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

Percentage Accurate: 41.2% → 89.6%

Time: 7.4s

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: 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: 89.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* 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(((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(((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(((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 (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(((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[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $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 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 4.0%

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

   2. Taylor expanded in im around 0 91.3%

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

      1. *-commutative 91.3%

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

   4. Simplified 91.3%

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

      1. expm1-log1p-u 91.2%

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

      2. expm1-udef 8.2%

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

      3. sqrt-unprod 8.2%

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

      4. metadata-eval 8.2%

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

      5. metadata-eval 8.2%

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

      6. *-rgt-identity 8.2%

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

      7. sqrt-div 8.2%

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

      8. metadata-eval 8.2%

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

      9. un-div-inv 8.2%

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

   6. Applied egg-rr 8.2%

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

      1. expm1-def 92.7%

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

      2. expm1-log1p 92.9%

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

   8. Simplified 92.9%

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

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

   1. Initial program 46.7%

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

      1. sqr-neg 46.7%

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

      2. sqr-neg 46.7%

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

      3. hypot-def 90.6%

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

   3. Simplified 90.6%

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

4. Final simplification 90.9%

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

Alternative 2: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e+136)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1.3e-5)
     (* 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 <= -1.9e+136) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.3e-5) {
		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 <= (-1.9d+136)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1.3d-5) 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 <= -1.9e+136) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.3e-5) {
		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 <= -1.9e+136:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1.3e-5:
		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 <= -1.9e+136)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1.3e-5)
		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 <= -1.9e+136)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1.3e-5)
		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, -1.9e+136], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.3e-5], 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 < -1.90000000000000007e136

   1. Initial program 17.1%

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

   2. Taylor expanded in re around -inf 97.0%

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

      1. *-commutative 97.0%

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

   4. Simplified 97.0%

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

   if -1.90000000000000007e136 < re < 1.29999999999999992e-5

   1. Initial program 59.9%

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

   2. Taylor expanded in re around 0 75.5%

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

   if 1.29999999999999992e-5 < re

   1. Initial program 4.8%

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

   2. Taylor expanded in im around 0 78.1%

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

      1. *-commutative 78.1%

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

   4. Simplified 78.1%

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

      1. expm1-log1p-u 78.0%

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

      2. expm1-udef 26.9%

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

      3. sqrt-unprod 26.9%

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

      4. metadata-eval 26.9%

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

      5. metadata-eval 26.9%

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

      6. *-rgt-identity 26.9%

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

      7. sqrt-div 26.9%

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

      8. metadata-eval 26.9%

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

      9. un-div-inv 26.9%

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

   6. Applied egg-rr 26.9%

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

      1. expm1-def 78.7%

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

      2. expm1-log1p 78.9%

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

   8. Simplified 78.9%

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

      1. div-inv 78.8%

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

      2. metadata-eval 78.8%

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

      3. sqrt-div 78.9%

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

      4. *-commutative 78.9%

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

      5. inv-pow 78.9%

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

      6. sqrt-pow1 79.0%

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

      7. metadata-eval 79.0%

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

   10. Applied egg-rr 79.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3: 73.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.4 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.4e+136)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1.5e-5) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.4e+136) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.5e-5) {
		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 <= (-1.4d+136)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1.5d-5) 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 <= -1.4e+136) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.5e-5) {
		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 <= -1.4e+136:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1.5e-5:
		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 <= -1.4e+136)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1.5e-5)
		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 <= -1.4e+136)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1.5e-5)
		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, -1.4e+136], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.5e-5], 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 < -1.4000000000000001e136

   1. Initial program 17.1%

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

   2. Taylor expanded in re around -inf 97.0%

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

      1. *-commutative 97.0%

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

   4. Simplified 97.0%

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

   if -1.4000000000000001e136 < re < 1.50000000000000004e-5

   1. Initial program 59.9%

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

   2. Taylor expanded in re around 0 73.4%

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

      1. expm1-log1p-u 69.3%

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

      2. expm1-udef 55.3%

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

      3. sqrt-unprod 55.3%

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

   4. Applied egg-rr 55.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im \cdot 2}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 69.4%

