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

Percentage Accurate: 41.2% → 89.4%

Time: 6.6s

Alternatives: 7

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.4% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.9%

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

   2. Taylor expanded in re around inf 41.5%

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

      1. unpow2 41.5%

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

   4. Simplified 41.5%

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

      1. add-log-exp 9.3%

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

      2. *-un-lft-identity 9.3%

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

      3. log-prod 9.3%

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

      4. metadata-eval 9.3%

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

      5. add-log-exp 41.5%

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

      6. sqrt-div 47.7%

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

      7. sqrt-prod 87.4%

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

      8. add-sqr-sqrt 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. +-lft-identity 87.8%

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

   8. Simplified 87.8%

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

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

   1. Initial program 40.9%

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

      1. hypot-def 89.3%

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

   3. Simplified 89.3%

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

      1. add-sqr-sqrt 88.6%

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

      2. sqrt-unprod 89.3%

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

      3. *-commutative 89.3%

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

      4. *-commutative 89.3%

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

      5. swap-sqr 89.3%

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

      6. add-sqr-sqrt 89.3%

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

      7. metadata-eval 89.3%

         \[\leadsto \sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \color{blue}{0.25}} \]

   5. Applied egg-rr 89.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}} \]
   6. Step-by-step derivation

      1. *-commutative 89.3%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}} \]

      2. associate-*r* 89.3%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      3. metadata-eval 89.3%

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

   7. Simplified 89.3%

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

4. Final simplification 89.1%

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

Alternative 2: 76.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -1.52 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -2.4 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -5.8 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 12.5:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0))))
        (t_1 (* 0.5 (sqrt (* 2.0 (- im re))))))
   (if (<= re -1.52e+97)
     t_0
     (if (<= re -2.4e-11)
       t_1
       (if (<= re -5.8e-25)
         t_0
         (if (<= re 12.5) t_1 (* im (/ 0.5 (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -1.52e+97) {
		tmp = t_0;
	} else if (re <= -2.4e-11) {
		tmp = t_1;
	} else if (re <= -5.8e-25) {
		tmp = t_0;
	} else if (re <= 12.5) {
		tmp = t_1;
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    if (re <= (-1.52d+97)) then
        tmp = t_0
    else if (re <= (-2.4d-11)) then
        tmp = t_1
    else if (re <= (-5.8d-25)) then
        tmp = t_0
    else if (re <= 12.5d0) then
        tmp = t_1
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -1.52e+97) {
		tmp = t_0;
	} else if (re <= -2.4e-11) {
		tmp = t_1;
	} else if (re <= -5.8e-25) {
		tmp = t_0;
	} else if (re <= 12.5) {
		tmp = t_1;
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((2.0 * (im - re)))
	tmp = 0
	if re <= -1.52e+97:
		tmp = t_0
	elif re <= -2.4e-11:
		tmp = t_1
	elif re <= -5.8e-25:
		tmp = t_0
	elif re <= 12.5:
		tmp = t_1
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	tmp = 0.0
	if (re <= -1.52e+97)
		tmp = t_0;
	elseif (re <= -2.4e-11)
		tmp = t_1;
	elseif (re <= -5.8e-25)
		tmp = t_0;
	elseif (re <= 12.5)
		tmp = t_1;
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * sqrt((2.0 * (im - re)));
	tmp = 0.0;
	if (re <= -1.52e+97)
		tmp = t_0;
	elseif (re <= -2.4e-11)
		tmp = t_1;
	elseif (re <= -5.8e-25)
		tmp = t_0;
	elseif (re <= 12.5)
		tmp = t_1;
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.52e+97], t$95$0, If[LessEqual[re, -2.4e-11], t$95$1, If[LessEqual[re, -5.8e-25], t$95$0, If[LessEqual[re, 12.5], t$95$1, N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.5200000000000001e97 or -2.4000000000000001e-11 < re < -5.8000000000000001e-25

   1. Initial program 23.1%

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

   2. Taylor expanded in re around -inf 91.4%

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

      1. *-commutative 91.4%

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

   4. Simplified 91.4%

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

   if -1.5200000000000001e97 < re < -2.4000000000000001e-11 or -5.8000000000000001e-25 < re < 12.5

