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

Percentage Accurate: 40.8% → 90.4%

Time: 7.7s

Alternatives: 5

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.8% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* 0.5 (* im (sqrt (/ 1.0 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((1.0 / 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((1.0 / 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((1.0 / 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(Float64(1.0 / 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((1.0 / 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[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.9%

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

   2. Taylor expanded in re around inf 46.4%

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

   3. Taylor expanded in im around 0 99.7%

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

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

   1. Initial program 45.2%

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

      1. sqr-neg 45.2%

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

      2. sqr-neg 45.2%

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

      3. hypot-def 89.8%

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

   3. Simplified 89.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]

Alternative 2: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-153}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{-96} \lor \neg \left(re \leq 1.9 \cdot 10^{+76}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.4e+42)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 2.2e-153)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (or (<= re 3.1e-96) (not (<= re 1.9e+76)))
       (* 0.5 (/ im (sqrt re)))
       (* 0.5 (sqrt (* 2.0 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.4e+42) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.2e-153) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if ((re <= 3.1e-96) || !(re <= 1.9e+76)) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.4d+42)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 2.2d-153) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if ((re <= 3.1d-96) .or. (.not. (re <= 1.9d+76))) then
        tmp = 0.5d0 * (im / sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.4e+42) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.2e-153) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if ((re <= 3.1e-96) || !(re <= 1.9e+76)) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.4e+42:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 2.2e-153:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif (re <= 3.1e-96) or not (re <= 1.9e+76):
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.4e+42)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 2.2e-153)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif ((re <= 3.1e-96) || !(re <= 1.9e+76))
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.4e+42)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 2.2e-153)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif ((re <= 3.1e-96) || ~((re <= 1.9e+76)))
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.4e+42], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.2e-153], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 3.1e-96], N[Not[LessEqual[re, 1.9e+76]], $MachinePrecision]], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.4e42

   1. Initial program 36.6%

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

   2. Taylor expanded in re around -inf 81.8%

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

      1. *-commutative 81.8%

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

   4. Simplified 81.8%

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

   if -1.4e42 < re < 2.20000000000000001e-153

   1. Initial program 59.6%

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

   2. Taylor expanded in re around 0 84.7%

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

   if 2.20000000000000001e-153 < re < 3.0999999999999999e-96 or 1.90000000000000012e76 < re

   1. Initial program 15.2%

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

   2. Taylor expanded in re around inf 47.7%

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

      1. expm1-log1p-u 47.2%

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

      2. expm1-udef 27.0%

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

      3. associate-*r* 27.0%

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

      4. metadata-eval 27.0%

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

      5. *-un-lft-identity 27.0%

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

      6. sqrt-div 27.1%

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

      7. unpow2 27.1%

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

      8. sqrt-prod 38.5%

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

      9. add-sqr-sqrt 38.5%

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

   4. Applied egg-rr 38.5%

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

      1. expm1-def 81.6%

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

      2. expm1-log1p 82.7%

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

   6. Simplified 82.7%

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

   if 3.0999999999999999e-96 < re < 1.90000000000000012e76

   1. Initial program 34.2%

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

      1. hypot-udef 68.2%

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

      2. add-cube-cbrt 67.6%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}} \]

      3. pow3 67.6%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   3. Applied egg-rr 67.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   4. Taylor expanded in re around 0 66.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left({1}^{0.3333333333333333} \cdot im\right)}} \]
   5. Step-by-step derivation

      1. pow-base-1 66.9%

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

      2. *-lft-identity 66.9%

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

   6. Simplified 66.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-153}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{-96} \lor \neg \left(re \leq 1.9 \cdot 10^{+76}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 3: 74.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.3e+38)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1.06e+72) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.3e+38) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.06e+72) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.3d+38)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1.06d+72) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.3e+38) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.06e+72) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.3e+38:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1.06e+72:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.3e+38)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1.06e+72)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.3e+38)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1.06e+72)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.3e+38], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.06e+72], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.2999999999999999e38

   1. Initial program 38.4%

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

   2. Taylor expanded in re around -inf 81.0%

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

      1. *-commutative 81.0%

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

   4. Simplified 81.0%

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

   if -3.2999999999999999e38 < re < 1.06e72

   1. Initial program 51.9%

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

      1. hypot-udef 86.1%

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

      2. add-cube-cbrt 85.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}} \]

      3. pow3 85.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   3. Applied egg-rr 85.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   4. Taylor expanded in re around 0 76.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left({1}^{0.3333333333333333} \cdot im\right)}} \]
   5. Step-by-step derivation

      1. pow-base-1 76.2%

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

      2. *-lft-identity 76.2%

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

   6. Simplified 76.2%

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

   if 1.06e72 < re

   1. Initial program 8.2%

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

   2. Taylor expanded in re around inf 61.5%

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

      1. expm1-log1p-u 60.8%

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

      2. expm1-udef 34.1%

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

      3. associate-*r* 34.1%

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

      4. metadata-eval 34.1%

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

      5. *-un-lft-identity 34.1%

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

      6. sqrt-div 34.1%

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

      7. unpow2 34.1%

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

      8. sqrt-prod 49.2%

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

      9. add-sqr-sqrt 49.2%

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

   4. Applied egg-rr 49.2%

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

      1. expm1-def 85.8%

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

      2. expm1-log1p 87.3%

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

   6. Simplified 87.3%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7.2 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7.2e+70) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7.2e+70) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 7.2d+70) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 7.2e+70) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7.2e+70:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7.2e+70)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 7.2e+70)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7.2e+70], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 7.1999999999999999e70

   1. Initial program 47.6%

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

      1. hypot-udef 90.5%

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

      2. add-cube-cbrt 89.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}} \]

      3. pow3 89.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   3. Applied egg-rr 89.5%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   4. Taylor expanded in re around 0 59.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left({1}^{0.3333333333333333} \cdot im\right)}} \]
   5. Step-by-step derivation

      1. pow-base-1 59.6%

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

      2. *-lft-identity 59.6%

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

   6. Simplified 59.6%

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

   if 7.1999999999999999e70 < re

   1. Initial program 8.2%

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

   2. Taylor expanded in re around inf 61.5%

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

      1. expm1-log1p-u 60.8%

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

      2. expm1-udef 34.1%

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

      3. associate-*r* 34.1%

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

      4. metadata-eval 34.1%

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

      5. *-un-lft-identity 34.1%

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

      6. sqrt-div 34.1%

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

      7. unpow2 34.1%

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

      8. sqrt-prod 49.2%

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

      9. add-sqr-sqrt 49.2%

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

   4. Applied egg-rr 49.2%

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

      1. expm1-def 85.8%

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

      2. expm1-log1p 87.3%

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

   6. Simplified 87.3%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.2 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 52.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.9%

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

   1. hypot-udef 82.5%

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

   2. add-cube-cbrt 81.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}} \]

   3. pow3 81.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

3. Applied egg-rr 81.6%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

4. Taylor expanded in re around 0 53.7%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left({1}^{0.3333333333333333} \cdot im\right)}} \]
5. Step-by-step derivation

   1. pow-base-1 53.7%

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

   2. *-lft-identity 53.7%

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

6. Simplified 53.7%

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

7. Final simplification 53.7%

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

Reproduce

herbie shell --seed 2023334 
(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)))))