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

      2. expm1-log1p 73.8%

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

   6. Simplified 73.8%

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

   if 1.50000000000000004e-5 < re

   1. Initial program 4.8%

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

   2. Taylor expanded in im around 0 78.1%

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

      1. *-commutative 78.1%

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

   4. Simplified 78.1%

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

      1. expm1-log1p-u 78.0%

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

      2. expm1-udef 26.9%

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

      3. sqrt-unprod 26.9%

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

      4. metadata-eval 26.9%

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

      5. metadata-eval 26.9%

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

      6. *-rgt-identity 26.9%

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

      7. sqrt-div 26.9%

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

      8. metadata-eval 26.9%

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

      9. un-div-inv 26.9%

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

   6. Applied egg-rr 26.9%

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

      1. expm1-def 78.7%

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

      2. expm1-log1p 78.9%

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

   8. Simplified 78.9%

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

      1. div-inv 78.8%

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

      2. metadata-eval 78.8%

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

      3. sqrt-div 78.9%

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

      4. *-commutative 78.9%

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

      5. inv-pow 78.9%

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

      6. sqrt-pow1 79.0%

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

      7. metadata-eval 79.0%

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

   10. Applied egg-rr 79.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.4 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 64.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.1e-5) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.1e-5) {
		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 <= 2.1d-5) 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 <= 2.1e-5) {
		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 <= 2.1e-5:
		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 <= 2.1e-5)
		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 <= 2.1e-5)
		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, 2.1e-5], 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 < 2.09999999999999988e-5

   1. Initial program 53.1%

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

   2. Taylor expanded in re around 0 63.0%

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

      1. expm1-log1p-u 59.5%

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

      2. expm1-udef 47.7%

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

      3. sqrt-unprod 47.7%

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

   4. Applied egg-rr 47.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im \cdot 2}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 59.6%

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

      2. expm1-log1p 63.4%

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

   6. Simplified 63.4%

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

   if 2.09999999999999988e-5 < re

   1. Initial program 4.8%

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

   2. Taylor expanded in im around 0 78.1%

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

      1. *-commutative 78.1%

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

   4. Simplified 78.1%

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

      1. expm1-log1p-u 78.0%

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

      2. expm1-udef 26.9%

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

      3. sqrt-unprod 26.9%

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

      4. metadata-eval 26.9%

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

      5. metadata-eval 26.9%

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

      6. *-rgt-identity 26.9%

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

      7. sqrt-div 26.9%

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

      8. metadata-eval 26.9%

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

      9. un-div-inv 26.9%

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

   6. Applied egg-rr 26.9%

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

      1. expm1-def 78.7%

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

      2. expm1-log1p 78.9%

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

   8. Simplified 78.9%

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

      1. div-inv 78.8%

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

      2. metadata-eval 78.8%

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

      3. sqrt-div 78.9%

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

      4. *-commutative 78.9%

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

      5. inv-pow 78.9%

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

      6. sqrt-pow1 79.0%

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

      7. metadata-eval 79.0%

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

   10. Applied egg-rr 79.0%

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

4. Final simplification 67.6%

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

Alternative 5: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.35e-5) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.35e-5) {
		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 <= 1.35d-5) 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 <= 1.35e-5) {
		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 <= 1.35e-5:
		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 <= 1.35e-5)
		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 <= 1.35e-5)
		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, 1.35e-5], 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 < 1.3499999999999999e-5

   1. Initial program 53.1%

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

   2. Taylor expanded in re around 0 63.0%

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

      1. expm1-log1p-u 59.5%

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

      2. expm1-udef 47.7%

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

      3. sqrt-unprod 47.7%

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

   4. Applied egg-rr 47.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im \cdot 2}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 59.6%

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

      2. expm1-log1p 63.4%

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

   6. Simplified 63.4%

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

   if 1.3499999999999999e-5 < re

   1. Initial program 4.8%

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

   2. Taylor expanded in im around 0 78.1%

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

      1. *-commutative 78.1%

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

   4. Simplified 78.1%

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

      1. expm1-log1p-u 78.0%

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

      2. expm1-udef 26.9%

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

      3. sqrt-unprod 26.9%

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

      4. metadata-eval 26.9%

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

      5. metadata-eval 26.9%

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

      6. *-rgt-identity 26.9%

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

      7. sqrt-div 26.9%

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

      8. metadata-eval 26.9%

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

      9. un-div-inv 26.9%

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

   6. Applied egg-rr 26.9%

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

      1. expm1-def 78.7%

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

      2. expm1-log1p 78.9%

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

   8. Simplified 78.9%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 6: 52.3% 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 40.0%

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

2. Taylor expanded in re around 0 53.1%

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

   1. expm1-log1p-u 50.1%

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

   2. expm1-udef 45.9%

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

   3. sqrt-unprod 45.9%

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

4. Applied egg-rr 45.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{im \cdot 2}\right)} - 1\right)} \]
5. Step-by-step derivation

   1. expm1-def 50.1%

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

   2. expm1-log1p 53.4%

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

6. Simplified 53.4%

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

7. Final simplification 53.4%

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

Reproduce

herbie shell --seed 2023318 
(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)))))