   1. Initial program 53.6%

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

   2. Taylor expanded in re around 0 80.1%

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

   if 12.5 < re

   1. Initial program 14.2%

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

   2. Taylor expanded in re around inf 41.4%

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

      1. unpow2 41.4%

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

   4. Simplified 41.4%

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

      1. associate-/l* 45.8%

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

      2. sqrt-div 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. add-log-exp 13.6%

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

      2. *-un-lft-identity 13.6%

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

      3. log-prod 13.6%

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

      4. metadata-eval 13.6%

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

      5. add-log-exp 59.8%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \frac{\sqrt{im}}{\sqrt{\frac{re}{im}}}} \]

      6. clear-num 59.7%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\frac{re}{im}}}{\sqrt{im}}}} \]

      7. un-div-inv 59.7%

         \[\leadsto 0 + \color{blue}{\frac{0.5}{\frac{\sqrt{\frac{re}{im}}}{\sqrt{im}}}} \]

      8. sqrt-div 72.4%

         \[\leadsto 0 + \frac{0.5}{\frac{\color{blue}{\frac{\sqrt{re}}{\sqrt{im}}}}{\sqrt{im}}} \]

      9. associate-/l/ 72.5%

         \[\leadsto 0 + \frac{0.5}{\color{blue}{\frac{\sqrt{re}}{\sqrt{im} \cdot \sqrt{im}}}} \]

      10. add-sqr-sqrt 72.7%

          \[\leadsto 0 + \frac{0.5}{\frac{\sqrt{re}}{\color{blue}{im}}} \]

   8. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{0 + \frac{0.5}{\frac{\sqrt{re}}{im}}} \]
   9. Step-by-step derivation

      1. +-lft-identity 72.7%

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

      2. associate-/r/ 73.8%

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

   10. Simplified 73.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.52 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -2.4 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -5.8 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 12.5:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -3.6 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4 - \frac{im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -2.2 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- im re))))))
   (if (<= re -3.6e+97)
     (* 0.5 (sqrt (- (* re -4.0) (/ im (/ re im)))))
     (if (<= re -1.9e-11)
       t_0
       (if (<= re -2.2e-25)
         (* 0.5 (sqrt (* re -4.0)))
         (if (<= re 1.45e-15) t_0 (* im (/ 0.5 (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -3.6e+97) {
		tmp = 0.5 * sqrt(((re * -4.0) - (im / (re / im))));
	} else if (re <= -1.9e-11) {
		tmp = t_0;
	} else if (re <= -2.2e-25) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.45e-15) {
		tmp = t_0;
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    if (re <= (-3.6d+97)) then
        tmp = 0.5d0 * sqrt(((re * (-4.0d0)) - (im / (re / im))))
    else if (re <= (-1.9d-11)) then
        tmp = t_0
    else if (re <= (-2.2d-25)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.45d-15) then
        tmp = t_0
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -3.6e+97) {
		tmp = 0.5 * Math.sqrt(((re * -4.0) - (im / (re / im))));
	} else if (re <= -1.9e-11) {
		tmp = t_0;
	} else if (re <= -2.2e-25) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.45e-15) {
		tmp = t_0;
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (im - re)))
	tmp = 0
	if re <= -3.6e+97:
		tmp = 0.5 * math.sqrt(((re * -4.0) - (im / (re / im))))
	elif re <= -1.9e-11:
		tmp = t_0
	elif re <= -2.2e-25:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.45e-15:
		tmp = t_0
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	tmp = 0.0
	if (re <= -3.6e+97)
		tmp = Float64(0.5 * sqrt(Float64(Float64(re * -4.0) - Float64(im / Float64(re / im)))));
	elseif (re <= -1.9e-11)
		tmp = t_0;
	elseif (re <= -2.2e-25)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.45e-15)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (im - re)));
	tmp = 0.0;
	if (re <= -3.6e+97)
		tmp = 0.5 * sqrt(((re * -4.0) - (im / (re / im))));
	elseif (re <= -1.9e-11)
		tmp = t_0;
	elseif (re <= -2.2e-25)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.45e-15)
		tmp = t_0;
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3.6e+97], N[(0.5 * N[Sqrt[N[(N[(re * -4.0), $MachinePrecision] - N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.9e-11], t$95$0, If[LessEqual[re, -2.2e-25], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.45e-15], t$95$0, N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.59999999999999966e97

