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

Percentage Accurate: 42.3% → 90.4%

Time: 9.4s

Alternatives: 6

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.3% 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, 0.5× 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 4.6%

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

   2. Taylor expanded in im around 0 98.0%

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

      1. associate-*l* 98.1%

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

      2. *-commutative 98.1%

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

   4. Simplified 98.1%

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

      1. expm1-log1p-u 98.0%

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

      2. expm1-udef 8.0%

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

      3. sqrt-unprod 8.0%

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

      4. metadata-eval 8.0%

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

      5. metadata-eval 8.0%

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

      6. *-un-lft-identity 8.0%

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

      7. sqrt-div 8.0%

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

      8. metadata-eval 8.0%

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

      9. un-div-inv 8.0%

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

   6. Applied egg-rr 8.0%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

   8. Simplified 99.6%

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

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

   1. Initial program 44.8%

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

      1. sqr-neg 44.8%

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

      2. sqr-neg 44.8%

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

      3. hypot-def 90.1%

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

   3. Simplified 90.1%

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

4. Final simplification 91.1%

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

Alternative 2: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + \left(0.5 \cdot \frac{re}{\frac{im}{re}} - re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e-102)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 3.6e+100)
     (* 0.5 (sqrt (* 2.0 (+ im (- (* 0.5 (/ re (/ im re))) re)))))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e-102) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.6e+100) {
		tmp = 0.5 * sqrt((2.0 * (im + ((0.5 * (re / (im / re))) - 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 <= (-3.6d-102)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 3.6d+100) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im + ((0.5d0 * (re / (im / re))) - 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 <= -3.6e-102) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.6e+100) {
		tmp = 0.5 * Math.sqrt((2.0 * (im + ((0.5 * (re / (im / re))) - re))));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.6e-102:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 3.6e+100:
		tmp = 0.5 * math.sqrt((2.0 * (im + ((0.5 * (re / (im / re))) - re))))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.6e-102)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 3.6e+100)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im + Float64(Float64(0.5 * Float64(re / Float64(im / re))) - 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 <= -3.6e-102)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 3.6e+100)
		tmp = 0.5 * sqrt((2.0 * (im + ((0.5 * (re / (im / re))) - re))));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e-102], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.6e+100], N[(0.5 * N[Sqrt[N[(2.0 * N[(im + N[(N[(0.5 * N[(re / N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - re), $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 < -3.6e-102

   1. Initial program 50.6%

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

   2. Taylor expanded in re around -inf 79.9%

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

      1. *-commutative 79.9%

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

   4. Simplified 79.9%

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

   if -3.6e-102 < re < 3.6e100

   1. Initial program 45.5%

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

   2. Taylor expanded in re around 0 78.8%

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

      1. neg-mul-1 78.8%

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

      2. +-commutative 78.8%

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

      3. unsub-neg 78.8%

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

      4. unpow2 78.8%

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

      5. associate-/l* 78.9%

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

   4. Simplified 78.9%

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

   if 3.6e100 < re

   1. Initial program 3.3%

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

   2. Taylor expanded in im around 0 83.5%

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

      1. associate-*l* 83.5%

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

      2. *-commutative 83.5%

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

   4. Simplified 83.5%

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

      1. expm1-log1p-u 82.9%

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

      2. expm1-udef 25.0%

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

      3. sqrt-unprod 25.0%

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

      4. metadata-eval 25.0%

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

      5. metadata-eval 25.0%

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

      6. *-un-lft-identity 25.0%

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

      7. sqrt-div 25.0%

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

      8. metadata-eval 25.0%

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

      9. un-div-inv 25.0%

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

   6. Applied egg-rr 25.0%

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

      1. expm1-def 83.9%

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

      2. expm1-log1p 84.6%

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

   8. Simplified 84.6%

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

      1. div-inv 84.5%

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

      2. metadata-eval 84.5%

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

      3. sqrt-div 84.7%

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

      4. *-commutative 84.7%

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

      5. inv-pow 84.7%

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

      6. sqrt-pow1 84.7%

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

      7. metadata-eval 84.7%

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

   10. Applied egg-rr 84.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + \left(0.5 \cdot \frac{re}{\frac{im}{re}} - re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.2e-98)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 3.5e+101)
     (* 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 <= -7.2e-98) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.5e+101) {
		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 <= (-7.2d-98)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 3.5d+101) 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 <= -7.2e-98) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.5e+101) {
		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 <= -7.2e-98:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 3.5e+101:
		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 <= -7.2e-98)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 3.5e+101)
		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 <= -7.2e-98)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 3.5e+101)
		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, -7.2e-98], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.5e+101], 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 < -7.2000000000000005e-98

