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

Percentage Accurate: 40.5% → 90.3%

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: 40.5% 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: 90.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.2%

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

   3. Taylor expanded in im around 0 98.2%

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

      1. *-commutative 98.2%

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

   5. Simplified 98.2%

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

      1. expm1-log1p-u 97.9%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      2. expm1-udef 20.8%

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

      3. sqrt-unprod 20.9%

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

      4. metadata-eval 20.9%

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

      5. metadata-eval 20.9%

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

      6. *-rgt-identity 20.9%

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

      7. sqrt-div 20.9%

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

      8. metadata-eval 20.9%

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

      9. un-div-inv 20.9%

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

   7. Applied egg-rr 20.9%

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

      1. expm1-def 99.4%

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

      2. expm1-log1p 99.7%

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

   9. Simplified 99.7%

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

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

   1. Initial program 42.8%

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

      1. sub-neg 42.8%

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

      2. sqr-neg 42.8%

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

      3. sub-neg 42.8%

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

      4. sqr-neg 42.8%

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

      5. hypot-def 89.6%

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

   3. Simplified 89.6%

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

4. Final simplification 90.8%

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

Alternative 2: 75.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e-29)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 5e-21)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-29) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5e-21) {
		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 <= (-2.4d-29)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5d-21) 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 <= -2.4e-29) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5e-21) {
		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 <= -2.4e-29:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5e-21:
		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 <= -2.4e-29)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5e-21)
		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 <= -2.4e-29)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5e-21)
		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, -2.4e-29], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e-21], 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 < -2.39999999999999992e-29

   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 78.1%

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

      1. *-commutative 78.1%

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

   5. Simplified 78.1%

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

   if -2.39999999999999992e-29 < re < 4.99999999999999973e-21

   1. Initial program 43.1%

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

   3. Taylor expanded in re around 0 74.3%

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

   if 4.99999999999999973e-21 < re

   1. Initial program 15.1%

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

   3. Taylor expanded in im around 0 71.1%

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

      1. *-commutative 71.1%

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

   5. Simplified 71.1%

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

      1. expm1-log1p-u 70.9%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      2. expm1-udef 21.7%

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

      3. sqrt-unprod 21.8%

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

      4. metadata-eval 21.8%

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

      5. metadata-eval 21.8%

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

      6. *-rgt-identity 21.8%

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

      7. sqrt-div 21.8%

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

      8. metadata-eval 21.8%

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

      9. un-div-inv 21.8%

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

   7. Applied egg-rr 21.8%

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

      1. expm1-def 71.8%

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

      2. expm1-log1p 72.1%

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

   9. Simplified 72.1%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.5e-45)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 5e-19) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.5e-45) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5e-19) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= (-7.5d-45)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5d-19) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= -7.5e-45) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5e-19) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.5e-45:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5e-19:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.5e-45)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5e-19)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= -7.5e-45)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5e-19)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.5e-45], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e-19], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.5000000000000006e-45

   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 78.1%

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

      1. *-commutative 78.1%

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

   5. Simplified 78.1%

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

   if -7.5000000000000006e-45 < re < 5.0000000000000004e-19

   1. Initial program 43.1%

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

   3. Taylor expanded in re around 0 73.6%

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

      1. *-commutative 73.6%

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

   5. Simplified 73.6%

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

   if 5.0000000000000004e-19 < re

   1. Initial program 15.1%

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

   3. Taylor expanded in im around 0 71.1%

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

      1. *-commutative 71.1%

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

   5. Simplified 71.1%

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

      1. expm1-log1p-u 70.9%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      2. expm1-udef 21.7%

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

      3. sqrt-unprod 21.8%

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

      4. metadata-eval 21.8%

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

      5. metadata-eval 21.8%

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

      6. *-rgt-identity 21.8%

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

      7. sqrt-div 21.8%

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

      8. metadata-eval 21.8%

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

      9. un-div-inv 21.8%

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

   7. Applied egg-rr 21.8%

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

      1. expm1-def 71.8%

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

      2. expm1-log1p 72.1%

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

   9. Simplified 72.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.5e-44) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.5e-44) {
		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 <= (-8.5d-44)) 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 <= -8.5e-44) {
		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 <= -8.5e-44:
		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 <= -8.5e-44)
		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 <= -8.5e-44)
		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, -8.5e-44], 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 < -8.5000000000000002e-44

   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 78.1%

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

      1. *-commutative 78.1%

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

   5. Simplified 78.1%

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

   if -8.5000000000000002e-44 < re

   1. Initial program 33.5%

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

   3. Taylor expanded in re around 0 60.2%

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

      1. *-commutative 60.2%

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

   5. Simplified 60.2%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.6% 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 38.6%

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

3. Taylor expanded in re around 0 48.7%

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

   1. *-commutative 48.7%

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

5. Simplified 48.7%

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

6. Final simplification 48.7%

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

Reproduce

herbie shell --seed 2024027 
(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)))))