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

Percentage Accurate: 40.4% → 89.5%

Time: 7.8s

Alternatives: 7

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.4% 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: 89.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.0%

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

   3. Taylor expanded in im around 0 90.1%

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

      1. associate-*l* 90.3%

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

      2. associate-*l* 90.6%

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

   5. Simplified 90.6%

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

      1. add-log-exp 12.7%

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

      2. *-un-lft-identity 12.7%

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

      3. log-prod 12.7%

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

      4. metadata-eval 12.7%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)}\right)\right) \]

      5. add-log-exp 90.6%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)}\right) \]

      6. associate-*r* 90.3%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      7. *-commutative 90.3%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      8. sqrt-unprod 91.2%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      9. metadata-eval 91.2%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      10. metadata-eval 91.2%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{1} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      11. *-un-lft-identity 91.2%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\sqrt{\frac{1}{re}}}\right) \]

      12. sqrt-div 90.9%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}\right) \]

      13. metadata-eval 90.9%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \frac{\color{blue}{1}}{\sqrt{re}}\right) \]

      14. un-div-inv 91.3%

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

   7. Applied egg-rr 91.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   8. Step-by-step derivation

      1. +-lft-identity 91.3%

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

   9. Simplified 91.3%

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

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

   1. Initial program 46.5%

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

      1. sub-neg 46.5%

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

      2. sqr-neg 46.5%

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

      3. sub-neg 46.5%

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

      4. sqr-neg 46.5%

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

      5. hypot-def 92.6%

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

   3. Simplified 92.6%

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

4. Final simplification 92.4%

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

Alternative 2: 75.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00082:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 4.55 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00082)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3.4e+17)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 8.2e+61)
       (* 0.5 (/ im (sqrt re)))
       (if (<= re 4.55e+111)
         (* 0.5 (sqrt (* im 2.0)))
         (* 0.5 (* im (pow re -0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00082) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3.4e+17) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 8.2e+61) {
		tmp = 0.5 * (im / sqrt(re));
	} else if (re <= 4.55e+111) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.00082d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3.4d+17) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 8.2d+61) then
        tmp = 0.5d0 * (im / sqrt(re))
    else if (re <= 4.55d+111) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00082) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3.4e+17) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 8.2e+61) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else if (re <= 4.55e+111) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00082:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3.4e+17:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 8.2e+61:
		tmp = 0.5 * (im / math.sqrt(re))
	elif re <= 4.55e+111:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00082)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3.4e+17)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 8.2e+61)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	elseif (re <= 4.55e+111)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00082)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3.4e+17)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 8.2e+61)
		tmp = 0.5 * (im / sqrt(re));
	elseif (re <= 4.55e+111)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00082], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.4e+17], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.2e+61], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.55e+111], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -8.1999999999999998e-4

   1. Initial program 39.6%

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

   3. Taylor expanded in re around -inf 77.1%

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

      1. *-commutative 77.1%

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

   5. Simplified 77.1%

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

   if -8.1999999999999998e-4 < re < 3.4e17

   1. Initial program 55.2%

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

   3. Taylor expanded in re around 0 84.0%

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

   if 3.4e17 < re < 8.19999999999999944e61

   1. Initial program 23.4%

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

   3. Taylor expanded in im around 0 71.0%

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

      1. associate-*l* 71.0%

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

      2. associate-*l* 71.3%

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

   5. Simplified 71.3%

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

      1. add-log-exp 6.4%

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

      2. *-un-lft-identity 6.4%

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

      3. log-prod 6.4%

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

      4. metadata-eval 6.4%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)}\right)\right) \]

      5. add-log-exp 71.3%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)}\right) \]

      6. associate-*r* 71.0%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      7. *-commutative 71.0%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      8. sqrt-unprod 71.6%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      9. metadata-eval 71.6%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      10. metadata-eval 71.6%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{1} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      11. *-un-lft-identity 71.6%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\sqrt{\frac{1}{re}}}\right) \]

      12. sqrt-div 71.3%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}\right) \]

