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

Percentage Accurate: 40.9% → 88.1%

Time: 6.6s

Alternatives: 6

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.9% 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: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.6 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.6e+49)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.6e+49) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.6e+49) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.6e+49:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.6e+49)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	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 <= 3.6e+49)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.6e+49], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.59999999999999996e49

   1. Initial program 48.4%

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

      1. hypot-def 93.5%

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

   3. Simplified 93.5%

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

   if 3.59999999999999996e49 < re

   1. Initial program 3.8%

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

   2. Taylor expanded in re around inf 51.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   3. Step-by-step derivation

      1. unpow2 51.3%

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

   4. Simplified 51.3%

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

   5. Taylor expanded in im around 0 85.6%

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

      1. unpow-1 85.6%

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

      2. metadata-eval 85.6%

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

      3. pow-sqr 85.6%

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

      4. rem-sqrt-square 85.6%

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

      5. rem-square-sqrt 85.3%

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

      6. fabs-sqr 85.3%

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

      7. rem-square-sqrt 85.6%

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

      8. exp-to-pow 80.5%

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

      9. metadata-eval 80.5%

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

      10. distribute-rgt-neg-in 80.5%

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

      11. exp-neg 80.5%

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

      12. exp-to-pow 85.5%

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

      13. unpow1/2 85.5%

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

      14. associate-*l/ 85.7%

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

      15. *-lft-identity 85.7%

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

   7. Simplified 85.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.6 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 2: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -3 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -8.6 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -6.8 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4 - \frac{im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- im re))))))
   (if (<= re -3e+77)
     (* 0.5 (sqrt (* re -4.0)))
     (if (<= re -8.6e-133)
       t_0
       (if (<= re -6.8e-164)
         (* 0.5 (sqrt (- (* re -4.0) (/ im (/ re im)))))
         (if (<= re 4.1e+48) t_0 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -3e+77) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= -8.6e-133) {
		tmp = t_0;
	} else if (re <= -6.8e-164) {
		tmp = 0.5 * sqrt(((re * -4.0) - (im / (re / im))));
	} else if (re <= 4.1e+48) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    if (re <= (-3d+77)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= (-8.6d-133)) then
        tmp = t_0
    else if (re <= (-6.8d-164)) then
        tmp = 0.5d0 * sqrt(((re * (-4.0d0)) - (im / (re / im))))
    else if (re <= 4.1d+48) then
        tmp = t_0
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -3e+77) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= -8.6e-133) {
		tmp = t_0;
	} else if (re <= -6.8e-164) {
		tmp = 0.5 * Math.sqrt(((re * -4.0) - (im / (re / im))));
	} else if (re <= 4.1e+48) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (im - re)))
	tmp = 0
	if re <= -3e+77:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= -8.6e-133:
		tmp = t_0
	elif re <= -6.8e-164:
		tmp = 0.5 * math.sqrt(((re * -4.0) - (im / (re / im))))
	elif re <= 4.1e+48:
		tmp = t_0
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	tmp = 0.0
	if (re <= -3e+77)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= -8.6e-133)
		tmp = t_0;
	elseif (re <= -6.8e-164)
		tmp = Float64(0.5 * sqrt(Float64(Float64(re * -4.0) - Float64(im / Float64(re / im)))));
	elseif (re <= 4.1e+48)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (im - re)));
	tmp = 0.0;
	if (re <= -3e+77)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= -8.6e-133)
		tmp = t_0;
	elseif (re <= -6.8e-164)
		tmp = 0.5 * sqrt(((re * -4.0) - (im / (re / im))));
	elseif (re <= 4.1e+48)
		tmp = t_0;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3e+77], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -8.6e-133], t$95$0, If[LessEqual[re, -6.8e-164], N[(0.5 * N[Sqrt[N[(N[(re * -4.0), $MachinePrecision] - N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.1e+48], t$95$0, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.9999999999999998e77

   1. Initial program 27.2%

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

   2. Taylor expanded in re around -inf 80.2%

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

      1. *-commutative 80.2%

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

   4. Simplified 80.2%

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

   if -2.9999999999999998e77 < re < -8.60000000000000032e-133 or -6.8e-164 < re < 4.1000000000000003e48

   1. Initial program 53.1%

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

   2. Taylor expanded in re around 0 78.8%

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

   if -8.60000000000000032e-133 < re < -6.8e-164

   1. Initial program 83.7%

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

   2. Taylor expanded in re around -inf 88.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re} + -4 \cdot re}} \]
   3. Step-by-step derivation

      1. +-commutative 88.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re + -1 \cdot \frac{{im}^{2}}{re}}} \]

      2. mul-1-neg 88.3%

         \[\leadsto 0.5 \cdot \sqrt{-4 \cdot re + \color{blue}{\left(-\frac{{im}^{2}}{re}\right)}} \]

      3. unsub-neg 88.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re - \frac{{im}^{2}}{re}}} \]

