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

Percentage Accurate: 41.2% → 90.4%

Time: 8.5s

Alternatives: 7

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: 41.2% 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: 90.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(0.5 * N[(im * N[Power[re, -0.5], $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 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 7.3%

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

   3. Taylor expanded in re around inf 49.7%

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

      1. associate-*r/ 49.7%

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

      2. *-commutative 49.7%

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

      3. associate-*r/ 49.7%

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

   5. Simplified 49.7%

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

   6. Taylor expanded in im around 0 97.3%

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

      1. unpow-1 97.3%

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

      2. metadata-eval 97.3%

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

      3. pow-sqr 97.4%

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

      4. rem-sqrt-square 97.4%

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

      5. rem-square-sqrt 96.8%

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

      6. fabs-sqr 96.8%

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

      7. rem-square-sqrt 97.4%

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

   8. Simplified 97.4%

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

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

   1. Initial program 52.5%

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

      1. sub-neg 52.5%

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

      2. sqr-neg 52.5%

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

      3. sub-neg 52.5%

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

      4. sqr-neg 52.5%

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

      5. hypot-define 90.2%

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

   3. Simplified 90.2%

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

4. Final simplification 91.3%

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

Alternative 2: 74.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.75 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re \cdot \left(0.5 \cdot \frac{re}{im} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e-80)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 2.75e-59)
     (* 0.5 (sqrt (* 2.0 (+ im (* re (+ (* 0.5 (/ re im)) -1.0))))))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-80) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.75e-59) {
		tmp = 0.5 * sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.1d-80)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 2.75d-59) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im + (re * ((0.5d0 * (re / im)) + (-1.0d0))))))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-80) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.75e-59) {
		tmp = 0.5 * Math.sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e-80:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 2.75e-59:
		tmp = 0.5 * math.sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e-80)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 2.75e-59)
		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 * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1e-80)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 2.75e-59)
		tmp = 0.5 * sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e-80], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.75e-59], 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[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.10000000000000016e-80

   1. Initial program 57.7%

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

   3. Taylor expanded in re around -inf 82.7%

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

      1. *-commutative 82.7%

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

   5. Simplified 82.7%

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

   if -3.10000000000000016e-80 < re < 2.75000000000000007e-59

   1. Initial program 60.7%

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

   3. Taylor expanded in re around 0 88.0%

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

   if 2.75000000000000007e-59 < re

   1. Initial program 14.0%

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

   3. Taylor expanded in re around inf 49.2%

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

      1. associate-*r/ 49.2%

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

      2. *-commutative 49.2%

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

      3. associate-*r/ 49.2%

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

   5. Simplified 49.2%

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

   6. Taylor expanded in im around 0 75.0%

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

      1. unpow-1 75.0%

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

      2. metadata-eval 75.0%

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

      3. pow-sqr 75.0%

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

      4. rem-sqrt-square 75.0%

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

      5. rem-square-sqrt 74.7%

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

      6. fabs-sqr 74.7%

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

      7. rem-square-sqrt 75.0%

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

   8. Simplified 75.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.75 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re \cdot \left(0.5 \cdot \frac{re}{im} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.1e-80)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1.85e-59)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.1e-80) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.85e-59) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.1d-80)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1.85d-59) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.1e-80) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.85e-59) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.1e-80:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1.85e-59:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.1e-80)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1.85e-59)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.1e-80)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1.85e-59)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.1e-80], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.85e-59], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.10000000000000001e-80

   1. Initial program 57.7%

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

   3. Taylor expanded in re around -inf 82.7%

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

      1. *-commutative 82.7%

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

   5. Simplified 82.7%

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

   if -2.10000000000000001e-80 < re < 1.85e-59

   1. Initial program 60.7%

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

   3. Taylor expanded in re around 0 87.9%

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

   if 1.85e-59 < re

   1. Initial program 14.0%

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

   3. Taylor expanded in re around inf 49.2%

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

      1. associate-*r/ 49.2%

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

      2. *-commutative 49.2%

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

      3. associate-*r/ 49.2%

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

   5. Simplified 49.2%

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

   6. Taylor expanded in im around 0 75.0%

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

      1. unpow-1 75.0%

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

      2. metadata-eval 75.0%

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

      3. pow-sqr 75.0%

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

      4. rem-sqrt-square 75.0%

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

      5. rem-square-sqrt 74.7%

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

      6. fabs-sqr 74.7%

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

      7. rem-square-sqrt 75.0%

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

   8. Simplified 75.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.42 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.42e-80)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 2.35e-60)
     (* 0.5 (sqrt (* im 2.0)))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.42e-80) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.35e-60) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.42d-80)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 2.35d-60) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.42e-80) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.35e-60) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.42e-80:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 2.35e-60:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.42e-80)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 2.35e-60)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.42e-80)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 2.35e-60)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.42e-80], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.35e-60], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.42000000000000004e-80

