math.sqrt on complex, real part

Percentage Accurate: 41.5% → 89.7%

Time: 10.7s

Alternatives: 5

Speedup: 2.0×

Specification

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: 41.5% 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: 89.7% accurate, 0.7× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im_m im_m)))) 0.0)
   (* 0.5 (/ im_m (sqrt (- re))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * (im_m / sqrt(-re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if ((re + Math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * (im_m / Math.sqrt(-re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if (re + math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0:
		tmp = 0.5 * (im_m / math.sqrt(-re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))) <= 0.0)
		tmp = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im_m)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0)
		tmp = 0.5 * (im_m / sqrt(-re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.0%

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

      1. sqr-neg 5.0%

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

      2. +-commutative 5.0%

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

      3. sqr-neg 5.0%

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

      4. +-commutative 5.0%

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

      5. distribute-rgt-in 5.0%

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

      6. cancel-sign-sub 5.0%

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

      7. distribute-rgt-out-- 5.0%

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

      8. sub-neg 5.0%

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

      9. remove-double-neg 5.0%

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

      10. +-commutative 5.0%

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

      11. hypot-define 13.8%

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

   3. Simplified 13.8%

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

   5. Taylor expanded in re around -inf 48.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re} + 0.25 \cdot \frac{{im}^{4}}{{re}^{3}}}} \]
   6. Step-by-step derivation

      1. +-commutative 48.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + -1 \cdot \frac{{im}^{2}}{re}}} \]

      2. mul-1-neg 48.2%

         \[\leadsto 0.5 \cdot \sqrt{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + \color{blue}{\left(-\frac{{im}^{2}}{re}\right)}} \]

      3. unsub-neg 48.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} - \frac{{im}^{2}}{re}}} \]

      4. *-commutative 48.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25} - \frac{{im}^{2}}{re}} \]

   7. Simplified 48.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 47.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}} \cdot \sqrt{\sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}}\right)} \]

      2. pow2 47.8%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}}\right)}^{2}} \]

      3. pow1/2 47.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{0.5}}}\right)}^{2} \]

      4. metadata-eval 47.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt{{\left(\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right)}^{2} \]

      5. sqrt-pow1 47.9%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}}^{2} \]

      6. div-inv 47.8%

         \[\leadsto 0.5 \cdot {\left({\left(\color{blue}{\left({im}^{4} \cdot \frac{1}{{re}^{3}}\right)} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      7. associate-*l* 47.8%

         \[\leadsto 0.5 \cdot {\left({\left(\color{blue}{{im}^{4} \cdot \left(\frac{1}{{re}^{3}} \cdot 0.25\right)} - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      8. pow-flip 47.8%

         \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left(\color{blue}{{re}^{\left(-3\right)}} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      9. metadata-eval 47.8%

         \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left({re}^{\color{blue}{-3}} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      10. metadata-eval 47.8%

          \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left({re}^{-3} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right)}^{2} \]

      11. metadata-eval 47.8%

          \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left({re}^{-3} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   9. Applied egg-rr 47.8%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left({im}^{4} \cdot \left({re}^{-3} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{0.25}\right)}^{2}} \]

   10. Taylor expanded in im around 0 48.9%

       \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(-1 \cdot \frac{{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]
   11. Step-by-step derivation

       1. mul-1-neg 48.9%

          \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(-\frac{{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]

       2. distribute-frac-neg 48.9%

          \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(\frac{-{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]

   12. Simplified 48.9%

       \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(\frac{-{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]
   13. Step-by-step derivation

       1. pow-pow 49.3%

          \[\leadsto 0.5 \cdot \color{blue}{{\left(\frac{-{im}^{2}}{re}\right)}^{\left(0.25 \cdot 2\right)}} \]

