math.sqrt on complex, real part

Percentage Accurate: 40.4% → 87.2%

Time: 10.5s

Alternatives: 7

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: 40.4% 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: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5.7 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \frac{im_m}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im_m} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq -4.9 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \left(\left(im_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-0.5}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5.7e+59)
   (* 0.5 (/ im_m (sqrt (- re))))
   (if (<= re -4.2e-10)
     (* 0.5 (* (sqrt im_m) (sqrt 2.0)))
     (if (<= re -4.9e-60)
       (* 0.5 (* (* im_m (sqrt 2.0)) (sqrt (/ -0.5 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 <= -5.7e+59) {
		tmp = 0.5 * (im_m / sqrt(-re));
	} else if (re <= -4.2e-10) {
		tmp = 0.5 * (sqrt(im_m) * sqrt(2.0));
	} else if (re <= -4.9e-60) {
		tmp = 0.5 * ((im_m * sqrt(2.0)) * sqrt((-0.5 / 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 <= -5.7e+59) {
		tmp = 0.5 * (im_m / Math.sqrt(-re));
	} else if (re <= -4.2e-10) {
		tmp = 0.5 * (Math.sqrt(im_m) * Math.sqrt(2.0));
	} else if (re <= -4.9e-60) {
		tmp = 0.5 * ((im_m * Math.sqrt(2.0)) * Math.sqrt((-0.5 / 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 <= -5.7e+59:
		tmp = 0.5 * (im_m / math.sqrt(-re))
	elif re <= -4.2e-10:
		tmp = 0.5 * (math.sqrt(im_m) * math.sqrt(2.0))
	elif re <= -4.9e-60:
		tmp = 0.5 * ((im_m * math.sqrt(2.0)) * math.sqrt((-0.5 / 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 (re <= -5.7e+59)
		tmp = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))));
	elseif (re <= -4.2e-10)
		tmp = Float64(0.5 * Float64(sqrt(im_m) * sqrt(2.0)));
	elseif (re <= -4.9e-60)
		tmp = Float64(0.5 * Float64(Float64(im_m * sqrt(2.0)) * sqrt(Float64(-0.5 / 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 <= -5.7e+59)
		tmp = 0.5 * (im_m / sqrt(-re));
	elseif (re <= -4.2e-10)
		tmp = 0.5 * (sqrt(im_m) * sqrt(2.0));
	elseif (re <= -4.9e-60)
		tmp = 0.5 * ((im_m * sqrt(2.0)) * sqrt((-0.5 / 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[re, -5.7e+59], N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -4.2e-10], N[(0.5 * N[(N[Sqrt[im$95$m], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -4.9e-60], N[(0.5 * N[(N[(im$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-0.5 / re), $MachinePrecision]], $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 4 regimes

2. if re < -5.7000000000000001e59

   1. Initial program 6.2%

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

      1. sqr-neg 6.2%

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

      2. +-commutative 6.2%

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

      3. sqr-neg 6.2%

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

      4. +-commutative 6.2%

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

      5. distribute-rgt-in 6.2%

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

      6. cancel-sign-sub 6.2%

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

      7. distribute-rgt-out-- 6.2%

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

      8. sub-neg 6.2%

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

      9. remove-double-neg 6.2%

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

      10. +-commutative 6.2%

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

   3. Simplified 30.0%

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

   5. Taylor expanded in re around -inf 58.8%

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

      1. associate-*r/ 58.8%

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

      2. associate-/l* 58.7%

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

   7. Simplified 58.7%

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

      1. *-un-lft-identity 58.7%

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

      2. unpow2 58.7%

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

      3. times-frac 64.0%

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

   9. Applied egg-rr 64.0%

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

       1. associate-*r/ 64.0%

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

       2. metadata-eval 64.0%

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

       3. frac-times 58.7%

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

       4. *-un-lft-identity 58.7%

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

       5. unpow2 58.7%

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

       6. associate-/l* 58.8%

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

       7. neg-mul-1 58.8%

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

       8. frac-2neg 58.8%

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

       9. sqrt-div 78.4%

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

       10. remove-double-neg 78.4%

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

       11. unpow2 78.4%

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

       12. sqrt-prod 40.6%

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

       13. add-sqr-sqrt 56.1%

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

   11. Applied egg-rr 56.1%

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

   if -5.7000000000000001e59 < re < -4.2e-10

   1. Initial program 42.5%

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

      1. sqr-neg 42.5%

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

      2. +-commutative 42.5%

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

      3. sqr-neg 42.5%

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

      4. +-commutative 42.5%

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

      5. distribute-rgt-in 42.5%

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

      6. cancel-sign-sub 42.5%

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

      7. distribute-rgt-out-- 42.5%

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

      8. sub-neg 42.5%

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

      9. remove-double-neg 42.5%

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

      10. +-commutative 42.5%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in re around 0 50.3%