   1. Initial program 57.7%

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

   3. Taylor expanded in re around -inf 82.7%

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

      1. *-commutative 82.7%

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

   5. Simplified 82.7%

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

   if -1.42000000000000004e-80 < re < 2.35e-60

   1. Initial program 60.7%

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

   3. Taylor expanded in re around 0 87.2%

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

      1. add-log-exp 8.8%

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

      2. *-un-lft-identity 8.8%

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

      3. log-prod 8.8%

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

      4. metadata-eval 8.8%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{im} \cdot \sqrt{2}}\right)\right) \]

      5. add-log-exp 87.2%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\sqrt{im} \cdot \sqrt{2}}\right) \]

      6. sqrt-unprod 87.8%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\sqrt{im \cdot 2}}\right) \]

   5. Applied egg-rr 87.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \sqrt{im \cdot 2}\right)} \]
   6. Step-by-step derivation

      1. +-lft-identity 87.8%

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

   7. Simplified 87.8%

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

   if 2.35e-60 < re

   1. Initial program 14.0%

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

   3. Taylor expanded in re around inf 49.2%

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

      1. associate-*r/ 49.2%

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

      2. *-commutative 49.2%

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

      3. associate-*r/ 49.2%

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

   5. Simplified 49.2%

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

   6. Taylor expanded in im around 0 75.0%

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

      1. unpow-1 75.0%

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

      2. metadata-eval 75.0%

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

      3. pow-sqr 75.0%

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

      4. rem-sqrt-square 75.0%

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

      5. rem-square-sqrt 74.7%

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

      6. fabs-sqr 74.7%

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

      7. rem-square-sqrt 75.0%

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

   8. Simplified 75.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.42 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.6 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.6e-59) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.6e-59) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 6.6d-59) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.6e-59) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.6e-59:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.6e-59)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.6e-59)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.6e-59], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.59999999999999964e-59

   1. Initial program 59.4%

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

   3. Taylor expanded in re around 0 59.5%

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

      1. add-log-exp 6.8%

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

      2. *-un-lft-identity 6.8%

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

      3. log-prod 6.8%

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

      4. metadata-eval 6.8%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{im} \cdot \sqrt{2}}\right)\right) \]

      5. add-log-exp 59.5%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\sqrt{im} \cdot \sqrt{2}}\right) \]

      6. sqrt-unprod 59.9%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\sqrt{im \cdot 2}}\right) \]

   5. Applied egg-rr 59.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \sqrt{im \cdot 2}\right)} \]
   6. Step-by-step derivation

      1. +-lft-identity 59.9%

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

   7. Simplified 59.9%

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

   if 6.59999999999999964e-59 < re

   1. Initial program 14.0%

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

   3. Taylor expanded in re around inf 49.2%

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

      1. associate-*r/ 49.2%

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

      2. *-commutative 49.2%

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

      3. associate-*r/ 49.2%

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

   5. Simplified 49.2%

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

   6. Taylor expanded in im around 0 75.0%

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

      1. unpow-1 75.0%

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

      2. metadata-eval 75.0%

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

      3. pow-sqr 75.0%

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

      4. rem-sqrt-square 75.0%

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

      5. rem-square-sqrt 74.7%

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

      6. fabs-sqr 74.7%

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

      7. rem-square-sqrt 75.0%

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

   8. Simplified 75.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.6 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.65 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.65e-59) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.65e-59) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= 3.65d-59) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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 <= 3.65e-59) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.65e-59:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.65e-59)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	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 <= 3.65e-59)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.65e-59], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.6500000000000002e-59

   1. Initial program 59.4%

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

   3. Taylor expanded in re around 0 59.5%

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

      1. add-log-exp 6.8%

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

      2. *-un-lft-identity 6.8%

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

      3. log-prod 6.8%

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

      4. metadata-eval 6.8%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{im} \cdot \sqrt{2}}\right)\right) \]

      5. add-log-exp 59.5%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\sqrt{im} \cdot \sqrt{2}}\right) \]

      6. sqrt-unprod 59.9%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\sqrt{im \cdot 2}}\right) \]

   5. Applied egg-rr 59.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \sqrt{im \cdot 2}\right)} \]
   6. Step-by-step derivation