       2. metadata-eval 49.3%

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

       3. pow1/2 49.3%

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

       4. frac-2neg 49.3%

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

       5. sqrt-div 52.6%

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

       6. remove-double-neg 52.6%

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

       7. unpow2 52.6%

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

       8. sqrt-prod 51.9%

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

       9. add-sqr-sqrt 53.9%

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

   14. Applied egg-rr 53.9%

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

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

   1. Initial program 44.9%

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

      1. sqr-neg 44.9%

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

      2. +-commutative 44.9%

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

      3. sqr-neg 44.9%

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

      4. +-commutative 44.9%

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

      5. distribute-rgt-in 44.9%

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

      6. cancel-sign-sub 44.9%

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

      7. distribute-rgt-out-- 44.9%

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

      8. sub-neg 44.9%

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

      9. remove-double-neg 44.9%

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

      10. +-commutative 44.9%

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

      11. hypot-define 90.3%

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

   3. Simplified 90.3%

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

4. Final simplification 83.5%

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

Alternative 2: 75.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \frac{im\_m}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{-76} \lor \neg \left(re \leq 7.2 \cdot 10^{-37}\right) \land re \leq 1.02 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5.2e-43)
   (* 0.5 (/ im_m (sqrt (- re))))
   (if (or (<= re 5.4e-76) (and (not (<= re 7.2e-37)) (<= re 1.02e+60)))
     (* 0.5 (sqrt (* 2.0 (+ re im_m))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5.2e-43) {
		tmp = 0.5 * (im_m / sqrt(-re));
	} else if ((re <= 5.4e-76) || (!(re <= 7.2e-37) && (re <= 1.02e+60))) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-5.2d-43)) then
        tmp = 0.5d0 * (im_m / sqrt(-re))
    else if ((re <= 5.4d-76) .or. (.not. (re <= 7.2d-37)) .and. (re <= 1.02d+60)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -5.2e-43) {
		tmp = 0.5 * (im_m / Math.sqrt(-re));
	} else if ((re <= 5.4e-76) || (!(re <= 7.2e-37) && (re <= 1.02e+60))) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -5.2e-43:
		tmp = 0.5 * (im_m / math.sqrt(-re))
	elif (re <= 5.4e-76) or (not (re <= 7.2e-37) and (re <= 1.02e+60)):
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -5.2e-43)
		tmp = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))));
	elseif ((re <= 5.4e-76) || (!(re <= 7.2e-37) && (re <= 1.02e+60)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -5.2e-43)
		tmp = 0.5 * (im_m / sqrt(-re));
	elseif ((re <= 5.4e-76) || (~((re <= 7.2e-37)) && (re <= 1.02e+60)))
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -5.2e-43], N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 5.4e-76], And[N[Not[LessEqual[re, 7.2e-37]], $MachinePrecision], LessEqual[re, 1.02e+60]]], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.2e-43

   1. Initial program 11.2%

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

      1. sqr-neg 11.2%

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

      2. +-commutative 11.2%

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

      3. sqr-neg 11.2%

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

      4. +-commutative 11.2%

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

      5. distribute-rgt-in 11.2%

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

      6. cancel-sign-sub 11.2%

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

      7. distribute-rgt-out-- 11.2%

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

      8. sub-neg 11.2%

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

      9. remove-double-neg 11.2%

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

      10. +-commutative 11.2%

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

      11. hypot-define 32.7%

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

   3. Simplified 32.7%

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

   5. Taylor expanded in re around -inf 51.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re} + 0.25 \cdot \frac{{im}^{4}}{{re}^{3}}}} \]
   6. Step-by-step derivation

      1. +-commutative 51.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + -1 \cdot \frac{{im}^{2}}{re}}} \]

      2. mul-1-neg 51.1%

         \[\leadsto 0.5 \cdot \sqrt{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + \color{blue}{\left(-\frac{{im}^{2}}{re}\right)}} \]

      3. unsub-neg 51.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} - \frac{{im}^{2}}{re}}} \]

      4. *-commutative 51.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25} - \frac{{im}^{2}}{re}} \]

   7. Simplified 51.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 50.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}} \cdot \sqrt{\sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}}\right)} \]

      2. pow2 50.7%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}}\right)}^{2}} \]

      3. pow1/2 50.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{0.5}}}\right)}^{2} \]

      4. metadata-eval 50.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt{{\left(\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right)}^{2} \]

      5. sqrt-pow1 50.8%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}}^{2} \]

      6. div-inv 50.8%

         \[\leadsto 0.5 \cdot {\left({\left(\color{blue}{\left({im}^{4} \cdot \frac{1}{{re}^{3}}\right)} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      7. associate-*l* 50.8%

         \[\leadsto 0.5 \cdot {\left({\left(\color{blue}{{im}^{4} \cdot \left(\frac{1}{{re}^{3}} \cdot 0.25\right)} - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      8. pow-flip 50.8%

         \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left(\color{blue}{{re}^{\left(-3\right)}} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      9. metadata-eval 50.8%

         \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left({re}^{\color{blue}{-3}} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      10. metadata-eval 50.8%

          \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left({re}^{-3} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right)}^{2} \]

      11. metadata-eval 50.8%

          \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left({re}^{-3} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   9. Applied egg-rr 50.8%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left({im}^{4} \cdot \left({re}^{-3} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{0.25}\right)}^{2}} \]

   10. Taylor expanded in im around 0 55.0%

       \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(-1 \cdot \frac{{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]
   11. Step-by-step derivation

