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

Percentage Accurate: 41.6% → 87.8%

Time: 9.1s

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: 41.6% 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: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9.2 \cdot 10^{-54} \lor \neg \left(re \leq 2300\right) \land re \leq 3.25 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 9.2e-54) (and (not (<= re 2300.0)) (<= re 3.25e+43)))
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 9.2e-54) || (!(re <= 2300.0) && (re <= 3.25e+43))) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 9.2e-54) || (!(re <= 2300.0) && (re <= 3.25e+43))) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 9.2e-54) or (not (re <= 2300.0) and (re <= 3.25e+43)):
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 9.2e-54) || (!(re <= 2300.0) && (re <= 3.25e+43)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	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 <= 9.2e-54) || (~((re <= 2300.0)) && (re <= 3.25e+43)))
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 9.2e-54], And[N[Not[LessEqual[re, 2300.0]], $MachinePrecision], LessEqual[re, 3.25e+43]]], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.1999999999999996e-54 or 2300 < re < 3.2499999999999999e43

   1. Initial program 51.6%

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

      1. sub-neg 51.6%

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

      2. sqr-neg 51.6%

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

      3. sub-neg 51.6%

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

      4. sqr-neg 51.6%

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

      5. hypot-define 94.8%

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

   3. Simplified 94.8%

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

   if 9.1999999999999996e-54 < re < 2300 or 3.2499999999999999e43 < re

   1. Initial program 8.2%

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

   3. Taylor expanded in re around inf 88.5%

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

      1. associate-*l* 88.5%

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

      2. *-commutative 88.5%

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

   5. Simplified 88.5%

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

      1. add-log-exp 16.4%

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

      2. *-un-lft-identity 16.4%

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

      3. log-prod 16.4%

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

      4. metadata-eval 16.4%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]

      5. add-log-exp 88.5%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      6. sqrt-unprod 89.6%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      7. metadata-eval 89.6%

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

      8. metadata-eval 89.6%

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

      9. *-un-lft-identity 89.6%

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

      10. sqrt-div 89.4%

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

      11. metadata-eval 89.4%

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

      12. un-div-inv 89.7%

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

   7. Applied egg-rr 89.7%

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

      1. +-lft-identity 89.7%

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

   9. Simplified 89.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9.2 \cdot 10^{-54} \lor \neg \left(re \leq 2300\right) \land re \leq 3.25 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-55} \lor \neg \left(re \leq 1620\right) \land re \leq 9.5 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.9e+26)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (or (<= re 7e-55) (and (not (<= re 1620.0)) (<= re 9.5e+31)))
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.9e+26) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 7e-55) || (!(re <= 1620.0) && (re <= 9.5e+31))) {
		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 <= (-2.9d+26)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if ((re <= 7d-55) .or. (.not. (re <= 1620.0d0)) .and. (re <= 9.5d+31)) 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 <= -2.9e+26) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 7e-55) || (!(re <= 1620.0) && (re <= 9.5e+31))) {
		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 <= -2.9e+26:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif (re <= 7e-55) or (not (re <= 1620.0) and (re <= 9.5e+31)):
		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 <= -2.9e+26)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif ((re <= 7e-55) || (!(re <= 1620.0) && (re <= 9.5e+31)))
		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 <= -2.9e+26)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif ((re <= 7e-55) || (~((re <= 1620.0)) && (re <= 9.5e+31)))
		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, -2.9e+26], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 7e-55], And[N[Not[LessEqual[re, 1620.0]], $MachinePrecision], LessEqual[re, 9.5e+31]]], 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 < -2.9e26

   1. Initial program 36.7%

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

   3. Taylor expanded in re around -inf 74.1%

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

      1. *-commutative 74.1%

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

   5. Simplified 74.1%

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

   if -2.9e26 < re < 7.00000000000000051e-55 or 1620 < re < 9.5000000000000008e31

   1. Initial program 57.5%

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

   3. Taylor expanded in re around 0 81.9%

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

      1. add-log-exp 9.2%

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

      2. *-un-lft-identity 9.2%

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

      3. log-prod 9.2%

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

      4. metadata-eval 9.2%

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

      5. add-log-exp 81.9%

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

      6. sqrt-unprod 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. +-lft-identity 82.5%

