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

Percentage Accurate: 49.4% → 92.5%

Time: 12.1s

Alternatives: 11

Speedup: 1.9×

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: 49.4% 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: 92.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

   1. *-commutative 53.5%

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

   2. hypot-udef 92.2%

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

   3. sqrt-prod 92.4%

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

4. Applied egg-rr 92.4%

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

5. Final simplification 92.4%

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

Alternative 2: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+206}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.2e+39)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.15e+32)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 2.1e+206)
       (* 0.5 (sqrt (/ (pow im 2.0) re)))
       (* 0.5 (* im (sqrt (/ 1.0 re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.2e+39) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.15e+32) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 2.1e+206) {
		tmp = 0.5 * sqrt((pow(im, 2.0) / re));
	} else {
		tmp = 0.5 * (im * sqrt((1.0 / 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 <= (-5.2d+39)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.15d+32) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 2.1d+206) then
        tmp = 0.5d0 * sqrt(((im ** 2.0d0) / re))
    else
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5.2e+39) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.15e+32) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 2.1e+206) {
		tmp = 0.5 * Math.sqrt((Math.pow(im, 2.0) / re));
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.2e+39:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.15e+32:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 2.1e+206:
		tmp = 0.5 * math.sqrt((math.pow(im, 2.0) / re))
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.2e+39)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.15e+32)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 2.1e+206)
		tmp = Float64(0.5 * sqrt(Float64((im ^ 2.0) / re)));
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5.2e+39)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.15e+32)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 2.1e+206)
		tmp = 0.5 * sqrt(((im ^ 2.0) / re));
	else
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.2e+39], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.15e+32], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+206], N[(0.5 * N[Sqrt[N[(N[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -5.2e39

   1. Initial program 49.5%

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

   3. Taylor expanded in re around -inf 83.2%

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

      1. *-commutative 83.2%

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

   5. Simplified 83.2%

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

   if -5.2e39 < re < 2.1499999999999999e32

   1. Initial program 62.4%

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

   3. Taylor expanded in re around 0 78.4%

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

   if 2.1499999999999999e32 < re < 2.09999999999999987e206

   1. Initial program 49.1%

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

   3. Taylor expanded in re around inf 62.2%

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

   if 2.09999999999999987e206 < re

   1. Initial program 3.0%

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

   3. Taylor expanded in im around 0 61.8%

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

      1. associate-*l* 61.7%

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

      2. associate-*l* 61.9%

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

   5. Simplified 61.9%

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

      1. associate-*r* 61.7%

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

      2. sqrt-unprod 62.5%

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

      3. metadata-eval 62.5%

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

      4. sqrt-prod 62.5%

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

      5. *-un-lft-identity 62.5%

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

   7. Applied egg-rr 62.5%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\sqrt{\frac{1}{re}}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+206}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{re \cdot -2}\right)\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+209}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4e+39)
   (* 0.5 (* (sqrt 2.0) (sqrt (* re -2.0))))
   (if (<= re 6.4e+32)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 8e+209)
       (* 0.5 (sqrt (/ (pow im 2.0) re)))
       (* 0.5 (* im (sqrt (/ 1.0 re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4e+39) {
		tmp = 0.5 * (sqrt(2.0) * sqrt((re * -2.0)));
	} else if (re <= 6.4e+32) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 8e+209) {
		tmp = 0.5 * sqrt((pow(im, 2.0) / re));
	} else {
		tmp = 0.5 * (im * sqrt((1.0 / 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 <= (-4d+39)) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt((re * (-2.0d0))))
    else if (re <= 6.4d+32) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 8d+209) then
        tmp = 0.5d0 * sqrt(((im ** 2.0d0) / re))
    else
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4e+39) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt((re * -2.0)));
	} else if (re <= 6.4e+32) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 8e+209) {
		tmp = 0.5 * Math.sqrt((Math.pow(im, 2.0) / re));
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4e+39:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt((re * -2.0)))
	elif re <= 6.4e+32:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 8e+209:
		tmp = 0.5 * math.sqrt((math.pow(im, 2.0) / re))
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4e+39)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(re * -2.0))));
	elseif (re <= 6.4e+32)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 8e+209)
		tmp = Float64(0.5 * sqrt(Float64((im ^ 2.0) / re)));
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4e+39)
		tmp = 0.5 * (sqrt(2.0) * sqrt((re * -2.0)));
	elseif (re <= 6.4e+32)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 8e+209)
		tmp = 0.5 * sqrt(((im ^ 2.0) / re));
	else
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4e+39], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(re * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.4e+32], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8e+209], N[(0.5 * N[Sqrt[N[(N[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.99999999999999976e39