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

   if -4.2e-10 < re < -4.89999999999999988e-60

   1. Initial program 18.2%

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

      1. sqr-neg 18.2%

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

      2. +-commutative 18.2%

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

      3. sqr-neg 18.2%

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

      4. +-commutative 18.2%

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

      5. distribute-rgt-in 18.2%

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

      6. cancel-sign-sub 18.2%

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

      7. distribute-rgt-out-- 18.2%

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

      8. sub-neg 18.2%

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

      9. remove-double-neg 18.2%

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

      10. +-commutative 18.2%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in re around -inf 44.1%

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

      1. associate-*r/ 44.1%

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

      2. associate-/l* 44.1%

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

   7. Simplified 44.1%

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

      1. sqrt-prod 43.9%

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

      2. associate-/r/ 43.9%

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

      3. sqrt-prod 43.9%

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

      4. sqrt-pow1 28.1%

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

      5. metadata-eval 28.1%

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

      6. pow1 28.1%

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

   9. Applied egg-rr 28.1%

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

       1. *-commutative 28.1%

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

       2. associate-*r* 28.1%

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

       3. *-commutative 28.1%

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

   11. Simplified 28.1%

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

   if -4.89999999999999988e-60 < re

   1. Initial program 54.6%

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

      1. sqr-neg 54.6%

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

      2. +-commutative 54.6%

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

      3. sqr-neg 54.6%

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

      4. +-commutative 54.6%

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

      5. distribute-rgt-in 54.6%

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

      6. cancel-sign-sub 54.6%

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

      7. distribute-rgt-out-- 54.6%

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

      8. sub-neg 54.6%

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

      9. remove-double-neg 54.6%

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

      10. +-commutative 54.6%

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

   3. Simplified 96.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.7 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq -4.9 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-0.5}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{im_m}{\sqrt{-re}}\\ \mathbf{if}\;re \leq -5.5 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -5.5 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im_m} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ im_m (sqrt (- re))))))
   (if (<= re -5.5e+57)
     t_0
     (if (<= re -5.5e-10)
       (* 0.5 (* (sqrt im_m) (sqrt 2.0)))
       (if (<= re -1.7e-54)
         t_0
         (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * (im_m / sqrt(-re));
	double tmp;
	if (re <= -5.5e+57) {
		tmp = t_0;
	} else if (re <= -5.5e-10) {
		tmp = 0.5 * (sqrt(im_m) * sqrt(2.0));
	} else if (re <= -1.7e-54) {
		tmp = t_0;
	} 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 t_0 = 0.5 * (im_m / Math.sqrt(-re));
	double tmp;
	if (re <= -5.5e+57) {
		tmp = t_0;
	} else if (re <= -5.5e-10) {
		tmp = 0.5 * (Math.sqrt(im_m) * Math.sqrt(2.0));
	} else if (re <= -1.7e-54) {
		tmp = t_0;
	} 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):
	t_0 = 0.5 * (im_m / math.sqrt(-re))
	tmp = 0
	if re <= -5.5e+57:
		tmp = t_0
	elif re <= -5.5e-10:
		tmp = 0.5 * (math.sqrt(im_m) * math.sqrt(2.0))
	elif re <= -1.7e-54:
		tmp = t_0
	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)
	t_0 = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))))
	tmp = 0.0
	if (re <= -5.5e+57)
		tmp = t_0;
	elseif (re <= -5.5e-10)
		tmp = Float64(0.5 * Float64(sqrt(im_m) * sqrt(2.0)));
	elseif (re <= -1.7e-54)
		tmp = t_0;
	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)
	t_0 = 0.5 * (im_m / sqrt(-re));
	tmp = 0.0;
	if (re <= -5.5e+57)
		tmp = t_0;
	elseif (re <= -5.5e-10)
		tmp = 0.5 * (sqrt(im_m) * sqrt(2.0));
	elseif (re <= -1.7e-54)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -5.5e+57], t$95$0, If[LessEqual[re, -5.5e-10], N[(0.5 * N[(N[Sqrt[im$95$m], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.7e-54], t$95$0, 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 3 regimes