   1. Initial program 13.8%

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

   2. Taylor expanded in re around -inf 79.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re} + -4 \cdot re}} \]
   3. Step-by-step derivation

      1. +-commutative 79.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re + -1 \cdot \frac{{im}^{2}}{re}}} \]

      2. mul-1-neg 79.0%

         \[\leadsto 0.5 \cdot \sqrt{-4 \cdot re + \color{blue}{\left(-\frac{{im}^{2}}{re}\right)}} \]

      3. unsub-neg 79.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re - \frac{{im}^{2}}{re}}} \]

      4. *-commutative 79.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4} - \frac{{im}^{2}}{re}} \]

      5. unpow2 79.0%

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

      6. associate-/l* 90.8%

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

   4. Simplified 90.8%

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

   if -3.59999999999999966e97 < re < -1.8999999999999999e-11 or -2.2000000000000002e-25 < re < 1.45000000000000009e-15

   1. Initial program 53.6%

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

   2. Taylor expanded in re around 0 80.1%

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

   if -1.8999999999999999e-11 < re < -2.2000000000000002e-25

   1. Initial program 100.0%

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

   2. Taylor expanded in re around -inf 100.0%

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

      1. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   if 1.45000000000000009e-15 < re

   1. Initial program 14.2%

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

   2. Taylor expanded in re around inf 41.4%

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

      1. unpow2 41.4%

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

   4. Simplified 41.4%

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

      1. associate-/l* 45.8%

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

      2. sqrt-div 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. add-log-exp 13.6%

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

      2. *-un-lft-identity 13.6%

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

      3. log-prod 13.6%

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

      4. metadata-eval 13.6%

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

      5. add-log-exp 59.8%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \frac{\sqrt{im}}{\sqrt{\frac{re}{im}}}} \]

      6. clear-num 59.7%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\frac{re}{im}}}{\sqrt{im}}}} \]

      7. un-div-inv 59.7%

         \[\leadsto 0 + \color{blue}{\frac{0.5}{\frac{\sqrt{\frac{re}{im}}}{\sqrt{im}}}} \]

      8. sqrt-div 72.4%

         \[\leadsto 0 + \frac{0.5}{\frac{\color{blue}{\frac{\sqrt{re}}{\sqrt{im}}}}{\sqrt{im}}} \]

      9. associate-/l/ 72.5%

         \[\leadsto 0 + \frac{0.5}{\color{blue}{\frac{\sqrt{re}}{\sqrt{im} \cdot \sqrt{im}}}} \]

      10. add-sqr-sqrt 72.7%

          \[\leadsto 0 + \frac{0.5}{\frac{\sqrt{re}}{\color{blue}{im}}} \]

   8. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{0 + \frac{0.5}{\frac{\sqrt{re}}{im}}} \]
   9. Step-by-step derivation

      1. +-lft-identity 72.7%

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

      2. associate-/r/ 73.8%

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

   10. Simplified 73.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4 - \frac{im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -2.2 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 76.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e-26)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 7.2e-5) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e-26) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 7.2e-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 <= (-3.6d-26)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 7.2d-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 <= -3.6e-26) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 7.2e-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 <= -3.6e-26:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 7.2e-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 <= -3.6e-26)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 7.2e-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 <= -3.6e-26)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 7.2e-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, -3.6e-26], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.2e-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 3 regimes