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. hypot-udef 98.3%

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

      3. sqrt-prod 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in re around -inf 84.3%

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

      1. *-commutative 84.3%

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

   7. Simplified 84.3%

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

   if -3.99999999999999976e39 < re < 6.3999999999999998e32

   1. Initial program 62.4%

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

   3. Taylor expanded in re around 0 78.4%

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

   if 6.3999999999999998e32 < re < 8.0000000000000006e209

   1. Initial program 49.1%

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

   3. Taylor expanded in re around inf 62.2%

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

   if 8.0000000000000006e209 < re

   1. Initial program 3.0%

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

   3. Taylor expanded in im around 0 61.8%

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

      1. associate-*l* 61.7%

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

      2. associate-*l* 61.9%

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

   5. Simplified 61.9%

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

      1. associate-*r* 61.7%

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

      2. sqrt-unprod 62.5%

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

      3. metadata-eval 62.5%

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

      4. sqrt-prod 62.5%

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

      5. *-un-lft-identity 62.5%

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

   7. Applied egg-rr 62.5%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\sqrt{\frac{1}{re}}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{re \cdot -2}\right)\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+209}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

   1. sub-neg 53.5%

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

   2. sqr-neg 53.5%

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

   3. sub-neg 53.5%

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

   4. sqr-neg 53.5%

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

   5. hypot-def 92.2%

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

3. Simplified 92.2%

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

5. Final simplification 92.2%

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

Alternative 5: 67.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq -1.3 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -1.75 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.16 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;0.5 \cdot \left(\left(im + 1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re -1.3e+31)
     t_0
     (if (<= re -1.75e-39)
       t_1
       (if (<= re -4.2e-129)
         t_0
         (if (<= re 1.16e+33)
           t_1
           (if (<= re 1.2e+167)
             (* 0.5 (+ (+ im 1.0) -1.0))
             (* 0.5 (* im (pow re -0.5))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((im * 2.0));
	double tmp;
	if (re <= -1.3e+31) {
		tmp = t_0;
	} else if (re <= -1.75e-39) {
		tmp = t_1;
	} else if (re <= -4.2e-129) {
		tmp = t_0;
	} else if (re <= 1.16e+33) {
		tmp = t_1;
	} else if (re <= 1.2e+167) {
		tmp = 0.5 * ((im + 1.0) + -1.0);
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((im * 2.0d0))
    if (re <= (-1.3d+31)) then
        tmp = t_0
    else if (re <= (-1.75d-39)) then
        tmp = t_1
    else if (re <= (-4.2d-129)) then
        tmp = t_0
    else if (re <= 1.16d+33) then
        tmp = t_1
    else if (re <= 1.2d+167) then
        tmp = 0.5d0 * ((im + 1.0d0) + (-1.0d0))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((im * 2.0));
	double tmp;
	if (re <= -1.3e+31) {
		tmp = t_0;
	} else if (re <= -1.75e-39) {
		tmp = t_1;
	} else if (re <= -4.2e-129) {
		tmp = t_0;
	} else if (re <= 1.16e+33) {
		tmp = t_1;
	} else if (re <= 1.2e+167) {
		tmp = 0.5 * ((im + 1.0) + -1.0);
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((im * 2.0))
	tmp = 0
	if re <= -1.3e+31:
		tmp = t_0
	elif re <= -1.75e-39:
		tmp = t_1
	elif re <= -4.2e-129:
		tmp = t_0
	elif re <= 1.16e+33:
		tmp = t_1
	elif re <= 1.2e+167:
		tmp = 0.5 * ((im + 1.0) + -1.0)
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	tmp = 0.0
	if (re <= -1.3e+31)
		tmp = t_0;
	elseif (re <= -1.75e-39)
		tmp = t_1;
	elseif (re <= -4.2e-129)
		tmp = t_0;
	elseif (re <= 1.16e+33)
		tmp = t_1;
	elseif (re <= 1.2e+167)
		tmp = Float64(0.5 * Float64(Float64(im + 1.0) + -1.0));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * sqrt((im * 2.0));
	tmp = 0.0;
	if (re <= -1.3e+31)
		tmp = t_0;
	elseif (re <= -1.75e-39)
		tmp = t_1;
	elseif (re <= -4.2e-129)
		tmp = t_0;
	elseif (re <= 1.16e+33)
		tmp = t_1;
	elseif (re <= 1.2e+167)
		tmp = 0.5 * ((im + 1.0) + -1.0);
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.3e+31], t$95$0, If[LessEqual[re, -1.75e-39], t$95$1, If[LessEqual[re, -4.2e-129], t$95$0, If[LessEqual[re, 1.16e+33], t$95$1, If[LessEqual[re, 1.2e+167], N[(0.5 * N[(N[(im + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.3e31 or -1.75e-39 < re < -4.2e-129