2. if re < -5.5000000000000002e57 or -5.4999999999999996e-10 < re < -1.69999999999999994e-54

   1. Initial program 8.5%

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

      1. sqr-neg 8.5%

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

      2. +-commutative 8.5%

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

      3. sqr-neg 8.5%

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

      4. +-commutative 8.5%

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

      5. distribute-rgt-in 8.5%

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

      6. cancel-sign-sub 8.5%

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

      7. distribute-rgt-out-- 8.5%

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

      8. sub-neg 8.5%

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

      9. remove-double-neg 8.5%

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

      10. +-commutative 8.5%

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

   3. Simplified 31.9%

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

   5. Taylor expanded in re around -inf 56.1%

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

      1. associate-*r/ 56.1%

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

      2. associate-/l* 56.0%

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

   7. Simplified 56.0%

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

      1. *-un-lft-identity 56.0%

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

      2. unpow2 56.0%

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

      3. times-frac 61.7%

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

   9. Applied egg-rr 61.7%

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

       1. associate-*r/ 61.7%

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

       2. metadata-eval 61.7%

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

       3. frac-times 56.0%

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

       4. *-un-lft-identity 56.0%

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

       5. unpow2 56.0%

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

       6. associate-/l* 56.1%

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

       7. neg-mul-1 56.1%

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

       8. frac-2neg 56.1%

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

       9. sqrt-div 72.0%

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

       10. remove-double-neg 72.0%

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

       11. unpow2 72.0%

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

       12. sqrt-prod 38.0%

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

       13. add-sqr-sqrt 50.9%

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

   11. Applied egg-rr 50.9%

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

   if -5.5000000000000002e57 < re < -5.4999999999999996e-10

   1. Initial program 42.5%

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

      1. sqr-neg 42.5%

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

      2. +-commutative 42.5%

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

      3. sqr-neg 42.5%

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

      4. +-commutative 42.5%

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

      5. distribute-rgt-in 42.5%

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

      6. cancel-sign-sub 42.5%

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

      7. distribute-rgt-out-- 42.5%

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

      8. sub-neg 42.5%

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

      9. remove-double-neg 42.5%

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

      10. +-commutative 42.5%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in re around 0 50.3%