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

   7. Simplified 82.5%

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

   if 7.00000000000000051e-55 < re < 1620 or 9.5000000000000008e31 < re

   1. Initial program 9.8%

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

   3. Taylor expanded in re around inf 87.3%

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

      1. associate-*l* 87.3%

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

      2. *-commutative 87.3%

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

   5. Simplified 87.3%

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

      1. add-log-exp 17.7%

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

      2. *-un-lft-identity 17.7%

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

      3. log-prod 17.7%

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

      4. metadata-eval 17.7%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]

      5. add-log-exp 87.3%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      6. sqrt-unprod 88.3%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      7. metadata-eval 88.3%

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

      8. metadata-eval 88.3%

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

      9. *-un-lft-identity 88.3%

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

      10. sqrt-div 88.2%

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

      11. metadata-eval 88.2%

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

      12. un-div-inv 88.4%

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

   7. Applied egg-rr 88.4%

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

      1. +-lft-identity 88.4%

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

   9. Simplified 88.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-55} \lor \neg \left(re \leq 1620\right) \land re \leq 9.5 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1620 \lor \neg \left(re \leq 1.45 \cdot 10^{+32}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e+26)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1.75e-122)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (or (<= re 1620.0) (not (<= re 1.45e+32)))
       (* 0.5 (/ im (sqrt re)))
       (* 0.5 (sqrt (* 2.0 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e+26) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.75e-122) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if ((re <= 1620.0) || !(re <= 1.45e+32)) {
		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 <= (-5d+26)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1.75d-122) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if ((re <= 1620.0d0) .or. (.not. (re <= 1.45d+32))) 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 <= -5e+26) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.75e-122) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if ((re <= 1620.0) || !(re <= 1.45e+32)) {
		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 <= -5e+26:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1.75e-122:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif (re <= 1620.0) or not (re <= 1.45e+32):
		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 <= -5e+26)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1.75e-122)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif ((re <= 1620.0) || !(re <= 1.45e+32))
		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 <= -5e+26)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1.75e-122)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif ((re <= 1620.0) || ~((re <= 1.45e+32)))
		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, -5e+26], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.75e-122], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1620.0], N[Not[LessEqual[re, 1.45e+32]], $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 < -5.0000000000000001e26

   1. Initial program 36.7%

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

   3. Taylor expanded in re around -inf 74.1%

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

      1. *-commutative 74.1%

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

   5. Simplified 74.1%

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

   if -5.0000000000000001e26 < re < 1.7500000000000001e-122

   1. Initial program 60.7%

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

   3. Taylor expanded in re around 0 86.9%

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

   if 1.7500000000000001e-122 < re < 1620 or 1.45000000000000001e32 < re

   1. Initial program 15.6%

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

   3. Taylor expanded in re around inf 80.4%

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

      1. associate-*l* 80.4%

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

      2. *-commutative 80.4%

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

   5. Simplified 80.4%

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

      1. add-log-exp 15.2%

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

      2. *-un-lft-identity 15.2%

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

      3. log-prod 15.2%

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

      4. metadata-eval 15.2%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]

      5. add-log-exp 80.4%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      6. sqrt-unprod 81.4%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      7. metadata-eval 81.4%

