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

Percentage Accurate: 42.4% → 88.2%

Time: 9.1s

Alternatives: 5

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: 42.4% 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: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.8 \cdot 10^{+73}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.8e+73)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* im (/ 0.5 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.8e+73) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.8e+73) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.8e+73:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.8e+73)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.8e+73)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.8e+73], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.80000000000000008e73

   1. Initial program 48.2%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right)\right) \]

      5. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right), re\right)\right)\right)\right) \]

      6. hypot-lowering-hypot.f64 90.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right)\right) \]

   3. Simplified 90.3%

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

   if 2.80000000000000008e73 < re

   1. Initial program 8.8%

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

   3. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{{im}^{2}}{re}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right) \]

      3. *-lowering-*.f64 43.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right) \]

   5. Simplified 43.2%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{\frac{re}{im \cdot im}}} \]

      2. associate-/r/ N/A

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

      3. sqrt-prod N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{re}}\right), \color{blue}{im}\right) \]

      8. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{\sqrt{1}}{\sqrt{re}}\right), im\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\sqrt{re}}\right), im\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{\sqrt{re}}\right), im\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{re}\right)\right), im\right) \]

      12. sqrt-lowering-sqrt.f64 82.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(re\right)\right), im\right) \]

   7. Applied egg-rr 82.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.8 \cdot 10^{+73}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{+90}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.9e+90)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3.9e+70)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.9e+90) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3.9e+70) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.9d+90)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3.9d+70) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.9e+90) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3.9e+70) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.9e+90:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3.9e+70:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.9e+90)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3.9e+70)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.9e+90)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3.9e+70)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.9e+90], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.9e+70], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.9000000000000002e90

   1. Initial program 19.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 83.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 83.1%

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

   if -3.9000000000000002e90 < re < 3.89999999999999975e70

   1. Initial program 54.2%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\color{blue}{im}, re\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 77.7%

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

   if 3.89999999999999975e70 < re

   1. Initial program 12.1%

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

   3. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{{im}^{2}}{re}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right) \]

      3. *-lowering-*.f64 43.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right) \]

   5. Simplified 43.6%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{\frac{re}{im \cdot im}}} \]

      2. associate-/r/ N/A

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

      3. sqrt-prod N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{re}}\right), \color{blue}{im}\right) \]

      8. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{\sqrt{1}}{\sqrt{re}}\right), im\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\sqrt{re}}\right), im\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{\sqrt{re}}\right), im\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{re}\right)\right), im\right) \]

      12. sqrt-lowering-sqrt.f64 81.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(re\right)\right), im\right) \]

   7. Applied egg-rr 81.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{+90}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.15e+89)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3.2e+70) (* 0.5 (sqrt (* 2.0 im))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.15e+89) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3.2e+70) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.15d+89)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3.2d+70) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.15e+89) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3.2e+70) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.15e+89:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3.2e+70:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.15e+89)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3.2e+70)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.15e+89)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3.2e+70)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.15e+89], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.2e+70], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.1499999999999999e89

   1. Initial program 19.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 83.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 83.1%

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

   if -1.1499999999999999e89 < re < 3.2000000000000002e70

   1. Initial program 54.2%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 77.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 77.2%

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

   if 3.2000000000000002e70 < re

   1. Initial program 12.1%

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

   3. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{{im}^{2}}{re}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right) \]

      3. *-lowering-*.f64 43.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right) \]

   5. Simplified 43.6%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{\frac{re}{im \cdot im}}} \]

      2. associate-/r/ N/A

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

      3. sqrt-prod N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{re}}\right), \color{blue}{im}\right) \]

      8. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{\sqrt{1}}{\sqrt{re}}\right), im\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\sqrt{re}}\right), im\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{\sqrt{re}}\right), im\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{re}\right)\right), im\right) \]

      12. sqrt-lowering-sqrt.f64 81.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(re\right)\right), im\right) \]

   7. Applied egg-rr 81.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.02 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.02e+96) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.02e+96) {
		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 (re <= (-1.02d+96)) 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 (re <= -1.02e+96) {
		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 re <= -1.02e+96:
		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 (re <= -1.02e+96)
		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 (re <= -1.02e+96)
		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[re, -1.02e+96], 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 re < -1.02000000000000001e96

   1. Initial program 19.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 83.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 83.1%

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

   if -1.02000000000000001e96 < re

   1. Initial program 43.6%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 63.3%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.02 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.5% 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 40.0%

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

3. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

   2. *-lowering-*.f64 57.1%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

5. Simplified 57.1%

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

6. Final simplification 57.1%

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

Reproduce

herbie shell --seed 2024163 
(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)))))