2. if re < -3.6000000000000001e-26

   1. Initial program 31.8%

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

   2. Taylor expanded in re around -inf 79.5%

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

      1. *-commutative 79.5%

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

   4. Simplified 79.5%

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

   if -3.6000000000000001e-26 < re < 7.20000000000000018e-5

   1. Initial program 52.6%

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

   2. Taylor expanded in re around 0 80.9%

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

      1. *-commutative 80.9%

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

   4. Simplified 80.9%

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

   if 7.20000000000000018e-5 < re

   1. Initial program 14.2%

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

   2. Taylor expanded in re around inf 41.4%

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

      1. unpow2 41.4%

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

   4. Simplified 41.4%

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

      1. add-log-exp 13.7%

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

      2. *-un-lft-identity 13.7%

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

      3. log-prod 13.7%

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

      4. metadata-eval 13.7%

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

      5. add-log-exp 41.4%

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

      6. sqrt-div 59.7%

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

      7. sqrt-prod 73.5%

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

      8. add-sqr-sqrt 73.8%

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

   6. Applied egg-rr 73.8%

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

      1. +-lft-identity 73.8%

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

   8. Simplified 73.8%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 76.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1e-27)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 5.2e-5) (* 0.5 (sqrt (* im 2.0))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1e-27) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5.2e-5) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = im * (0.5 / 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 <= (-1.1d-27)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5.2d-5) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.1e-27) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5.2e-5) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.1e-27:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5.2e-5:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1e-27)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5.2e-5)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.1e-27)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5.2e-5)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1e-27], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.2e-5], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.09999999999999993e-27

   1. Initial program 31.8%

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

   2. Taylor expanded in re around -inf 79.5%

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

      1. *-commutative 79.5%

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

   4. Simplified 79.5%

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

   if -1.09999999999999993e-27 < re < 5.19999999999999968e-5

   1. Initial program 52.6%

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

   2. Taylor expanded in re around 0 80.9%

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

      1. *-commutative 80.9%

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

   4. Simplified 80.9%

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

   if 5.19999999999999968e-5 < re

   1. Initial program 14.2%

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

   2. Taylor expanded in re around inf 41.4%

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

      1. unpow2 41.4%

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

   4. Simplified 41.4%

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

      1. associate-/l* 45.8%

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

      2. sqrt-div 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. add-log-exp 13.6%

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

      2. *-un-lft-identity 13.6%

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

      3. log-prod 13.6%

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

      4. metadata-eval 13.6%

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

      5. add-log-exp 59.8%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \frac{\sqrt{im}}{\sqrt{\frac{re}{im}}}} \]

      6. clear-num 59.7%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\frac{1}{\frac{\sqrt{\frac{re}{im}}}{\sqrt{im}}}} \]

      7. un-div-inv 59.7%

         \[\leadsto 0 + \color{blue}{\frac{0.5}{\frac{\sqrt{\frac{re}{im}}}{\sqrt{im}}}} \]

      8. sqrt-div 72.4%

         \[\leadsto 0 + \frac{0.5}{\frac{\color{blue}{\frac{\sqrt{re}}{\sqrt{im}}}}{\sqrt{im}}} \]

      9. associate-/l/ 72.5%

         \[\leadsto 0 + \frac{0.5}{\color{blue}{\frac{\sqrt{re}}{\sqrt{im} \cdot \sqrt{im}}}} \]

      10. add-sqr-sqrt 72.7%

          \[\leadsto 0 + \frac{0.5}{\frac{\sqrt{re}}{\color{blue}{im}}} \]

   8. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{0 + \frac{0.5}{\frac{\sqrt{re}}{im}}} \]
   9. Step-by-step derivation

      1. +-lft-identity 72.7%

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

      2. associate-/r/ 73.8%

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

   10. Simplified 73.8%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 6: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e-25) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e-25) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	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-25)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.6e-25) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.6e-25:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.6e-25)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.6e-25)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e-25], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.5999999999999999e-25

   1. Initial program 31.8%

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

   2. Taylor expanded in re around -inf 79.5%

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

      1. *-commutative 79.5%

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

   4. Simplified 79.5%

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

   if -3.5999999999999999e-25 < re

   1. Initial program 37.7%

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

   2. Taylor expanded in re around 0 61.8%

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

      1. *-commutative 61.8%

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

   4. Simplified 61.8%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7: 52.1% 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 36.0%

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

2. Taylor expanded in re around 0 51.2%

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

   1. *-commutative 51.2%

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

4. Simplified 51.2%

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

5. Final simplification 51.2%

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

Reproduce

herbie shell --seed 2023173 
(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)))))