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

      8. metadata-eval 81.4%

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

      9. *-un-lft-identity 81.4%

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

      10. sqrt-div 81.2%

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

      11. metadata-eval 81.2%

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

      12. un-div-inv 81.4%

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

   7. Applied egg-rr 81.4%

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

      1. +-lft-identity 81.4%

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

   9. Simplified 81.4%

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

   if 1620 < re < 1.45000000000000001e32

   1. Initial program 31.6%

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

   3. Taylor expanded in re around 0 85.9%

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

      1. add-log-exp 4.0%

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

      2. *-un-lft-identity 4.0%

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

      3. log-prod 4.0%

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

      4. metadata-eval 4.0%

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

      5. add-log-exp 85.9%

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

      6. sqrt-unprod 86.8%

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

   5. Applied egg-rr 86.8%

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

      1. +-lft-identity 86.8%

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

   7. Simplified 86.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1620 \lor \neg \left(re \leq 1.45 \cdot 10^{+32}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \left(1 - \frac{re}{im}\right)\right)}\\ \mathbf{elif}\;re \leq 2400 \lor \neg \left(re \leq 2.15 \cdot 10^{+33}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.35e+27)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1.5e-122)
     (* 0.5 (sqrt (* 2.0 (* im (- 1.0 (/ re im))))))
     (if (or (<= re 2400.0) (not (<= re 2.15e+33)))
       (* 0.5 (/ im (sqrt re)))
       (* 0.5 (sqrt (* 2.0 im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.35e+27) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.5e-122) {
		tmp = 0.5 * sqrt((2.0 * (im * (1.0 - (re / im)))));
	} else if ((re <= 2400.0) || !(re <= 2.15e+33)) {
		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 <= (-1.35d+27)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1.5d-122) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im * (1.0d0 - (re / im)))))
    else if ((re <= 2400.0d0) .or. (.not. (re <= 2.15d+33))) 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 <= -1.35e+27) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.5e-122) {
		tmp = 0.5 * Math.sqrt((2.0 * (im * (1.0 - (re / im)))));
	} else if ((re <= 2400.0) || !(re <= 2.15e+33)) {
		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 <= -1.35e+27:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1.5e-122:
		tmp = 0.5 * math.sqrt((2.0 * (im * (1.0 - (re / im)))))
	elif (re <= 2400.0) or not (re <= 2.15e+33):
		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 <= -1.35e+27)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1.5e-122)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im * Float64(1.0 - Float64(re / im))))));
	elseif ((re <= 2400.0) || !(re <= 2.15e+33))
		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 <= -1.35e+27)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1.5e-122)
		tmp = 0.5 * sqrt((2.0 * (im * (1.0 - (re / im)))));
	elseif ((re <= 2400.0) || ~((re <= 2.15e+33)))
		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, -1.35e+27], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.5e-122], N[(0.5 * N[Sqrt[N[(2.0 * N[(im * N[(1.0 - N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 2400.0], N[Not[LessEqual[re, 2.15e+33]], $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 < -1.3499999999999999e27

   1. Initial program 36.7%

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

   3. Taylor expanded in re around -inf 74.1%

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

      1. *-commutative 74.1%

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

   5. Simplified 74.1%

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

   if -1.3499999999999999e27 < re < 1.50000000000000002e-122

   1. Initial program 60.7%

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

   3. Taylor expanded in im around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

   5. Simplified 86.9%

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

   if 1.50000000000000002e-122 < re < 2400 or 2.15000000000000014e33 < re

   1. Initial program 15.6%

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

   3. Taylor expanded in re around inf 80.4%

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

      1. associate-*l* 80.4%

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

      2. *-commutative 80.4%

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

   5. Simplified 80.4%

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

      1. add-log-exp 15.2%

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

      2. *-un-lft-identity 15.2%

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

      3. log-prod 15.2%

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

      4. metadata-eval 15.2%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]

      5. add-log-exp 80.4%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      6. sqrt-unprod 81.4%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      7. metadata-eval 81.4%