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

   if -1.69999999999999994e-54 < re

   1. Initial program 54.6%

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

      1. sqr-neg 54.6%

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

      2. +-commutative 54.6%

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

      3. sqr-neg 54.6%

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

      4. +-commutative 54.6%

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

      5. distribute-rgt-in 54.6%

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

      6. cancel-sign-sub 54.6%

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

      7. distribute-rgt-out-- 54.6%

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

      8. sub-neg 54.6%

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

      9. remove-double-neg 54.6%

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

      10. +-commutative 54.6%

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

   3. Simplified 96.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq -5.5 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{im_m}{\sqrt{-re}}\\ \mathbf{if}\;re \leq -1.02 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -7 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im_m} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq -1.3 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ im_m (sqrt (- re))))))
   (if (<= re -1.02e+59)
     t_0
     (if (<= re -7e-10)
       (* 0.5 (* (sqrt im_m) (sqrt 2.0)))
       (if (<= re -1.3e-57)
         t_0
         (if (<= re 7e-44)
           (* 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 t_0 = 0.5 * (im_m / sqrt(-re));
	double tmp;
	if (re <= -1.02e+59) {
		tmp = t_0;
	} else if (re <= -7e-10) {
		tmp = 0.5 * (sqrt(im_m) * sqrt(2.0));
	} else if (re <= -1.3e-57) {
		tmp = t_0;
	} else if (re <= 7e-44) {
		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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im_m / sqrt(-re))
    if (re <= (-1.02d+59)) then
        tmp = t_0
    else if (re <= (-7d-10)) then
        tmp = 0.5d0 * (sqrt(im_m) * sqrt(2.0d0))
    else if (re <= (-1.3d-57)) then
        tmp = t_0
    else if (re <= 7d-44) 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 t_0 = 0.5 * (im_m / Math.sqrt(-re));
	double tmp;
	if (re <= -1.02e+59) {
		tmp = t_0;
	} else if (re <= -7e-10) {
		tmp = 0.5 * (Math.sqrt(im_m) * Math.sqrt(2.0));
	} else if (re <= -1.3e-57) {
		tmp = t_0;
	} else if (re <= 7e-44) {
		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):
	t_0 = 0.5 * (im_m / math.sqrt(-re))
	tmp = 0
	if re <= -1.02e+59:
		tmp = t_0
	elif re <= -7e-10:
		tmp = 0.5 * (math.sqrt(im_m) * math.sqrt(2.0))
	elif re <= -1.3e-57:
		tmp = t_0
	elif re <= 7e-44:
		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)
	t_0 = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))))
	tmp = 0.0
	if (re <= -1.02e+59)
		tmp = t_0;
	elseif (re <= -7e-10)
		tmp = Float64(0.5 * Float64(sqrt(im_m) * sqrt(2.0)));
	elseif (re <= -1.3e-57)
		tmp = t_0;
	elseif (re <= 7e-44)
		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)
	t_0 = 0.5 * (im_m / sqrt(-re));
	tmp = 0.0;
	if (re <= -1.02e+59)
		tmp = t_0;
	elseif (re <= -7e-10)
		tmp = 0.5 * (sqrt(im_m) * sqrt(2.0));
	elseif (re <= -1.3e-57)
		tmp = t_0;
	elseif (re <= 7e-44)
		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_] := Block[{t$95$0 = N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.02e+59], t$95$0, If[LessEqual[re, -7e-10], N[(0.5 * N[(N[Sqrt[im$95$m], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.3e-57], t$95$0, If[LessEqual[re, 7e-44], 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 4 regimes

2. if re < -1.02000000000000002e59 or -6.99999999999999961e-10 < re < -1.29999999999999993e-57

   1. Initial program 8.5%

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

      1. sqr-neg 8.5%

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

      2. +-commutative 8.5%

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

      3. sqr-neg 8.5%

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

      4. +-commutative 8.5%

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

      5. distribute-rgt-in 8.5%

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

      6. cancel-sign-sub 8.5%

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

      7. distribute-rgt-out-- 8.5%

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

      8. sub-neg 8.5%

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

      9. remove-double-neg 8.5%

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

      10. +-commutative 8.5%

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

   3. Simplified 31.9%

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

   5. Taylor expanded in re around -inf 56.1%

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

      1. associate-*r/ 56.1%

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

      2. associate-/l* 56.0%

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

   7. Simplified 56.0%

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

      1. *-un-lft-identity 56.0%

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

      2. unpow2 56.0%

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

      3. times-frac 61.7%

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

   9. Applied egg-rr 61.7%

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

       1. associate-*r/ 61.7%

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

       2. metadata-eval 61.7%

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

       3. frac-times 56.0%

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

       4. *-un-lft-identity 56.0%

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

       5. unpow2 56.0%

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

       6. associate-/l* 56.1%

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

       7. neg-mul-1 56.1%

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

       8. frac-2neg 56.1%

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

       9. sqrt-div 72.0%

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

       10. remove-double-neg 72.0%

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

       11. unpow2 72.0%

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

       12. sqrt-prod 38.0%

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

       13. add-sqr-sqrt 50.9%

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

   11. Applied egg-rr 50.9%

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

   if -1.02000000000000002e59 < re < -6.99999999999999961e-10

   1. Initial program 42.5%

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

      1. sqr-neg 42.5%

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

      2. +-commutative 42.5%

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

      3. sqr-neg 42.5%

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

      4. +-commutative 42.5%

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

      5. distribute-rgt-in 42.5%

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

      6. cancel-sign-sub 42.5%

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

      7. distribute-rgt-out-- 42.5%

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

      8. sub-neg 42.5%

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

      9. remove-double-neg 42.5%

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

      10. +-commutative 42.5%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in re around 0 50.3%

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

   if -1.29999999999999993e-57 < re < 6.9999999999999995e-44

   1. Initial program 61.9%

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

      1. sqr-neg 61.9%

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

      2. +-commutative 61.9%

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

      3. sqr-neg 61.9%

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

      4. +-commutative 61.9%

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

      5. distribute-rgt-in 61.9%

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

      6. cancel-sign-sub 61.9%

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

      7. distribute-rgt-out-- 61.9%

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

      8. sub-neg 61.9%

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

      9. remove-double-neg 61.9%

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

      10. +-commutative 61.9%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in re around 0 41.4%

