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 6e-164) {
		tmp = t_0;
	} else if (re <= 5.4e-89) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else if (re <= 9.6e-49) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((im * 2.0d0))
    if (re <= (-5.4d+60)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 6d-164) then
        tmp = t_0
    else if (re <= 5.4d-89) then
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    else if (re <= 9.6d-49) then
        tmp = t_0
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((im * 2.0));
	double tmp;
	if (re <= -5.4e+60) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 6e-164) {
		tmp = t_0;
	} else if (re <= 5.4e-89) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else if (re <= 9.6e-49) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((im * 2.0))
	tmp = 0
	if re <= -5.4e+60:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 6e-164:
		tmp = t_0
	elif re <= 5.4e-89:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	elif re <= 9.6e-49:
		tmp = t_0
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	tmp = 0.0
	if (re <= -5.4e+60)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 6e-164)
		tmp = t_0;
	elseif (re <= 5.4e-89)
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	elseif (re <= 9.6e-49)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((im * 2.0));
	tmp = 0.0;
	if (re <= -5.4e+60)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 6e-164)
		tmp = t_0;
	elseif (re <= 5.4e-89)
		tmp = 0.5 * (im * (re ^ -0.5));
	elseif (re <= 9.6e-49)
		tmp = t_0;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -5.4e+60], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6e-164], t$95$0, If[LessEqual[re, 5.4e-89], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.6e-49], t$95$0, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -5.3999999999999999e60

   1. Initial program 30.0%

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

   3. Taylor expanded in re around -inf 84.3%

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

      1. *-commutative 84.3%

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

   5. Simplified 84.3%

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

   if -5.3999999999999999e60 < re < 6.0000000000000002e-164 or 5.39999999999999975e-89 < re < 9.59999999999999969e-49

   1. Initial program 60.9%

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

   3. Taylor expanded in re around 0 80.8%

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

   4. Taylor expanded in im around inf 80.3%

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

   if 6.0000000000000002e-164 < re < 5.39999999999999975e-89

   1. Initial program 20.2%

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

   3. Taylor expanded in re around inf 8.5%

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

      1. associate-*r* 8.5%

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

      2. metadata-eval 8.5%

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

      3. *-un-lft-identity 8.5%

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

      4. div-inv 8.5%

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

      5. sqrt-prod 8.5%

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

      6. unpow2 8.5%

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

      7. sqrt-prod 63.4%

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

      8. add-log-exp 4.4%

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

      9. add-sqr-sqrt 4.4%

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

      10. pow1 4.4%

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

      11. metadata-eval 4.4%

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

      12. metadata-eval 4.4%

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

      13. sqrt-unprod 4.4%

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

      14. exp-prod 4.4%

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

      15. associate-*l* 4.4%

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

      16. add-log-exp 62.8%

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

      17. *-commutative 62.8%

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

   5. Applied egg-rr 63.5%

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

   if 9.59999999999999969e-49 < re

   1. Initial program 14.6%

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

   3. Taylor expanded in re around inf 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. associate-*r* 45.5%

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

      3. metadata-eval 45.5%

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

      4. *-un-lft-identity 45.5%

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

      5. sqrt-div 58.1%

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

      6. sqrt-pow1 76.9%

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

      7. metadata-eval 76.9%

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

      8. pow1 76.9%

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

   5. Applied egg-rr 76.9%

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

      1. *-lft-identity 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.4 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 9.6 \cdot 10^{-49}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{+87}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 5.7 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 5.9 \cdot 10^{-49}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6e+87)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 5.7e-164)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 5.4e-89)
       (* 0.5 (* im (pow re -0.5)))
       (if (<= re 5.9e-49)
         (* 0.5 (sqrt (* im 2.0)))
         (* 0.5 (/ im (sqrt re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6e+87) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 5.7e-164) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 5.4e-89) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else if (re <= 5.9e-49) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6d+87)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 5.7d-164) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 5.4d-89) then
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    else if (re <= 5.9d-49) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6e+87) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 5.7e-164) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 5.4e-89) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else if (re <= 5.9e-49) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6e+87:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 5.7e-164:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 5.4e-89:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	elif re <= 5.9e-49:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6e+87)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 5.7e-164)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 5.4e-89)
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	elseif (re <= 5.9e-49)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6e+87)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 5.7e-164)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 5.4e-89)
		tmp = 0.5 * (im * (re ^ -0.5));
	elseif (re <= 5.9e-49)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6e+87], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.7e-164], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.4e-89], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.9e-49], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -5.9999999999999998e87