   1. Initial program 60.7%

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

   3. Taylor expanded in re around -inf 77.5%

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

      1. *-commutative 77.5%

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

   5. Simplified 77.5%

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

   if -1.3e31 < re < -1.75e-39 or -4.2e-129 < re < 1.16000000000000001e33

   1. Initial program 57.7%

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

   3. Taylor expanded in re around 0 83.8%

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

      1. *-commutative 83.8%

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

   5. Simplified 83.8%

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

   if 1.16000000000000001e33 < re < 1.19999999999999999e167

   1. Initial program 57.7%

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

   3. Taylor expanded in im around 0 25.2%

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

      1. associate-*l* 25.3%

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

      2. associate-*l* 25.4%

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

   5. Simplified 25.4%

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

      1. add-cbrt-cube 70.6%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}} \]

      2. pow3 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}^{3}}} \]

      3. associate-*r* 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)}^{3}} \]

      4. sqrt-unprod 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\color{blue}{\sqrt{0.5 \cdot 2}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

      5. metadata-eval 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

      6. sqrt-prod 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\sqrt{1 \cdot \frac{1}{re}}}\right)}^{3}} \]

      7. *-un-lft-identity 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{\frac{1}{re}}}\right)}^{3}} \]

      8. inv-pow 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{{re}^{-1}}}\right)}^{3}} \]

      9. sqrt-pow1 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right)}^{3}} \]