      13. metadata-eval 71.3%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \frac{\color{blue}{1}}{\sqrt{re}}\right) \]

      14. un-div-inv 71.9%

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

   7. Applied egg-rr 71.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   8. Step-by-step derivation

      1. +-lft-identity 71.9%

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

   9. Simplified 71.9%

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

   if 8.19999999999999944e61 < re < 4.55000000000000002e111

   1. Initial program 41.2%

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

   3. Taylor expanded in re around 0 85.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   4. Step-by-step derivation

      1. *-commutative 85.4%

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

   5. Simplified 85.4%

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

   if 4.55000000000000002e111 < re

   1. Initial program 6.1%

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

   3. Taylor expanded in im around 0 81.2%

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

      1. associate-*l* 81.3%

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

      2. associate-*l* 81.5%

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

   5. Simplified 81.5%

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

      1. associate-*r* 81.3%

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

      2. *-commutative 81.3%

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

      3. sqrt-unprod 82.3%

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

      4. metadata-eval 82.3%

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

      5. metadata-eval 82.3%

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

      6. *-un-lft-identity 82.3%

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

      7. add-log-exp 18.1%

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

      8. *-un-lft-identity 18.1%

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

      9. log-prod 18.1%

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

      10. metadata-eval 18.1%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{\frac{1}{re}}}\right)\right)\right) \]

      11. add-log-exp 82.3%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \color{blue}{\sqrt{\frac{1}{re}}}\right)\right) \]

      12. inv-pow 82.3%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \sqrt{\color{blue}{{re}^{-1}}}\right)\right) \]

      13. sqrt-pow1 82.4%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right)\right) \]

      14. metadata-eval 82.4%

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

   7. Applied egg-rr 82.4%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(0 + {re}^{-0.5}\right)}\right) \]
   8. Step-by-step derivation

      1. +-lft-identity 82.4%

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

   9. Simplified 82.4%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{re}^{-0.5}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00082:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 4.55 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.095:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 4.55 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.095)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.2e+17)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 1.25e+65)
       (* 0.5 (/ 1.0 (/ (sqrt re) im)))
       (if (<= re 4.55e+111)
         (* 0.5 (sqrt (* im 2.0)))
         (* 0.5 (* im (pow re -0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.095) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.2e+17) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 1.25e+65) {
		tmp = 0.5 * (1.0 / (sqrt(re) / im));
	} else if (re <= 4.55e+111) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.095d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.2d+17) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 1.25d+65) then
        tmp = 0.5d0 * (1.0d0 / (sqrt(re) / im))
    else if (re <= 4.55d+111) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.095) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.2e+17) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 1.25e+65) {
		tmp = 0.5 * (1.0 / (Math.sqrt(re) / im));
	} else if (re <= 4.55e+111) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.095:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.2e+17:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 1.25e+65:
		tmp = 0.5 * (1.0 / (math.sqrt(re) / im))
	elif re <= 4.55e+111:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.095)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.2e+17)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 1.25e+65)
		tmp = Float64(0.5 * Float64(1.0 / Float64(sqrt(re) / im)));
	elseif (re <= 4.55e+111)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.095)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.2e+17)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 1.25e+65)
		tmp = 0.5 * (1.0 / (sqrt(re) / im));
	elseif (re <= 4.55e+111)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.095], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.2e+17], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.25e+65], N[(0.5 * N[(1.0 / N[(N[Sqrt[re], $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.55e+111], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -0.095000000000000001

   1. Initial program 39.6%

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

   3. Taylor expanded in re around -inf 77.1%

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

      1. *-commutative 77.1%

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

   5. Simplified 77.1%

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

   if -0.095000000000000001 < re < 4.2e17

   1. Initial program 55.2%

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

   3. Taylor expanded in re around 0 84.0%

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

   if 4.2e17 < re < 1.24999999999999993e65

   1. Initial program 23.4%

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

   3. Taylor expanded in im around 0 71.0%

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

      1. associate-*l* 71.0%

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

      2. associate-*l* 71.3%

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

   5. Simplified 71.3%

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

      1. associate-*r* 71.0%

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

      2. *-commutative 71.0%

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

      3. sqrt-unprod 71.6%

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

      4. metadata-eval 71.6%

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

      5. metadata-eval 71.6%

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

      6. *-un-lft-identity 71.6%

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

      7. add-log-exp 11.3%

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

      8. *-un-lft-identity 11.3%

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

      9. log-prod 11.3%

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

      10. metadata-eval 11.3%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{\frac{1}{re}}}\right)\right)\right) \]