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

      8. metadata-eval 81.4%

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

      9. *-un-lft-identity 81.4%

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

      10. sqrt-div 81.2%

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

      11. metadata-eval 81.2%

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

      12. un-div-inv 81.4%

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

   7. Applied egg-rr 81.4%

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

      1. +-lft-identity 81.4%

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

   9. Simplified 81.4%

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

   if 2400 < re < 2.15000000000000014e33

   1. Initial program 31.6%

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

   3. Taylor expanded in re around 0 85.9%

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

      1. add-log-exp 4.0%

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

      2. *-un-lft-identity 4.0%

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

      3. log-prod 4.0%

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

      4. metadata-eval 4.0%

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

      5. add-log-exp 85.9%

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

      6. sqrt-unprod 86.8%

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

   5. Applied egg-rr 86.8%

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

      1. +-lft-identity 86.8%

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

   7. Simplified 86.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \left(1 - \frac{re}{im}\right)\right)}\\ \mathbf{elif}\;re \leq 2400 \lor \neg \left(re \leq 2.15 \cdot 10^{+33}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.2 \cdot 10^{-52} \lor \neg \left(re \leq 850\right) \land re \leq 3.9 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 3.2e-52) (and (not (<= re 850.0)) (<= re 3.9e+34)))
   (* 0.5 (sqrt (* 2.0 im)))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 3.2e-52) || (!(re <= 850.0) && (re <= 3.9e+34))) {
		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 <= 3.2d-52) .or. (.not. (re <= 850.0d0)) .and. (re <= 3.9d+34)) 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 <= 3.2e-52) || (!(re <= 850.0) && (re <= 3.9e+34))) {
		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 <= 3.2e-52) or (not (re <= 850.0) and (re <= 3.9e+34)):
		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 <= 3.2e-52) || (!(re <= 850.0) && (re <= 3.9e+34)))
		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 <= 3.2e-52) || (~((re <= 850.0)) && (re <= 3.9e+34)))
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 3.2e-52], And[N[Not[LessEqual[re, 850.0]], $MachinePrecision], LessEqual[re, 3.9e+34]]], 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 < 3.2000000000000001e-52 or 850 < re < 3.90000000000000019e34

   1. Initial program 51.6%

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

   3. Taylor expanded in re around 0 66.5%

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

      1. add-log-exp 7.6%

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

      2. *-un-lft-identity 7.6%

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

      3. log-prod 7.6%

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

      4. metadata-eval 7.6%

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

      5. add-log-exp 66.5%

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

      6. sqrt-unprod 67.0%

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

   5. Applied egg-rr 67.0%

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

      1. +-lft-identity 67.0%

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

   7. Simplified 67.0%

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

   if 3.2000000000000001e-52 < re < 850 or 3.90000000000000019e34 < re

   1. Initial program 9.8%

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

   3. Taylor expanded in re around inf 87.3%

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

      1. associate-*l* 87.3%

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

      2. *-commutative 87.3%

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

   5. Simplified 87.3%

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

      1. add-log-exp 17.7%

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

      2. *-un-lft-identity 17.7%

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

      3. log-prod 17.7%

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

      4. metadata-eval 17.7%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]

      5. add-log-exp 87.3%

         \[\leadsto 0.5 \cdot \left(0 + \color{blue}{im \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      6. sqrt-unprod 88.3%

         \[\leadsto 0.5 \cdot \left(0 + im \cdot \left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      7. metadata-eval 88.3%

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

      8. metadata-eval 88.3%

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

      9. *-un-lft-identity 88.3%

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

      10. sqrt-div 88.2%

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

      11. metadata-eval 88.2%

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

      12. un-div-inv 88.4%

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

   7. Applied egg-rr 88.4%

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

      1. +-lft-identity 88.4%

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

   9. Simplified 88.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.2 \cdot 10^{-52} \lor \neg \left(re \leq 850\right) \land re \leq 3.9 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% 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 42.3%

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

3. Taylor expanded in re around 0 55.6%

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

   1. add-log-exp 9.6%

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

   2. *-un-lft-identity 9.6%

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

   3. log-prod 9.6%

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

   4. metadata-eval 9.6%

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

   5. add-log-exp 55.6%

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

   6. sqrt-unprod 55.9%

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

5. Applied egg-rr 55.9%

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

   1. +-lft-identity 55.9%

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

7. Simplified 55.9%

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

8. Final simplification 55.9%

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

Reproduce

herbie shell --seed 2024066 
(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)))))