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

Percentage Accurate: 41.5% → 88.6%

Time: 9.0s

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: 41.5% 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: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.2e+27)
   (* 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 <= 4.2e+27) {
		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 <= 4.2e+27) {
		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 <= 4.2e+27:
		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 <= 4.2e+27)
		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 <= 4.2e+27)
		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, 4.2e+27], 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 < 4.19999999999999989e27

   1. Initial program 53.3%

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

      1. sub-neg 53.3%

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

      2. sqr-neg 53.3%

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

      3. sub-neg 53.3%

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

      4. sqr-neg 53.3%

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

      5. hypot-define 93.5%

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

   3. Simplified 93.5%

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

   if 4.19999999999999989e27 < re

   1. Initial program 9.3%

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

   3. Taylor expanded in re around inf 82.1%

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

      1. *-commutative 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*r* 82.1%

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

      4. *-commutative 82.1%

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

      5. associate-*l* 82.0%

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

      6. *-commutative 82.0%

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

   5. Simplified 82.0%

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

      1. sqrt-unprod 83.0%

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

      2. metadata-eval 83.0%

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

      3. metadata-eval 83.0%

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

   7. Applied egg-rr 83.0%

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

      1. *-commutative 83.0%

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

      2. sqrt-div 82.9%

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

      3. metadata-eval 82.9%

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

      4. un-div-inv 83.0%

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

   9. Applied egg-rr 83.0%

      \[\leadsto 0.5 \cdot \left(1 \cdot \color{blue}{\frac{im}{\sqrt{re}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.2 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re \cdot \left(0.5 \cdot \frac{re}{im} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.2e-84)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 3e+27)
     (* 0.5 (sqrt (* 2.0 (+ im (* re (+ (* 0.5 (/ re im)) -1.0))))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.2e-84) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3e+27) {
		tmp = 0.5 * sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.2d-84)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 3d+27) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im + (re * ((0.5d0 * (re / im)) + (-1.0d0))))))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.2e-84) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3e+27) {
		tmp = 0.5 * Math.sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.2e-84:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 3e+27:
		tmp = 0.5 * math.sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.2e-84)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 3e+27)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im + Float64(re * Float64(Float64(0.5 * Float64(re / im)) + -1.0))))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.2e-84)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 3e+27)
		tmp = 0.5 * sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.2e-84], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3e+27], N[(0.5 * N[Sqrt[N[(2.0 * N[(im + N[(re * N[(N[(0.5 * N[(re / im), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.20000000000000009e-84

   1. Initial program 49.7%

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

   3. Taylor expanded in re around -inf 70.4%

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

      1. *-commutative 70.4%

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

   5. Simplified 70.4%

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

   if -1.20000000000000009e-84 < re < 2.99999999999999976e27

   1. Initial program 55.9%

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

   3. Taylor expanded in re around 0 86.4%

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

   if 2.99999999999999976e27 < re

   1. Initial program 9.3%

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

   3. Taylor expanded in re around inf 82.1%

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

      1. *-commutative 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*r* 82.1%

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

      4. *-commutative 82.1%

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

      5. associate-*l* 82.0%

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

      6. *-commutative 82.0%

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

   5. Simplified 82.0%

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

      1. sqrt-unprod 83.0%

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

      2. metadata-eval 83.0%

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

      3. metadata-eval 83.0%

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

   7. Applied egg-rr 83.0%

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

      1. *-commutative 83.0%

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

      2. sqrt-div 82.9%

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

      3. metadata-eval 82.9%

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

      4. un-div-inv 83.0%

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

   9. Applied egg-rr 83.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.2 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re \cdot \left(0.5 \cdot \frac{re}{im} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.2 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.2e-84)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 2.9e+27)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.2e-84) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.9e+27) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.2d-84)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 2.9d+27) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.2e-84) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.9e+27) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.2e-84:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 2.9e+27:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.2e-84)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 2.9e+27)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(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 <= -1.2e-84)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 2.9e+27)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.2e-84], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.9e+27], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.20000000000000009e-84

   1. Initial program 49.7%

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

   3. Taylor expanded in re around -inf 70.4%

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

      1. *-commutative 70.4%

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

   5. Simplified 70.4%

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

   if -1.20000000000000009e-84 < re < 2.9000000000000001e27

   1. Initial program 55.9%

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

   3. Taylor expanded in re around 0 86.2%

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

   if 2.9000000000000001e27 < re

   1. Initial program 9.3%

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

   3. Taylor expanded in re around inf 82.1%

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

      1. *-commutative 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*r* 82.1%

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

      4. *-commutative 82.1%

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

      5. associate-*l* 82.0%

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

      6. *-commutative 82.0%

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

   5. Simplified 82.0%

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

      1. sqrt-unprod 83.0%

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

      2. metadata-eval 83.0%

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

      3. metadata-eval 83.0%

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

   7. Applied egg-rr 83.0%

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

      1. *-commutative 83.0%

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

      2. sqrt-div 82.9%

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

      3. metadata-eval 82.9%

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

      4. un-div-inv 83.0%

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

   9. Applied egg-rr 83.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.2 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.5e-85)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.5e-85) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} 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 <= (-7.5d-85)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    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 <= -7.5e-85) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.5e-85:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.5e-85)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	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 <= -7.5e-85)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.5e-85], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $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 < -7.5000000000000003e-85

   1. Initial program 49.7%

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

   3. Taylor expanded in re around -inf 70.4%

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

      1. *-commutative 70.4%

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

   5. Simplified 70.4%

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

   if -7.5000000000000003e-85 < re

   1. Initial program 38.4%

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

   3. Taylor expanded in re around 0 62.0%

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

      1. pow1 62.0%

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

      2. sqrt-unprod 62.4%

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

   5. Applied egg-rr 62.4%

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

      1. unpow1 62.4%

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

   7. Simplified 62.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.1% 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 42.0%

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

3. Taylor expanded in re around 0 53.2%

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

   1. pow1 53.2%

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

   2. sqrt-unprod 53.2%

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

5. Applied egg-rr 53.2%

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

   1. unpow1 53.2%

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

7. Simplified 53.2%

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

8. Final simplification 53.2%

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

Reproduce

herbie shell --seed 2024069 
(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)))))