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

   if 6.9999999999999995e-44 < re

   1. Initial program 42.8%

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

      1. sqr-neg 42.8%

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

      2. +-commutative 42.8%

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

      3. sqr-neg 42.8%

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

      4. +-commutative 42.8%

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

      5. distribute-rgt-in 42.8%

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

      6. cancel-sign-sub 42.8%

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

      7. distribute-rgt-out-- 42.8%

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

      8. sub-neg 42.8%

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

      9. remove-double-neg 42.8%

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

      10. +-commutative 42.8%

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

   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 74.5%

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

      1. *-commutative 74.5%

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

      2. unpow2 74.5%

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

      3. rem-square-sqrt 76.0%

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

   7. Simplified 76.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot \sqrt{re}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.02 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq -7 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{im} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq -1.3 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{im_m}{\sqrt{-re}}\\ \mathbf{if}\;re \leq -7.4 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -3.8 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{im_m \cdot 2}\\ \mathbf{elif}\;re \leq -1.3 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ im_m (sqrt (- re))))))
   (if (<= re -7.4e+57)
     t_0
     (if (<= re -3.8e-10)
       (* 0.5 (sqrt (* im_m 2.0)))
       (if (<= re -1.3e-57)
         t_0
         (if (<= re 8.2e-43)
           (* 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 t_0 = 0.5 * (im_m / sqrt(-re));
	double tmp;
	if (re <= -7.4e+57) {
		tmp = t_0;
	} else if (re <= -3.8e-10) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else if (re <= -1.3e-57) {
		tmp = t_0;
	} else if (re <= 8.2e-43) {
		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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im_m / sqrt(-re))
    if (re <= (-7.4d+57)) then
        tmp = t_0
    else if (re <= (-3.8d-10)) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else if (re <= (-1.3d-57)) then
        tmp = t_0
    else if (re <= 8.2d-43) 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 t_0 = 0.5 * (im_m / Math.sqrt(-re));
	double tmp;
	if (re <= -7.4e+57) {
		tmp = t_0;
	} else if (re <= -3.8e-10) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else if (re <= -1.3e-57) {
		tmp = t_0;
	} else if (re <= 8.2e-43) {
		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):
	t_0 = 0.5 * (im_m / math.sqrt(-re))
	tmp = 0
	if re <= -7.4e+57:
		tmp = t_0
	elif re <= -3.8e-10:
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	elif re <= -1.3e-57:
		tmp = t_0
	elif re <= 8.2e-43:
		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)
	t_0 = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))))
	tmp = 0.0
	if (re <= -7.4e+57)
		tmp = t_0;
	elseif (re <= -3.8e-10)
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	elseif (re <= -1.3e-57)
		tmp = t_0;
	elseif (re <= 8.2e-43)
		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)
	t_0 = 0.5 * (im_m / sqrt(-re));
	tmp = 0.0;
	if (re <= -7.4e+57)
		tmp = t_0;
	elseif (re <= -3.8e-10)
		tmp = 0.5 * sqrt((im_m * 2.0));
	elseif (re <= -1.3e-57)
		tmp = t_0;
	elseif (re <= 8.2e-43)
		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_] := Block[{t$95$0 = N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -7.4e+57], t$95$0, If[LessEqual[re, -3.8e-10], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.3e-57], t$95$0, If[LessEqual[re, 8.2e-43], 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 4 regimes

2. if re < -7.40000000000000011e57 or -3.7999999999999998e-10 < re < -1.29999999999999993e-57