      10. metadata-eval 70.6%

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

   7. Applied egg-rr 70.6%

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

   9. Applied egg-rr 64.9%

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

   if 1.19999999999999999e167 < re

   1. Initial program 3.1%

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

   3. Taylor expanded in im around 0 56.0%

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

      1. associate-*l* 56.0%

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

      2. associate-*l* 56.1%

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

   5. Simplified 56.1%

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

      1. associate-*r* 56.0%

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

      2. sqrt-unprod 56.7%

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

      3. metadata-eval 56.7%

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

      4. metadata-eval 56.7%

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

      5. expm1-log1p-u 56.7%

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

      6. *-un-lft-identity 56.7%

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

      7. expm1-udef 35.1%

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

      8. inv-pow 35.1%

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

      9. sqrt-pow1 35.1%

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

      10. metadata-eval 35.1%

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

   7. Applied egg-rr 35.1%

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

      1. expm1-def 56.7%

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

      2. expm1-log1p 56.7%

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

   9. Simplified 56.7%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{re}^{-0.5}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.75 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-129}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.16 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;0.5 \cdot \left(\left(im + 1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq -2.5 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -5.1 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im + 1\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re -2.5e+23)
     t_0
     (if (<= re -5.1e-39)
       t_1
       (if (<= re -4.2e-129)
         t_0
         (if (<= re 4.2e+26) t_1 (* 0.5 (+ (+ im 1.0) -1.0))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((im * 2.0));
	double tmp;
	if (re <= -2.5e+23) {
		tmp = t_0;
	} else if (re <= -5.1e-39) {
		tmp = t_1;
	} else if (re <= -4.2e-129) {
		tmp = t_0;
	} else if (re <= 4.2e+26) {
		tmp = t_1;
	} else {
		tmp = 0.5 * ((im + 1.0) + -1.0);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((im * 2.0d0))
    if (re <= (-2.5d+23)) then
        tmp = t_0
    else if (re <= (-5.1d-39)) then
        tmp = t_1
    else if (re <= (-4.2d-129)) then
        tmp = t_0
    else if (re <= 4.2d+26) then
        tmp = t_1
    else
        tmp = 0.5d0 * ((im + 1.0d0) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((im * 2.0));
	double tmp;
	if (re <= -2.5e+23) {
		tmp = t_0;
	} else if (re <= -5.1e-39) {
		tmp = t_1;
	} else if (re <= -4.2e-129) {
		tmp = t_0;
	} else if (re <= 4.2e+26) {
		tmp = t_1;
	} else {
		tmp = 0.5 * ((im + 1.0) + -1.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((im * 2.0))
	tmp = 0
	if re <= -2.5e+23:
		tmp = t_0
	elif re <= -5.1e-39:
		tmp = t_1
	elif re <= -4.2e-129:
		tmp = t_0
	elif re <= 4.2e+26:
		tmp = t_1
	else:
		tmp = 0.5 * ((im + 1.0) + -1.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	tmp = 0.0
	if (re <= -2.5e+23)
		tmp = t_0;
	elseif (re <= -5.1e-39)
		tmp = t_1;
	elseif (re <= -4.2e-129)
		tmp = t_0;
	elseif (re <= 4.2e+26)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(Float64(im + 1.0) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * sqrt((im * 2.0));
	tmp = 0.0;
	if (re <= -2.5e+23)
		tmp = t_0;
	elseif (re <= -5.1e-39)
		tmp = t_1;
	elseif (re <= -4.2e-129)
		tmp = t_0;
	elseif (re <= 4.2e+26)
		tmp = t_1;
	else
		tmp = 0.5 * ((im + 1.0) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.5e+23], t$95$0, If[LessEqual[re, -5.1e-39], t$95$1, If[LessEqual[re, -4.2e-129], t$95$0, If[LessEqual[re, 4.2e+26], t$95$1, N[(0.5 * N[(N[(im + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.5e23 or -5.09999999999999988e-39 < re < -4.2e-129

   1. Initial program 60.7%

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

   3. Taylor expanded in re around -inf 77.5%

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

      1. *-commutative 77.5%

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

   5. Simplified 77.5%

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

   if -2.5e23 < re < -5.09999999999999988e-39 or -4.2e-129 < re < 4.2000000000000002e26

   1. Initial program 57.7%

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

   3. Taylor expanded in re around 0 83.8%

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

      1. *-commutative 83.8%

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

   5. Simplified 83.8%

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

   if 4.2000000000000002e26 < re

   1. Initial program 34.3%

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

   3. Taylor expanded in im around 0 38.4%

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

      1. associate-*l* 38.4%

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

      2. associate-*l* 38.5%

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

   5. Simplified 38.5%

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

      1. add-cbrt-cube 67.7%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}} \]

      2. pow3 67.8%

         \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}^{3}}} \]

      3. associate-*r* 67.7%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)}^{3}} \]

      4. sqrt-unprod 67.8%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\color{blue}{\sqrt{0.5 \cdot 2}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

      5. metadata-eval 67.8%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

      6. sqrt-prod 67.8%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\sqrt{1 \cdot \frac{1}{re}}}\right)}^{3}} \]

      7. *-un-lft-identity 67.8%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{\frac{1}{re}}}\right)}^{3}} \]

      8. inv-pow 67.8%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{{re}^{-1}}}\right)}^{3}} \]

      9. sqrt-pow1 67.8%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right)}^{3}} \]