       1. mul-1-neg 55.0%

          \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(-\frac{{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]

       2. distribute-frac-neg 55.0%

          \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(\frac{-{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]

   12. Simplified 55.0%

       \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(\frac{-{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]
   13. Step-by-step derivation

       1. pow-pow 55.4%

          \[\leadsto 0.5 \cdot \color{blue}{{\left(\frac{-{im}^{2}}{re}\right)}^{\left(0.25 \cdot 2\right)}} \]

       2. metadata-eval 55.4%

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

       3. pow1/2 55.4%

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

       4. frac-2neg 55.4%

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

       5. sqrt-div 64.6%

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

       6. remove-double-neg 64.6%

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

       7. unpow2 64.6%

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

       8. sqrt-prod 51.1%

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

       9. add-sqr-sqrt 56.3%

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

   14. Applied egg-rr 56.3%

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

   if -5.2e-43 < re < 5.4000000000000001e-76 or 7.20000000000000014e-37 < re < 1.0200000000000001e60

   1. Initial program 50.5%

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

      1. sqr-neg 50.5%

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

      2. +-commutative 50.5%

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

      3. sqr-neg 50.5%

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

      4. +-commutative 50.5%

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

      5. distribute-rgt-in 50.5%

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

      6. cancel-sign-sub 50.5%

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

      7. distribute-rgt-out-- 50.5%

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

      8. sub-neg 50.5%

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

      9. remove-double-neg 50.5%

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

      10. +-commutative 50.5%

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

      11. hypot-define 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in re around 0 44.9%

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

      1. distribute-lft-out 44.9%

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

      2. *-commutative 44.9%

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

   7. Simplified 44.9%

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

   if 5.4000000000000001e-76 < re < 7.20000000000000014e-37 or 1.0200000000000001e60 < re

   1. Initial program 42.6%

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

      1. sqr-neg 42.6%

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

      2. +-commutative 42.6%

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

      3. sqr-neg 42.6%

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

      4. +-commutative 42.6%

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

      5. distribute-rgt-in 42.6%

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

      6. cancel-sign-sub 42.6%

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

      7. distribute-rgt-out-- 42.6%

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

      8. sub-neg 42.6%

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

      9. remove-double-neg 42.6%

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

      10. +-commutative 42.6%

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

      11. hypot-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 81.7%

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

      1. *-commutative 81.7%

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

      2. unpow2 81.7%

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

      3. rem-square-sqrt 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{-76} \lor \neg \left(re \leq 7.2 \cdot 10^{-37}\right) \land re \leq 1.02 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \frac{im\_m}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{-76} \lor \neg \left(re \leq 1.26 \cdot 10^{-35}\right) \land re \leq 7.5 \cdot 10^{+54}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.5e-39)
   (* 0.5 (/ im_m (sqrt (- re))))
   (if (or (<= re 5.2e-76) (and (not (<= re 1.26e-35)) (<= re 7.5e+54)))
     (* 0.5 (sqrt (* im_m 2.0)))
     (* 0.5 (* 2.0 (sqrt re))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.5e-39) {
		tmp = 0.5 * (im_m / sqrt(-re));
	} else if ((re <= 5.2e-76) || (!(re <= 1.26e-35) && (re <= 7.5e+54))) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-3.5d-39)) then
        tmp = 0.5d0 * (im_m / sqrt(-re))
    else if ((re <= 5.2d-76) .or. (.not. (re <= 1.26d-35)) .and. (re <= 7.5d+54)) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -3.5e-39) {
		tmp = 0.5 * (im_m / Math.sqrt(-re));
	} else if ((re <= 5.2e-76) || (!(re <= 1.26e-35) && (re <= 7.5e+54))) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3.5e-39:
		tmp = 0.5 * (im_m / math.sqrt(-re))
	elif (re <= 5.2e-76) or (not (re <= 1.26e-35) and (re <= 7.5e+54)):
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.5e-39)
		tmp = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))));
	elseif ((re <= 5.2e-76) || (!(re <= 1.26e-35) && (re <= 7.5e+54)))
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -3.5e-39)
		tmp = 0.5 * (im_m / sqrt(-re));
	elseif ((re <= 5.2e-76) || (~((re <= 1.26e-35)) && (re <= 7.5e+54)))
		tmp = 0.5 * sqrt((im_m * 2.0));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.5e-39], N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 5.2e-76], And[N[Not[LessEqual[re, 1.26e-35]], $MachinePrecision], LessEqual[re, 7.5e+54]]], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.5e-39