      11. add-log-exp 71.6%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \color{blue}{\sqrt{\frac{1}{re}}}\right)\right) \]

      12. inv-pow 71.6%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \sqrt{\color{blue}{{re}^{-1}}}\right)\right) \]

      13. sqrt-pow1 71.6%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right)\right) \]

      14. metadata-eval 71.6%

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

   7. Applied egg-rr 71.6%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(0 + {re}^{-0.5}\right)}\right) \]
   8. Step-by-step derivation

      1. +-lft-identity 71.6%

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

   9. Simplified 71.6%

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

       1. *-commutative 71.6%

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

       2. metadata-eval 71.6%

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

       3. pow-flip 71.3%

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

       4. pow1/2 71.3%

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

       5. associate-/r/ 72.4%

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

   11. Applied egg-rr 72.4%

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

   if 1.24999999999999993e65 < re < 4.55000000000000002e111

   1. Initial program 41.2%

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

   3. Taylor expanded in re around 0 85.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   4. Step-by-step derivation

      1. *-commutative 85.4%

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

   5. Simplified 85.4%

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

   if 4.55000000000000002e111 < re

   1. Initial program 6.1%

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

   3. Taylor expanded in im around 0 81.2%

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

      1. associate-*l* 81.3%

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

      2. associate-*l* 81.5%

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

   5. Simplified 81.5%

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

      1. associate-*r* 81.3%

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

      2. *-commutative 81.3%

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

      3. sqrt-unprod 82.3%

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

      4. metadata-eval 82.3%

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

      5. metadata-eval 82.3%

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

      6. *-un-lft-identity 82.3%

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

      7. add-log-exp 18.1%

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

      8. *-un-lft-identity 18.1%

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

      9. log-prod 18.1%

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

      10. metadata-eval 18.1%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{\frac{1}{re}}}\right)\right)\right) \]

      11. add-log-exp 82.3%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \color{blue}{\sqrt{\frac{1}{re}}}\right)\right) \]

      12. inv-pow 82.3%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \sqrt{\color{blue}{{re}^{-1}}}\right)\right) \]

      13. sqrt-pow1 82.4%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right)\right) \]

      14. metadata-eval 82.4%

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

   7. Applied egg-rr 82.4%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(0 + {re}^{-0.5}\right)}\right) \]
   8. Step-by-step derivation

      1. +-lft-identity 82.4%

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

   9. Simplified 82.4%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{re}^{-0.5}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.095:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 4.55 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0215:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.55 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0215)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.55e+111)
     (* 0.5 (sqrt (* im 2.0)))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0215) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.55e+111) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0215d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.55d+111) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0215) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.55e+111) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0215:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.55e+111:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0215)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.55e+111)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0215)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.55e+111)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0215], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.55e+111], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.021499999999999998

   1. Initial program 39.6%

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

   3. Taylor expanded in re around -inf 77.1%

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

      1. *-commutative 77.1%

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

   5. Simplified 77.1%

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

   if -0.021499999999999998 < re < 4.55000000000000002e111

   1. Initial program 51.9%

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

   3. Taylor expanded in re around 0 80.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   4. Step-by-step derivation

      1. *-commutative 80.2%

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

   5. Simplified 80.2%

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

   if 4.55000000000000002e111 < re

   1. Initial program 6.1%

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

   3. Taylor expanded in im around 0 81.2%

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

      1. associate-*l* 81.3%

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

      2. associate-*l* 81.5%

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

   5. Simplified 81.5%

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

      1. associate-*r* 81.3%

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

      2. *-commutative 81.3%

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

      3. sqrt-unprod 82.3%

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

      4. metadata-eval 82.3%

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

      5. metadata-eval 82.3%

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

      6. *-un-lft-identity 82.3%

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

      7. add-log-exp 18.1%

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

      8. *-un-lft-identity 18.1%

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

      9. log-prod 18.1%

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

      10. metadata-eval 18.1%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{\frac{1}{re}}}\right)\right)\right) \]