      10. metadata-eval 67.8%

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

   7. Applied egg-rr 67.8%

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

   9. Applied egg-rr 49.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im + 1\right) - 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -5.1 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-129}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im + 1\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.7 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 8.1 \cdot 10^{+170}:\\ \;\;\;\;0.5 \cdot \left(\left(im + 1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.7e+39)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 8e+31)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 8.1e+170)
       (* 0.5 (+ (+ im 1.0) -1.0))
       (* 0.5 (* im (sqrt (/ 1.0 re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.7e+39) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 8e+31) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 8.1e+170) {
		tmp = 0.5 * ((im + 1.0) + -1.0);
	} else {
		tmp = 0.5 * (im * sqrt((1.0 / 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 <= (-4.7d+39)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 8d+31) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 8.1d+170) then
        tmp = 0.5d0 * ((im + 1.0d0) + (-1.0d0))
    else
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.7e+39) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 8e+31) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 8.1e+170) {
		tmp = 0.5 * ((im + 1.0) + -1.0);
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.7e+39:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 8e+31:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 8.1e+170:
		tmp = 0.5 * ((im + 1.0) + -1.0)
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.7e+39)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 8e+31)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 8.1e+170)
		tmp = Float64(0.5 * Float64(Float64(im + 1.0) + -1.0));
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.7e+39)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 8e+31)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 8.1e+170)
		tmp = 0.5 * ((im + 1.0) + -1.0);
	else
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.7e+39], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8e+31], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.1e+170], N[(0.5 * N[(N[(im + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.6999999999999999e39

   1. Initial program 49.5%

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

   3. Taylor expanded in re around -inf 83.2%

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

      1. *-commutative 83.2%

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

   5. Simplified 83.2%

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

   if -4.6999999999999999e39 < re < 7.9999999999999997e31

   1. Initial program 62.4%

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

   3. Taylor expanded in re around 0 78.4%

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

   if 7.9999999999999997e31 < re < 8.09999999999999965e170

   1. Initial program 57.7%

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

   3. Taylor expanded in im around 0 25.2%

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

      1. associate-*l* 25.3%

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

      2. associate-*l* 25.4%

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

   5. Simplified 25.4%

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

      1. add-cbrt-cube 70.6%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}} \]

      2. pow3 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}^{3}}} \]

      3. associate-*r* 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)}^{3}} \]

      4. sqrt-unprod 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\color{blue}{\sqrt{0.5 \cdot 2}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

      5. metadata-eval 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

      6. sqrt-prod 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\sqrt{1 \cdot \frac{1}{re}}}\right)}^{3}} \]

      7. *-un-lft-identity 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{\frac{1}{re}}}\right)}^{3}} \]

      8. inv-pow 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{{re}^{-1}}}\right)}^{3}} \]

      9. sqrt-pow1 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right)}^{3}} \]

      10. metadata-eval 70.6%

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

   7. Applied egg-rr 70.6%

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

   9. Applied egg-rr 64.9%

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

   if 8.09999999999999965e170 < re

   1. Initial program 3.1%

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

   3. Taylor expanded in im around 0 56.0%

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

      1. associate-*l* 56.0%

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

      2. associate-*l* 56.1%

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

   5. Simplified 56.1%

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

      1. associate-*r* 56.0%

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

      2. sqrt-unprod 56.7%

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

      3. metadata-eval 56.7%

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

      4. sqrt-prod 56.7%

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

      5. *-un-lft-identity 56.7%

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

   7. Applied egg-rr 56.7%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\sqrt{\frac{1}{re}}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.7 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 8.1 \cdot 10^{+170}:\\ \;\;\;\;0.5 \cdot \left(\left(im + 1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+170}:\\ \;\;\;\;0.5 \cdot \left(\left(im + 1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.6e+39)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.2e+29)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 2.7e+170)
       (* 0.5 (+ (+ im 1.0) -1.0))
       (* 0.5 (* im (pow re -0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.6e+39) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.2e+29) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 2.7e+170) {
		tmp = 0.5 * ((im + 1.0) + -1.0);
	} 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 <= (-4.6d+39)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.2d+29) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 2.7d+170) then
        tmp = 0.5d0 * ((im + 1.0d0) + (-1.0d0))
    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 <= -4.6e+39) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.2e+29) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 2.7e+170) {
		tmp = 0.5 * ((im + 1.0) + -1.0);
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.6e+39:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.2e+29:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 2.7e+170:
		tmp = 0.5 * ((im + 1.0) + -1.0)
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.6e+39)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.2e+29)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 2.7e+170)
		tmp = Float64(0.5 * Float64(Float64(im + 1.0) + -1.0));
	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 <= -4.6e+39)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.2e+29)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 2.7e+170)
		tmp = 0.5 * ((im + 1.0) + -1.0);
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.6e+39], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.2e+29], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.7e+170], N[(0.5 * N[(N[(im + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.60000000000000024e39