   1. Initial program 26.2%

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

   3. Taylor expanded in re around -inf 86.8%

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

      1. *-commutative 86.8%

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

   5. Simplified 86.8%

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

   if -5.9999999999999998e87 < re < 5.70000000000000023e-164

   1. Initial program 63.0%

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

   3. Taylor expanded in re around 0 81.4%

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

   if 5.70000000000000023e-164 < re < 5.39999999999999975e-89

   1. Initial program 20.2%

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

   3. Taylor expanded in re around inf 8.5%

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

      1. associate-*r* 8.5%

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

      2. metadata-eval 8.5%

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

      3. *-un-lft-identity 8.5%

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

      4. div-inv 8.5%

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

      5. sqrt-prod 8.5%

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

      6. unpow2 8.5%

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

      7. sqrt-prod 63.4%

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

      8. add-log-exp 4.4%

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

      9. add-sqr-sqrt 4.4%

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

      10. pow1 4.4%

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

      11. metadata-eval 4.4%

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

      12. metadata-eval 4.4%

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

      13. sqrt-unprod 4.4%

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

      14. exp-prod 4.4%

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

      15. associate-*l* 4.4%

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

      16. add-log-exp 62.8%

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

      17. *-commutative 62.8%

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

   5. Applied egg-rr 63.5%

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

   if 5.39999999999999975e-89 < re < 5.90000000000000037e-49

   1. Initial program 40.1%

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

   3. Taylor expanded in re around 0 88.1%

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

   4. Taylor expanded in im around inf 88.4%

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

   if 5.90000000000000037e-49 < re

   1. Initial program 14.6%

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

   3. Taylor expanded in re around inf 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. associate-*r* 45.5%

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

      3. metadata-eval 45.5%

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

      4. *-un-lft-identity 45.5%

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

      5. sqrt-div 58.1%

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

      6. sqrt-pow1 76.9%

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

      7. metadata-eval 76.9%

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

      8. pow1 76.9%

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

   5. Applied egg-rr 76.9%

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

      1. *-lft-identity 76.9%

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

   7. Simplified 76.9%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{+87}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 5.7 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 5.9 \cdot 10^{-49}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6 \cdot 10^{-164} \lor \neg \left(re \leq 6 \cdot 10^{-89}\right) \land re \leq 4 \cdot 10^{-49}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 6e-164) (and (not (<= re 6e-89)) (<= re 4e-49)))
   (* 0.5 (sqrt (* im 2.0)))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 6e-164) || (!(re <= 6e-89) && (re <= 4e-49))) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= 6d-164) .or. (.not. (re <= 6d-89)) .and. (re <= 4d-49)) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 6e-164) || (!(re <= 6e-89) && (re <= 4e-49))) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 6e-164) or (not (re <= 6e-89) and (re <= 4e-49)):
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 6e-164) || (!(re <= 6e-89) && (re <= 4e-49)))
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 6e-164) || (~((re <= 6e-89)) && (re <= 4e-49)))
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 6e-164], And[N[Not[LessEqual[re, 6e-89]], $MachinePrecision], LessEqual[re, 4e-49]]], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.0000000000000002e-164 or 5.9999999999999999e-89 < re < 3.99999999999999975e-49

   1. Initial program 51.1%

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

   3. Taylor expanded in re around 0 61.9%

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

   4. Taylor expanded in im around inf 61.4%

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

   if 6.0000000000000002e-164 < re < 5.9999999999999999e-89 or 3.99999999999999975e-49 < re