   1. Initial program 11.2%

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

      1. sqr-neg 11.2%

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

      2. +-commutative 11.2%

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

      3. sqr-neg 11.2%

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

      4. +-commutative 11.2%

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

      5. distribute-rgt-in 11.2%

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

      6. cancel-sign-sub 11.2%

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

      7. distribute-rgt-out-- 11.2%

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

      8. sub-neg 11.2%

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

      9. remove-double-neg 11.2%

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

      10. +-commutative 11.2%

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

      11. hypot-define 32.7%

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

   3. Simplified 32.7%

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

   5. Taylor expanded in re around -inf 51.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re} + 0.25 \cdot \frac{{im}^{4}}{{re}^{3}}}} \]
   6. Step-by-step derivation

      1. +-commutative 51.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + -1 \cdot \frac{{im}^{2}}{re}}} \]

      2. mul-1-neg 51.1%

         \[\leadsto 0.5 \cdot \sqrt{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + \color{blue}{\left(-\frac{{im}^{2}}{re}\right)}} \]

      3. unsub-neg 51.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} - \frac{{im}^{2}}{re}}} \]

      4. *-commutative 51.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25} - \frac{{im}^{2}}{re}} \]

   7. Simplified 51.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 50.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}} \cdot \sqrt{\sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}}\right)} \]

      2. pow2 50.7%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}}}\right)}^{2}} \]

      3. pow1/2 50.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt{\color{blue}{{\left(\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{0.5}}}\right)}^{2} \]

      4. metadata-eval 50.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt{{\left(\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right)}^{2} \]

      5. sqrt-pow1 50.8%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}}^{2} \]

      6. div-inv 50.8%

         \[\leadsto 0.5 \cdot {\left({\left(\color{blue}{\left({im}^{4} \cdot \frac{1}{{re}^{3}}\right)} \cdot 0.25 - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      7. associate-*l* 50.8%

         \[\leadsto 0.5 \cdot {\left({\left(\color{blue}{{im}^{4} \cdot \left(\frac{1}{{re}^{3}} \cdot 0.25\right)} - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      8. pow-flip 50.8%

         \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left(\color{blue}{{re}^{\left(-3\right)}} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      9. metadata-eval 50.8%

         \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left({re}^{\color{blue}{-3}} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

      10. metadata-eval 50.8%

          \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left({re}^{-3} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right)}^{2} \]

      11. metadata-eval 50.8%

          \[\leadsto 0.5 \cdot {\left({\left({im}^{4} \cdot \left({re}^{-3} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   9. Applied egg-rr 50.8%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left({im}^{4} \cdot \left({re}^{-3} \cdot 0.25\right) - \frac{{im}^{2}}{re}\right)}^{0.25}\right)}^{2}} \]

   10. Taylor expanded in im around 0 55.0%

       \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(-1 \cdot \frac{{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]
   11. Step-by-step derivation

       1. mul-1-neg 55.0%

          \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(-\frac{{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]

       2. distribute-frac-neg 55.0%

          \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(\frac{-{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]

   12. Simplified 55.0%

       \[\leadsto 0.5 \cdot {\left({\color{blue}{\left(\frac{-{im}^{2}}{re}\right)}}^{0.25}\right)}^{2} \]
   13. Step-by-step derivation

       1. pow-pow 55.4%

          \[\leadsto 0.5 \cdot \color{blue}{{\left(\frac{-{im}^{2}}{re}\right)}^{\left(0.25 \cdot 2\right)}} \]

       2. metadata-eval 55.4%

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

       3. pow1/2 55.4%

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

       4. frac-2neg 55.4%

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

       5. sqrt-div 64.6%

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

       6. remove-double-neg 64.6%

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

       7. unpow2 64.6%

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

       8. sqrt-prod 51.1%

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

       9. add-sqr-sqrt 56.3%

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

   14. Applied egg-rr 56.3%

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

   if -3.5e-39 < re < 5.1999999999999999e-76 or 1.26e-35 < re < 7.50000000000000042e54

   1. Initial program 50.5%

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

      1. sqr-neg 50.5%

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

      2. +-commutative 50.5%

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

      3. sqr-neg 50.5%

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

      4. +-commutative 50.5%

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

      5. distribute-rgt-in 50.5%

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

      6. cancel-sign-sub 50.5%

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

      7. distribute-rgt-out-- 50.5%

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

      8. sub-neg 50.5%

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

      9. remove-double-neg 50.5%

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

      10. +-commutative 50.5%

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

      11. hypot-define 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in re around 0 43.1%