   1. Initial program 49.5%

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

   3. Taylor expanded in re around -inf 83.2%

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

      1. *-commutative 83.2%

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

   5. Simplified 83.2%

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

   if -4.60000000000000024e39 < re < 4.2000000000000003e29

   1. Initial program 62.4%

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

   3. Taylor expanded in re around 0 78.4%

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

   if 4.2000000000000003e29 < re < 2.7000000000000002e170

   1. Initial program 57.7%

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

   3. Taylor expanded in im around 0 25.2%

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

      1. associate-*l* 25.3%

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

      2. associate-*l* 25.4%

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

   5. Simplified 25.4%

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

      1. add-cbrt-cube 70.6%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}} \]

      2. pow3 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}^{3}}} \]

      3. associate-*r* 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)}^{3}} \]

      4. sqrt-unprod 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\color{blue}{\sqrt{0.5 \cdot 2}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

      5. metadata-eval 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

      6. sqrt-prod 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\sqrt{1 \cdot \frac{1}{re}}}\right)}^{3}} \]

      7. *-un-lft-identity 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{\frac{1}{re}}}\right)}^{3}} \]

      8. inv-pow 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{{re}^{-1}}}\right)}^{3}} \]

      9. sqrt-pow1 70.6%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right)}^{3}} \]

      10. metadata-eval 70.6%

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

   7. Applied egg-rr 70.6%

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

   9. Applied egg-rr 64.9%

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

   if 2.7000000000000002e170 < re

   1. Initial program 3.1%

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

   3. Taylor expanded in im around 0 56.0%

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

      1. associate-*l* 56.0%

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

      2. associate-*l* 56.1%

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

   5. Simplified 56.1%

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

      1. associate-*r* 56.0%

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

      2. sqrt-unprod 56.7%

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

      3. metadata-eval 56.7%

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

      4. metadata-eval 56.7%

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

      5. expm1-log1p-u 56.7%

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

      6. *-un-lft-identity 56.7%

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

      7. expm1-udef 35.1%

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

      8. inv-pow 35.1%

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

      9. sqrt-pow1 35.1%

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

      10. metadata-eval 35.1%

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

   7. Applied egg-rr 35.1%

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

      1. expm1-def 56.7%

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

      2. expm1-log1p 56.7%

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

   9. Simplified 56.7%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{re}^{-0.5}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+170}:\\ \;\;\;\;0.5 \cdot \left(\left(im + 1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{-92}:\\ \;\;\;\;0.5 \cdot \left(\left(im + 1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.5e-92) (* 0.5 (+ (+ im 1.0) -1.0)) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.5e-92) {
		tmp = 0.5 * ((im + 1.0) + -1.0);
	} 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 (im <= 1.5d-92) then
        tmp = 0.5d0 * ((im + 1.0d0) + (-1.0d0))
    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 (im <= 1.5e-92) {
		tmp = 0.5 * ((im + 1.0) + -1.0);
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.5e-92:
		tmp = 0.5 * ((im + 1.0) + -1.0)
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.5e-92)
		tmp = Float64(0.5 * Float64(Float64(im + 1.0) + -1.0));
	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 (im <= 1.5e-92)
		tmp = 0.5 * ((im + 1.0) + -1.0);
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.5e-92], N[(0.5 * N[(N[(im + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.50000000000000007e-92

   1. Initial program 70.2%

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

   3. Taylor expanded in im around 0 7.6%

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

      1. associate-*l* 7.6%

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

      2. associate-*l* 7.6%

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

   5. Simplified 7.6%

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

      1. add-cbrt-cube 37.4%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}} \]

      2. pow3 37.4%

         \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}^{3}}} \]

      3. associate-*r* 37.4%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)}^{3}} \]