   1. Initial program 8.5%

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

      1. sqr-neg 8.5%

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

      2. +-commutative 8.5%

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

      3. sqr-neg 8.5%

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

      4. +-commutative 8.5%

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

      5. distribute-rgt-in 8.5%

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

      6. cancel-sign-sub 8.5%

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

      7. distribute-rgt-out-- 8.5%

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

      8. sub-neg 8.5%

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

      9. remove-double-neg 8.5%

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

      10. +-commutative 8.5%

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

   3. Simplified 31.9%

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

   5. Taylor expanded in re around -inf 56.1%

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

      1. associate-*r/ 56.1%

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

      2. associate-/l* 56.0%

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

   7. Simplified 56.0%

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

      1. *-un-lft-identity 56.0%

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

      2. unpow2 56.0%

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

      3. times-frac 61.7%

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

   9. Applied egg-rr 61.7%

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

       1. associate-*r/ 61.7%

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

       2. metadata-eval 61.7%

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

       3. frac-times 56.0%

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

       4. *-un-lft-identity 56.0%

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

       5. unpow2 56.0%

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

       6. associate-/l* 56.1%

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

       7. neg-mul-1 56.1%

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

       8. frac-2neg 56.1%

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

       9. sqrt-div 72.0%

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

       10. remove-double-neg 72.0%

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

       11. unpow2 72.0%

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

       12. sqrt-prod 38.0%

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

       13. add-sqr-sqrt 50.9%

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

   11. Applied egg-rr 50.9%

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

   if -7.40000000000000011e57 < re < -3.7999999999999998e-10

   1. Initial program 42.5%

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

      1. sqr-neg 42.5%

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

      2. +-commutative 42.5%

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

      3. sqr-neg 42.5%

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

      4. +-commutative 42.5%

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

      5. distribute-rgt-in 42.5%

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

      6. cancel-sign-sub 42.5%

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

      7. distribute-rgt-out-- 42.5%

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

      8. sub-neg 42.5%

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

      9. remove-double-neg 42.5%

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

      10. +-commutative 42.5%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in re around 0 41.1%

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

   if -1.29999999999999993e-57 < re < 8.1999999999999996e-43

   1. Initial program 61.9%

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

      1. sqr-neg 61.9%

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

      2. +-commutative 61.9%

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

      3. sqr-neg 61.9%

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

      4. +-commutative 61.9%

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

      5. distribute-rgt-in 61.9%

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

      6. cancel-sign-sub 61.9%

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

      7. distribute-rgt-out-- 61.9%

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

      8. sub-neg 61.9%

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

      9. remove-double-neg 61.9%

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

      10. +-commutative 61.9%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in re around 0 41.4%

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

   if 8.1999999999999996e-43 < re

   1. Initial program 42.8%

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

      1. sqr-neg 42.8%

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

      2. +-commutative 42.8%

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

      3. sqr-neg 42.8%

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

      4. +-commutative 42.8%

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

      5. distribute-rgt-in 42.8%

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

      6. cancel-sign-sub 42.8%

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

      7. distribute-rgt-out-- 42.8%

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

      8. sub-neg 42.8%

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

      9. remove-double-neg 42.8%

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

      10. +-commutative 42.8%

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

   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 74.5%

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

      1. *-commutative 74.5%

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

      2. unpow2 74.5%

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

      3. rem-square-sqrt 76.0%

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

   7. Simplified 76.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot \sqrt{re}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.4 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq -3.8 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq -1.3 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{im_m}{\sqrt{-re}}\\ t_1 := 0.5 \cdot \sqrt{im_m \cdot 2}\\ \mathbf{if}\;re \leq -2.05 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -7.8 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -2.05 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ im_m (sqrt (- re))))) (t_1 (* 0.5 (sqrt (* im_m 2.0)))))
   (if (<= re -2.05e+60)
     t_0
     (if (<= re -7.8e-10)
       t_1
       (if (<= re -2.05e-60)
         t_0
         (if (<= re 3e-43) t_1 (* 0.5 (* 2.0 (sqrt re)))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * (im_m / sqrt(-re));
	double t_1 = 0.5 * sqrt((im_m * 2.0));
	double tmp;
	if (re <= -2.05e+60) {
		tmp = t_0;
	} else if (re <= -7.8e-10) {
		tmp = t_1;
	} else if (re <= -2.05e-60) {
		tmp = t_0;
	} else if (re <= 3e-43) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (im_m / sqrt(-re))
    t_1 = 0.5d0 * sqrt((im_m * 2.0d0))
    if (re <= (-2.05d+60)) then
        tmp = t_0
    else if (re <= (-7.8d-10)) then
        tmp = t_1
    else if (re <= (-2.05d-60)) then
        tmp = t_0
    else if (re <= 3d-43) then
        tmp = t_1
    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 t_0 = 0.5 * (im_m / Math.sqrt(-re));
	double t_1 = 0.5 * Math.sqrt((im_m * 2.0));
	double tmp;
	if (re <= -2.05e+60) {
		tmp = t_0;
	} else if (re <= -7.8e-10) {
		tmp = t_1;
	} else if (re <= -2.05e-60) {
		tmp = t_0;
	} else if (re <= 3e-43) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = 0.5 * (im_m / math.sqrt(-re))
	t_1 = 0.5 * math.sqrt((im_m * 2.0))
	tmp = 0
	if re <= -2.05e+60:
		tmp = t_0
	elif re <= -7.8e-10:
		tmp = t_1
	elif re <= -2.05e-60:
		tmp = t_0
	elif re <= 3e-43:
		tmp = t_1
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))))
	t_1 = Float64(0.5 * sqrt(Float64(im_m * 2.0)))
	tmp = 0.0
	if (re <= -2.05e+60)
		tmp = t_0;
	elseif (re <= -7.8e-10)
		tmp = t_1;
	elseif (re <= -2.05e-60)
		tmp = t_0;
	elseif (re <= 3e-43)
		tmp = t_1;
	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)
	t_0 = 0.5 * (im_m / sqrt(-re));
	t_1 = 0.5 * sqrt((im_m * 2.0));
	tmp = 0.0;
	if (re <= -2.05e+60)
		tmp = t_0;
	elseif (re <= -7.8e-10)
		tmp = t_1;
	elseif (re <= -2.05e-60)
		tmp = t_0;
	elseif (re <= 3e-43)
		tmp = t_1;
	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_] := Block[{t$95$0 = N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.05e+60], t$95$0, If[LessEqual[re, -7.8e-10], t$95$1, If[LessEqual[re, -2.05e-60], t$95$0, If[LessEqual[re, 3e-43], t$95$1, N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.05e60 or -7.7999999999999999e-10 < re < -2.05000000000000006e-60