      4. *-commutative 88.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4} - \frac{{im}^{2}}{re}} \]

      5. unpow2 88.3%

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

      6. associate-/l* 88.3%

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

   4. Simplified 88.3%

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

   if 4.1000000000000003e48 < re

   1. Initial program 3.8%

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

   2. Taylor expanded in re around inf 51.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   3. Step-by-step derivation

      1. unpow2 51.3%

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

   4. Simplified 51.3%

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

   5. Taylor expanded in im around 0 85.6%

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

      1. unpow-1 85.6%

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

      2. metadata-eval 85.6%

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

      3. pow-sqr 85.6%

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

      4. rem-sqrt-square 85.6%

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

      5. rem-square-sqrt 85.3%

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

      6. fabs-sqr 85.3%

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

      7. rem-square-sqrt 85.6%

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

      8. exp-to-pow 80.5%

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

      9. metadata-eval 80.5%

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

      10. distribute-rgt-neg-in 80.5%

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

      11. exp-neg 80.5%

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

      12. exp-to-pow 85.5%

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

      13. unpow1/2 85.5%

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

      14. associate-*l/ 85.7%

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

      15. *-lft-identity 85.7%

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

   7. Simplified 85.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -8.6 \cdot 10^{-133}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -6.8 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4 - \frac{im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -9.2 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -8.6 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -6.8 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0))))
        (t_1 (* 0.5 (sqrt (* 2.0 (- im re))))))
   (if (<= re -9.2e+77)
     t_0
     (if (<= re -8.6e-133)
       t_1
       (if (<= re -6.8e-164)
         t_0
         (if (<= re 7e+48) t_1 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -9.2e+77) {
		tmp = t_0;
	} else if (re <= -8.6e-133) {
		tmp = t_1;
	} else if (re <= -6.8e-164) {
		tmp = t_0;
	} else if (re <= 7e+48) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    if (re <= (-9.2d+77)) then
        tmp = t_0
    else if (re <= (-8.6d-133)) then
        tmp = t_1
    else if (re <= (-6.8d-164)) then
        tmp = t_0
    else if (re <= 7d+48) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -9.2e+77) {
		tmp = t_0;
	} else if (re <= -8.6e-133) {
		tmp = t_1;
	} else if (re <= -6.8e-164) {
		tmp = t_0;
	} else if (re <= 7e+48) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((2.0 * (im - re)))
	tmp = 0
	if re <= -9.2e+77:
		tmp = t_0
	elif re <= -8.6e-133:
		tmp = t_1
	elif re <= -6.8e-164:
		tmp = t_0
	elif re <= 7e+48:
		tmp = t_1
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	tmp = 0.0
	if (re <= -9.2e+77)
		tmp = t_0;
	elseif (re <= -8.6e-133)
		tmp = t_1;
	elseif (re <= -6.8e-164)
		tmp = t_0;
	elseif (re <= 7e+48)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * sqrt((2.0 * (im - re)));
	tmp = 0.0;
	if (re <= -9.2e+77)
		tmp = t_0;
	elseif (re <= -8.6e-133)
		tmp = t_1;
	elseif (re <= -6.8e-164)
		tmp = t_0;
	elseif (re <= 7e+48)
		tmp = t_1;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -9.2e+77], t$95$0, If[LessEqual[re, -8.6e-133], t$95$1, If[LessEqual[re, -6.8e-164], t$95$0, If[LessEqual[re, 7e+48], t$95$1, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -9.19999999999999979e77 or -8.60000000000000032e-133 < re < -6.8e-164

   1. Initial program 35.5%

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

   2. Taylor expanded in re around -inf 81.4%

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

      1. *-commutative 81.4%

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

   4. Simplified 81.4%

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

   if -9.19999999999999979e77 < re < -8.60000000000000032e-133 or -6.8e-164 < re < 6.9999999999999995e48

   1. Initial program 53.1%

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

   2. Taylor expanded in re around 0 78.8%

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

   if 6.9999999999999995e48 < re

   1. Initial program 3.8%

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

   2. Taylor expanded in re around inf 51.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   3. Step-by-step derivation