      4. sqrt-unprod 37.4%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\color{blue}{\sqrt{0.5 \cdot 2}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

      5. metadata-eval 37.4%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

      6. sqrt-prod 37.4%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\sqrt{1 \cdot \frac{1}{re}}}\right)}^{3}} \]

      7. *-un-lft-identity 37.4%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{\frac{1}{re}}}\right)}^{3}} \]

      8. inv-pow 37.4%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{{re}^{-1}}}\right)}^{3}} \]

      9. sqrt-pow1 37.4%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right)}^{3}} \]

      10. metadata-eval 37.4%

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

   7. Applied egg-rr 37.4%

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

   9. Applied egg-rr 38.9%

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

   if 1.50000000000000007e-92 < im

   1. Initial program 44.9%

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

   3. Taylor expanded in re around 0 70.9%

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

      1. *-commutative 70.9%

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

   5. Simplified 70.9%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{-92}:\\ \;\;\;\;0.5 \cdot \left(\left(im + 1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 20.5% accurate, 30.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(im + 1\right) + -1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (+ (+ im 1.0) -1.0)))

C

double code(double re, double im) {
	return 0.5 * ((im + 1.0) + -1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * ((im + 1.0d0) + (-1.0d0))
end function

Java

public static double code(double re, double im) {
	return 0.5 * ((im + 1.0) + -1.0);
}

Python

def code(re, im):
	return 0.5 * ((im + 1.0) + -1.0)

Julia

function code(re, im)
	return Float64(0.5 * Float64(Float64(im + 1.0) + -1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * ((im + 1.0) + -1.0);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[(im + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in im around 0 11.2%

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

   1. associate-*l* 11.2%

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

   2. associate-*l* 11.3%

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

5. Simplified 11.3%

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

   1. add-cbrt-cube 20.9%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}} \]

   2. pow3 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}^{3}}} \]

   3. associate-*r* 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)}^{3}} \]

   4. sqrt-unprod 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\color{blue}{\sqrt{0.5 \cdot 2}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

   5. metadata-eval 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

   6. sqrt-prod 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\sqrt{1 \cdot \frac{1}{re}}}\right)}^{3}} \]

   7. *-un-lft-identity 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{\frac{1}{re}}}\right)}^{3}} \]

   8. inv-pow 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{{re}^{-1}}}\right)}^{3}} \]

   9. sqrt-pow1 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right)}^{3}} \]

   10. metadata-eval 20.9%

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

7. Applied egg-rr 20.9%

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

9. Applied egg-rr 18.7%

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

10. Final simplification 18.7%

    \[\leadsto 0.5 \cdot \left(\left(im + 1\right) + -1\right) \]
11. Add Preprocessing

Alternative 11: 6.3% accurate, 71.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in im around 0 11.2%

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

   1. associate-*l* 11.2%

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

   2. associate-*l* 11.3%

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

5. Simplified 11.3%

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

   1. add-cbrt-cube 20.9%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)\right) \cdot \left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}} \]

   2. pow3 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)}^{3}}} \]

   3. associate-*r* 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)}^{3}} \]

   4. sqrt-unprod 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\color{blue}{\sqrt{0.5 \cdot 2}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

   5. metadata-eval 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \left(\sqrt{\color{blue}{1}} \cdot \sqrt{\frac{1}{re}}\right)\right)}^{3}} \]

   6. sqrt-prod 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{\sqrt{1 \cdot \frac{1}{re}}}\right)}^{3}} \]

   7. *-un-lft-identity 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{\frac{1}{re}}}\right)}^{3}} \]

   8. inv-pow 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \sqrt{\color{blue}{{re}^{-1}}}\right)}^{3}} \]

   9. sqrt-pow1 20.9%

      \[\leadsto 0.5 \cdot \sqrt[3]{{\left(im \cdot \color{blue}{{re}^{\left(\frac{-1}{2}\right)}}\right)}^{3}} \]

   10. metadata-eval 20.9%

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

7. Applied egg-rr 20.9%

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

9. Applied egg-rr 6.3%

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

    1. +-lft-identity 6.3%

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

11. Simplified 6.3%

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

12. Final simplification 6.3%

    \[\leadsto 0.5 \cdot im \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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)))))