      11. add-log-exp 82.3%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \color{blue}{\sqrt{\frac{1}{re}}}\right)\right) \]

      12. inv-pow 82.3%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \sqrt{\color{blue}{{re}^{-1}}}\right)\right) \]

      13. sqrt-pow1 82.4%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(0 + \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right)\right) \]

      14. metadata-eval 82.4%

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

   7. Applied egg-rr 82.4%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(0 + {re}^{-0.5}\right)}\right) \]
   8. Step-by-step derivation

      1. +-lft-identity 82.4%

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

   9. Simplified 82.4%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0215:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.55 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.55 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.8e-7)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.55e+111) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-7) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.55e+111) {
		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 <= (-4.8d-7)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.55d+111) 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 <= -4.8e-7) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.55e+111) {
		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 <= -4.8e-7:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.55e+111:
		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 <= -4.8e-7)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.55e+111)
		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 <= -4.8e-7)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.55e+111)
		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, -4.8e-7], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.55e+111], 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 3 regimes

2. if re < -4.79999999999999957e-7

   1. Initial program 39.6%

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

   3. Taylor expanded in re around -inf 77.1%

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

      1. *-commutative 77.1%

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

   5. Simplified 77.1%

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

   if -4.79999999999999957e-7 < re < 4.55000000000000002e111

   1. Initial program 51.9%

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

   3. Taylor expanded in re around 0 80.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   4. Step-by-step derivation

      1. *-commutative 80.2%

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

   5. Simplified 80.2%

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

   if 4.55000000000000002e111 < re

   1. Initial program 6.1%

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

   3. Taylor expanded in im around 0 81.2%

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

      1. associate-*l* 81.3%

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

      2. associate-*l* 81.5%

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

   5. Simplified 81.5%

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

      1. add-log-exp 21.5%

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

      2. *-un-lft-identity 21.5%

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

      3. log-prod 21.5%

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

      4. metadata-eval 21.5%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)}\right)\right) \]

      5. add-log-exp 81.5%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)}\right) \]

      6. associate-*r* 81.3%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      7. *-commutative 81.3%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right)} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      8. sqrt-unprod 82.3%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      9. metadata-eval 82.3%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      10. metadata-eval 82.3%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{1} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      11. *-un-lft-identity 82.3%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\sqrt{\frac{1}{re}}}\right) \]

      12. sqrt-div 82.1%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}\right) \]

      13. metadata-eval 82.1%

          \[\leadsto 0.5 \cdot \left(0 + im \cdot \frac{\color{blue}{1}}{\sqrt{re}}\right) \]

      14. un-div-inv 82.3%

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

   7. Applied egg-rr 82.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   8. Step-by-step derivation

      1. +-lft-identity 82.3%

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

   9. Simplified 82.3%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.55 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0072:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0072) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0072) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0072d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0072) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0072:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0072)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0072)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0072], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0071999999999999998

   1. Initial program 39.6%

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

   3. Taylor expanded in re around -inf 77.1%

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

      1. *-commutative 77.1%

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

   5. Simplified 77.1%

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

   if -0.0071999999999999998 < re

   1. Initial program 42.0%

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

   3. Taylor expanded in re around 0 67.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   4. Step-by-step derivation

      1. *-commutative 67.6%

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

   5. Simplified 67.6%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0072:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.3% 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 41.4%

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

3. Taylor expanded in re around 0 57.6%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
4. Step-by-step derivation

   1. *-commutative 57.6%

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

5. Simplified 57.6%

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

6. Final simplification 57.6%

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

Reproduce

herbie shell --seed 2024019 
(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)))))