      1. unpow2 51.3%

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

   4. Simplified 51.3%

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

   5. Taylor expanded in im around 0 85.6%

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

      1. unpow-1 85.6%

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

      2. metadata-eval 85.6%

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

      3. pow-sqr 85.6%

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

      4. rem-sqrt-square 85.6%

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

      5. rem-square-sqrt 85.3%

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

      6. fabs-sqr 85.3%

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

      7. rem-square-sqrt 85.6%

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

      8. exp-to-pow 80.5%

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

      9. metadata-eval 80.5%

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

      10. distribute-rgt-neg-in 80.5%

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

      11. exp-neg 80.5%

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

      12. exp-to-pow 85.5%

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

      13. unpow1/2 85.5%

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

      14. associate-*l/ 85.7%

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

      15. *-lft-identity 85.7%

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

   7. Simplified 85.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -8.6 \cdot 10^{-133}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -6.8 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -1.75 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -4.3 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -6.8 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (* 0.5 (sqrt (* 2.0 im)))))
   (if (<= re -1.75e+77)
     t_0
     (if (<= re -4.3e-132)
       t_1
       (if (<= re -6.8e-164)
         t_0
         (if (<= re 2.3e+49) t_1 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((2.0 * im));
	double tmp;
	if (re <= -1.75e+77) {
		tmp = t_0;
	} else if (re <= -4.3e-132) {
		tmp = t_1;
	} else if (re <= -6.8e-164) {
		tmp = t_0;
	} else if (re <= 2.3e+49) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((2.0d0 * im))
    if (re <= (-1.75d+77)) then
        tmp = t_0
    else if (re <= (-4.3d-132)) then
        tmp = t_1
    else if (re <= (-6.8d-164)) then
        tmp = t_0
    else if (re <= 2.3d+49) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((2.0 * im));
	double tmp;
	if (re <= -1.75e+77) {
		tmp = t_0;
	} else if (re <= -4.3e-132) {
		tmp = t_1;
	} else if (re <= -6.8e-164) {
		tmp = t_0;
	} else if (re <= 2.3e+49) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((2.0 * im))
	tmp = 0
	if re <= -1.75e+77:
		tmp = t_0
	elif re <= -4.3e-132:
		tmp = t_1
	elif re <= -6.8e-164:
		tmp = t_0
	elif re <= 2.3e+49:
		tmp = t_1
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * im)))
	tmp = 0.0
	if (re <= -1.75e+77)
		tmp = t_0;
	elseif (re <= -4.3e-132)
		tmp = t_1;
	elseif (re <= -6.8e-164)
		tmp = t_0;
	elseif (re <= 2.3e+49)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * sqrt((2.0 * im));
	tmp = 0.0;
	if (re <= -1.75e+77)
		tmp = t_0;
	elseif (re <= -4.3e-132)
		tmp = t_1;
	elseif (re <= -6.8e-164)
		tmp = t_0;
	elseif (re <= 2.3e+49)
		tmp = t_1;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.75e+77], t$95$0, If[LessEqual[re, -4.3e-132], t$95$1, If[LessEqual[re, -6.8e-164], t$95$0, If[LessEqual[re, 2.3e+49], t$95$1, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.7500000000000001e77 or -4.2999999999999997e-132 < re < -6.8e-164

   1. Initial program 35.5%

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

   2. Taylor expanded in re around -inf 81.4%

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

      1. *-commutative 81.4%

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

   4. Simplified 81.4%

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

   if -1.7500000000000001e77 < re < -4.2999999999999997e-132 or -6.8e-164 < re < 2.30000000000000002e49

   1. Initial program 53.1%

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

   2. Taylor expanded in re around 0 77.1%

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

      1. *-commutative 77.1%

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

   4. Simplified 77.1%

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

   if 2.30000000000000002e49 < re

   1. Initial program 3.8%

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

   2. Taylor expanded in re around inf 51.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   3. Step-by-step derivation

      1. unpow2 51.3%

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

   4. Simplified 51.3%

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

   5. Taylor expanded in im around 0 85.6%

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

      1. unpow-1 85.6%

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

      2. metadata-eval 85.6%

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

      3. pow-sqr 85.6%

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

      4. rem-sqrt-square 85.6%

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

      5. rem-square-sqrt 85.3%

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

      6. fabs-sqr 85.3%

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

      7. rem-square-sqrt 85.6%

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

      8. exp-to-pow 80.5%

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

      9. metadata-eval 80.5%

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

      10. distribute-rgt-neg-in 80.5%

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

      11. exp-neg 80.5%

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

      12. exp-to-pow 85.5%

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

      13. unpow1/2 85.5%

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

      14. associate-*l/ 85.7%

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

      15. *-lft-identity 85.7%

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

   7. Simplified 85.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -4.3 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq -6.8 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 58.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{-170}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.8e-170) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.8e-170) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.8d-170) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.8e-170) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.8e-170:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.8e-170)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.8e-170)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.8e-170], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.7999999999999998e-170

   1. Initial program 39.6%

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

   2. Taylor expanded in re around -inf 57.4%

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

      1. *-commutative 57.4%

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

   4. Simplified 57.4%

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

   if 3.7999999999999998e-170 < im

   1. Initial program 38.5%

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

   2. Taylor expanded in re around 0 65.1%

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

      1. *-commutative 65.1%

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

   4. Simplified 65.1%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{-170}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 6: 51.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

2. Taylor expanded in re around 0 53.4%

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

   1. *-commutative 53.4%

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

4. Simplified 53.4%

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

5. Final simplification 53.4%

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

Reproduce

herbie shell --seed 2023213 
(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)))))