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

      1. *-commutative 43.1%

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

   7. Simplified 43.1%

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

   if 5.1999999999999999e-76 < re < 1.26e-35 or 7.50000000000000042e54 < re

   1. Initial program 42.6%

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

      1. sqr-neg 42.6%

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

      2. +-commutative 42.6%

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

      3. sqr-neg 42.6%

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

      4. +-commutative 42.6%

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

      5. distribute-rgt-in 42.6%

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

      6. cancel-sign-sub 42.6%

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

      7. distribute-rgt-out-- 42.6%

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

      8. sub-neg 42.6%

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

      9. remove-double-neg 42.6%

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

      10. +-commutative 42.6%

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

      11. hypot-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 81.7%

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

      1. *-commutative 81.7%

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

      2. unpow2 81.7%

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

      3. rem-square-sqrt 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{-76} \lor \neg \left(re \leq 1.26 \cdot 10^{-35}\right) \land re \leq 7.5 \cdot 10^{+54}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{-76} \lor \neg \left(re \leq 1.25 \cdot 10^{-36}\right) \land re \leq 1.1 \cdot 10^{+56}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (or (<= re 2e-76) (and (not (<= re 1.25e-36)) (<= re 1.1e+56)))
   (* 0.5 (sqrt (* im_m 2.0)))
   (* 0.5 (* 2.0 (sqrt re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re <= 2e-76) || (!(re <= 1.25e-36) && (re <= 1.1e+56))) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((re <= 2d-76) .or. (.not. (re <= 1.25d-36)) .and. (re <= 1.1d+56)) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if ((re <= 2e-76) || (!(re <= 1.25e-36) && (re <= 1.1e+56))) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if (re <= 2e-76) or (not (re <= 1.25e-36) and (re <= 1.1e+56)):
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if ((re <= 2e-76) || (!(re <= 1.25e-36) && (re <= 1.1e+56)))
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if ((re <= 2e-76) || (~((re <= 1.25e-36)) && (re <= 1.1e+56)))
		tmp = 0.5 * sqrt((im_m * 2.0));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[Or[LessEqual[re, 2e-76], And[N[Not[LessEqual[re, 1.25e-36]], $MachinePrecision], LessEqual[re, 1.1e+56]]], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.99999999999999985e-76 or 1.25000000000000001e-36 < re < 1.10000000000000008e56

   1. Initial program 36.0%

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

      1. sqr-neg 36.0%

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

      2. +-commutative 36.0%

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

      3. sqr-neg 36.0%

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

      4. +-commutative 36.0%

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

      5. distribute-rgt-in 36.0%

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

      6. cancel-sign-sub 36.0%

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

      7. distribute-rgt-out-- 36.0%

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

      8. sub-neg 36.0%

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

      9. remove-double-neg 36.0%

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

      10. +-commutative 36.0%

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

      11. hypot-define 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in re around 0 33.4%

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

      1. *-commutative 33.4%

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

   7. Simplified 33.4%

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

   if 1.99999999999999985e-76 < re < 1.25000000000000001e-36 or 1.10000000000000008e56 < re

   1. Initial program 42.6%

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

      1. sqr-neg 42.6%

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

      2. +-commutative 42.6%

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

      3. sqr-neg 42.6%

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

      4. +-commutative 42.6%

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

      5. distribute-rgt-in 42.6%

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

      6. cancel-sign-sub 42.6%

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

      7. distribute-rgt-out-- 42.6%

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

      8. sub-neg 42.6%

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

      9. remove-double-neg 42.6%

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

      10. +-commutative 42.6%

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

      11. hypot-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 81.7%

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

      1. *-commutative 81.7%

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

      2. unpow2 81.7%

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

      3. rem-square-sqrt 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{-76} \lor \neg \left(re \leq 1.25 \cdot 10^{-36}\right) \land re \leq 1.1 \cdot 10^{+56}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.5 \cdot \sqrt{im\_m \cdot 2} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* 0.5 (sqrt (* im_m 2.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

   1. sqr-neg 37.4%

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

   2. +-commutative 37.4%

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

   3. sqr-neg 37.4%

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

   4. +-commutative 37.4%

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

   5. distribute-rgt-in 37.4%

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

   6. cancel-sign-sub 37.4%

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

   7. distribute-rgt-out-- 37.4%

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

   8. sub-neg 37.4%

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

   9. remove-double-neg 37.4%

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

   10. +-commutative 37.4%

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

   11. hypot-define 76.0%

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

3. Simplified 76.0%

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

5. Taylor expanded in re around 0 29.6%

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

   1. *-commutative 29.6%

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

7. Simplified 29.6%

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

8. Final simplification 29.6%

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

Developer target: 48.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t\_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t\_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + 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 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024039 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))