   1. Initial program 15.7%

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

   3. Taylor expanded in re around inf 38.3%

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

      1. *-un-lft-identity 38.3%

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

      2. associate-*r* 38.3%

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

      3. metadata-eval 38.3%

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

      4. *-un-lft-identity 38.3%

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

      5. sqrt-div 48.5%

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

      6. sqrt-pow1 74.3%

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

      7. metadata-eval 74.3%

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

      8. pow1 74.3%

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

   5. Applied egg-rr 74.3%

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

      1. *-lft-identity 74.3%

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

   7. Simplified 74.3%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6 \cdot 10^{-164} \lor \neg \left(re \leq 6 \cdot 10^{-89}\right) \land re \leq 4 \cdot 10^{-49}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq 6 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-50}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re 6e-164)
     t_0
     (if (<= re 7e-89)
       (* 0.5 (* im (pow re -0.5)))
       (if (<= re 1.6e-50) t_0 (* 0.5 (/ im (sqrt re))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((im * 2.0));
	double tmp;
	if (re <= 6e-164) {
		tmp = t_0;
	} else if (re <= 7e-89) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else if (re <= 1.6e-50) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((im * 2.0d0))
    if (re <= 6d-164) then
        tmp = t_0
    else if (re <= 7d-89) then
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    else if (re <= 1.6d-50) then
        tmp = t_0
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((im * 2.0));
	double tmp;
	if (re <= 6e-164) {
		tmp = t_0;
	} else if (re <= 7e-89) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else if (re <= 1.6e-50) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((im * 2.0))
	tmp = 0
	if re <= 6e-164:
		tmp = t_0
	elif re <= 7e-89:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	elif re <= 1.6e-50:
		tmp = t_0
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	tmp = 0.0
	if (re <= 6e-164)
		tmp = t_0;
	elseif (re <= 7e-89)
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	elseif (re <= 1.6e-50)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((im * 2.0));
	tmp = 0.0;
	if (re <= 6e-164)
		tmp = t_0;
	elseif (re <= 7e-89)
		tmp = 0.5 * (im * (re ^ -0.5));
	elseif (re <= 1.6e-50)
		tmp = t_0;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 6e-164], t$95$0, If[LessEqual[re, 7e-89], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.6e-50], t$95$0, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 6.0000000000000002e-164 or 6.9999999999999994e-89 < re < 1.6e-50

   1. Initial program 51.1%

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

   3. Taylor expanded in re around 0 61.9%

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

   4. Taylor expanded in im around inf 61.4%

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

   if 6.0000000000000002e-164 < re < 6.9999999999999994e-89

   1. Initial program 20.2%

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

   3. Taylor expanded in re around inf 8.5%

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

      1. associate-*r* 8.5%

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

      2. metadata-eval 8.5%

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

      3. *-un-lft-identity 8.5%

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

      4. div-inv 8.5%

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

      5. sqrt-prod 8.5%

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

      6. unpow2 8.5%

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

      7. sqrt-prod 63.4%

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

      8. add-log-exp 4.4%

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

      9. add-sqr-sqrt 4.4%

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

      10. pow1 4.4%

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

      11. metadata-eval 4.4%

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

      12. metadata-eval 4.4%

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

      13. sqrt-unprod 4.4%

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

      14. exp-prod 4.4%

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

      15. associate-*l* 4.4%

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

      16. add-log-exp 62.8%

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

      17. *-commutative 62.8%

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

   5. Applied egg-rr 63.5%

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

   if 1.6e-50 < re

   1. Initial program 14.6%

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

   3. Taylor expanded in re around inf 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. associate-*r* 45.5%

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

      3. metadata-eval 45.5%

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

      4. *-un-lft-identity 45.5%

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

      5. sqrt-div 58.1%

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

      6. sqrt-pow1 76.9%

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

      7. metadata-eval 76.9%

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

      8. pow1 76.9%

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

   5. Applied egg-rr 76.9%

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

      1. *-lft-identity 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

3. Taylor expanded in re around 0 50.1%

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

4. Taylor expanded in im around inf 50.1%

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

5. Final simplification 50.1%

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

Reproduce

herbie shell --seed 2024112 
(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)))))