   1. Initial program 50.6%

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

   2. Taylor expanded in re around -inf 79.9%

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

      1. *-commutative 79.9%

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

   4. Simplified 79.9%

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

   if -7.2000000000000005e-98 < re < 3.50000000000000023e101

   1. Initial program 45.5%

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

   2. Taylor expanded in re around 0 78.0%

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

   if 3.50000000000000023e101 < re

   1. Initial program 3.3%

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

   2. Taylor expanded in im around 0 83.5%

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

      1. associate-*l* 83.5%

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

      2. *-commutative 83.5%

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

   4. Simplified 83.5%

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

      1. expm1-log1p-u 82.9%

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

      2. expm1-udef 25.0%

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

      3. sqrt-unprod 25.0%

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

      4. metadata-eval 25.0%

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

      5. metadata-eval 25.0%

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

      6. *-un-lft-identity 25.0%

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

      7. sqrt-div 25.0%

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

      8. metadata-eval 25.0%

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

      9. un-div-inv 25.0%

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

   6. Applied egg-rr 25.0%

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

      1. expm1-def 83.9%

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

      2. expm1-log1p 84.6%

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

   8. Simplified 84.6%

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

      1. div-inv 84.5%

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

      2. metadata-eval 84.5%

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

      3. sqrt-div 84.7%

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

      4. *-commutative 84.7%

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

      5. inv-pow 84.7%

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

      6. sqrt-pow1 84.7%

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

      7. metadata-eval 84.7%

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

   10. Applied egg-rr 84.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-309}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1e-309)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e-309) {
		tmp = 0.5 * sqrt((2.0 * (re * -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 <= (-1d-309)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-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 <= -1e-309) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1e-309:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e-309)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -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 <= -1e-309)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e-309], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $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 < -1.000000000000002e-309

   1. Initial program 54.0%

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

   2. Taylor expanded in re around -inf 64.0%

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

      1. *-commutative 64.0%

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

   4. Simplified 64.0%

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

   if -1.000000000000002e-309 < re

   1. Initial program 26.4%

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

   2. Taylor expanded in im around 0 47.7%

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

      1. associate-*l* 47.8%

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

      2. *-commutative 47.8%

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

   4. Simplified 47.8%

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

      1. expm1-log1p-u 47.5%

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

      2. expm1-udef 12.9%

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

      3. sqrt-unprod 12.9%

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

      4. metadata-eval 12.9%

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

      5. metadata-eval 12.9%

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

      6. *-un-lft-identity 12.9%

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

      7. sqrt-div 12.9%

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

      8. metadata-eval 12.9%

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

      9. un-div-inv 12.9%

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

   6. Applied egg-rr 12.9%

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

      1. expm1-def 48.1%

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

      2. expm1-log1p 48.4%

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

   8. Simplified 48.4%

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

      1. div-inv 48.3%

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

      2. metadata-eval 48.3%

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

      3. sqrt-div 48.4%

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

      4. *-commutative 48.4%

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

      5. inv-pow 48.4%

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

      6. sqrt-pow1 48.4%

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

      7. metadata-eval 48.4%

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

   10. Applied egg-rr 48.4%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-309}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 5: 26.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* im (pow re -0.5))))

C

double code(double re, double im) {
	return 0.5 * (im * pow(re, -0.5));
}

Fortran

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

Java

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

Python

def code(re, im):
	return 0.5 * (im * math.pow(re, -0.5))

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * (re ^ -0.5)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * (re ^ -0.5));
end

Wolfram

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

Derivation

1. Initial program 40.4%

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

2. Taylor expanded in im around 0 23.5%

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

   1. associate-*l* 23.5%

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

   2. *-commutative 23.5%

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

4. Simplified 23.5%

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

   1. expm1-log1p-u 23.4%

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

   2. expm1-udef 6.4%

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

   3. sqrt-unprod 6.4%

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

   4. metadata-eval 6.4%

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

   5. metadata-eval 6.4%

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

   6. *-un-lft-identity 6.4%

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

   7. sqrt-div 6.4%

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

   8. metadata-eval 6.4%

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

   9. un-div-inv 6.4%

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

6. Applied egg-rr 6.4%

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

   1. expm1-def 23.7%

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

   2. expm1-log1p 23.8%

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

8. Simplified 23.8%

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

   1. div-inv 23.8%

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

   2. metadata-eval 23.8%

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

   3. sqrt-div 23.8%

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

   4. *-commutative 23.8%

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

   5. inv-pow 23.8%

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

   6. sqrt-pow1 23.8%

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

   7. metadata-eval 23.8%

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

10. Applied egg-rr 23.8%

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

11. Final simplification 23.8%

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

Alternative 6: 26.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

2. Taylor expanded in im around 0 23.5%

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

   1. associate-*l* 23.5%

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

   2. *-commutative 23.5%

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

4. Simplified 23.5%

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

   1. expm1-log1p-u 23.4%

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

   2. expm1-udef 6.4%

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

   3. sqrt-unprod 6.4%

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

   4. metadata-eval 6.4%

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

   5. metadata-eval 6.4%

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

   6. *-un-lft-identity 6.4%

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

   7. sqrt-div 6.4%

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

   8. metadata-eval 6.4%

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

   9. un-div-inv 6.4%

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

6. Applied egg-rr 6.4%

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

   1. expm1-def 23.7%

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

   2. expm1-log1p 23.8%

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

8. Simplified 23.8%

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

9. Final simplification 23.8%

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

Reproduce

herbie shell --seed 2023279 
(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)))))