      1. +-lft-identity 59.9%

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

   7. Simplified 59.9%

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

   if 3.6500000000000002e-59 < re

   1. Initial program 14.0%

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

   3. Taylor expanded in re around inf 49.2%

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

      1. associate-*r/ 49.2%

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

      2. *-commutative 49.2%

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

      3. associate-*r/ 49.2%

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

   5. Simplified 49.2%

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

      1. pow1/2 49.2%

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

      2. sqr-pow 49.0%

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

      3. clear-num 49.0%

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

      4. un-div-inv 49.0%

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

      5. div-inv 49.0%

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

      6. metadata-eval 49.0%

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

      7. metadata-eval 49.0%

         \[\leadsto 0.5 \cdot \left({\left(2 \cdot \frac{{im}^{2}}{re \cdot 2}\right)}^{\color{blue}{0.25}} \cdot {\left(2 \cdot \left({im}^{2} \cdot \frac{0.5}{re}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right) \]

      8. clear-num 49.0%

         \[\leadsto 0.5 \cdot \left({\left(2 \cdot \frac{{im}^{2}}{re \cdot 2}\right)}^{0.25} \cdot {\left(2 \cdot \left({im}^{2} \cdot \color{blue}{\frac{1}{\frac{re}{0.5}}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right) \]

      9. un-div-inv 49.0%

         \[\leadsto 0.5 \cdot \left({\left(2 \cdot \frac{{im}^{2}}{re \cdot 2}\right)}^{0.25} \cdot {\left(2 \cdot \color{blue}{\frac{{im}^{2}}{\frac{re}{0.5}}}\right)}^{\left(\frac{0.5}{2}\right)}\right) \]

      10. div-inv 49.0%

          \[\leadsto 0.5 \cdot \left({\left(2 \cdot \frac{{im}^{2}}{re \cdot 2}\right)}^{0.25} \cdot {\left(2 \cdot \frac{{im}^{2}}{\color{blue}{re \cdot \frac{1}{0.5}}}\right)}^{\left(\frac{0.5}{2}\right)}\right) \]

      11. metadata-eval 49.0%

          \[\leadsto 0.5 \cdot \left({\left(2 \cdot \frac{{im}^{2}}{re \cdot 2}\right)}^{0.25} \cdot {\left(2 \cdot \frac{{im}^{2}}{re \cdot \color{blue}{2}}\right)}^{\left(\frac{0.5}{2}\right)}\right) \]

      12. metadata-eval 49.0%

          \[\leadsto 0.5 \cdot \left({\left(2 \cdot \frac{{im}^{2}}{re \cdot 2}\right)}^{0.25} \cdot {\left(2 \cdot \frac{{im}^{2}}{re \cdot 2}\right)}^{\color{blue}{0.25}}\right) \]

   7. Applied egg-rr 49.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left({\left(2 \cdot \frac{{im}^{2}}{re \cdot 2}\right)}^{0.25} \cdot {\left(2 \cdot \frac{{im}^{2}}{re \cdot 2}\right)}^{0.25}\right)} \]
   8. Step-by-step derivation

      1. pow-sqr 49.2%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot \frac{{im}^{2}}{re \cdot 2}\right)}^{\left(2 \cdot 0.25\right)}} \]

      2. metadata-eval 49.2%

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

      3. unpow1/2 49.2%

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

      4. associate-*r/ 49.2%

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

      5. *-commutative 49.2%

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

      6. times-frac 49.2%

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

      7. metadata-eval 49.2%

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

      8. associate-*r/ 49.2%

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

      9. *-lft-identity 49.2%

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

      10. unpow2 49.2%

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

      11. rem-square-sqrt 49.0%

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

      12. times-frac 55.4%

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

      13. rem-sqrt-square 74.9%

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

      14. rem-square-sqrt 74.6%

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

      15. fabs-sqr 74.6%

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

      16. rem-square-sqrt 74.9%

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

   9. Simplified 74.9%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.65 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.6%

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

3. Taylor expanded in re around 0 50.4%

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

   1. add-log-exp 8.7%

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

   2. *-un-lft-identity 8.7%

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

   3. log-prod 8.7%

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

   4. metadata-eval 8.7%

      \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{im} \cdot \sqrt{2}}\right)\right) \]

   5. add-log-exp 50.4%

      \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\sqrt{im} \cdot \sqrt{2}}\right) \]

   6. sqrt-unprod 50.7%

      \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\sqrt{im \cdot 2}}\right) \]

5. Applied egg-rr 50.7%

   \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \sqrt{im \cdot 2}\right)} \]
6. Step-by-step derivation

   1. +-lft-identity 50.7%

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

7. Simplified 50.7%

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

8. Final simplification 50.7%

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

Reproduce

herbie shell --seed 2024081 
(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)))))