   1. Initial program 8.5%

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

      1. sqr-neg 8.5%

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

      2. +-commutative 8.5%

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

      3. sqr-neg 8.5%

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

      4. +-commutative 8.5%

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

      5. distribute-rgt-in 8.5%

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

      6. cancel-sign-sub 8.5%

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

      7. distribute-rgt-out-- 8.5%

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

      8. sub-neg 8.5%

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

      9. remove-double-neg 8.5%

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

      10. +-commutative 8.5%

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

   3. Simplified 31.9%

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

   5. Taylor expanded in re around -inf 56.1%

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

      1. associate-*r/ 56.1%

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

      2. associate-/l* 56.0%

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

   7. Simplified 56.0%

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

      1. *-un-lft-identity 56.0%

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

      2. unpow2 56.0%

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

      3. times-frac 61.7%

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

   9. Applied egg-rr 61.7%

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

       1. associate-*r/ 61.7%

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

       2. metadata-eval 61.7%

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

       3. frac-times 56.0%

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

       4. *-un-lft-identity 56.0%

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

       5. unpow2 56.0%

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

       6. associate-/l* 56.1%

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

       7. neg-mul-1 56.1%

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

       8. frac-2neg 56.1%

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

       9. sqrt-div 72.0%

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

       10. remove-double-neg 72.0%

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

       11. unpow2 72.0%

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

       12. sqrt-prod 38.0%

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

       13. add-sqr-sqrt 50.9%

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

   11. Applied egg-rr 50.9%

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

   if -2.05e60 < re < -7.7999999999999999e-10 or -2.05000000000000006e-60 < re < 3.00000000000000003e-43

   1. Initial program 60.3%

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

      1. sqr-neg 60.3%

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

      2. +-commutative 60.3%

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

      3. sqr-neg 60.3%

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

      4. +-commutative 60.3%

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

      5. distribute-rgt-in 60.3%

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

      6. cancel-sign-sub 60.3%

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

      7. distribute-rgt-out-- 60.3%

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

      8. sub-neg 60.3%

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

      9. remove-double-neg 60.3%

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

      10. +-commutative 60.3%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in re around 0 40.4%

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

   if 3.00000000000000003e-43 < re

   1. Initial program 42.8%

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

      1. sqr-neg 42.8%

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

      2. +-commutative 42.8%

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

      3. sqr-neg 42.8%

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

      4. +-commutative 42.8%

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

      5. distribute-rgt-in 42.8%

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

      6. cancel-sign-sub 42.8%

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

      7. distribute-rgt-out-- 42.8%

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

      8. sub-neg 42.8%

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

      9. remove-double-neg 42.8%

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

      10. +-commutative 42.8%

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

   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 74.5%

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

      1. *-commutative 74.5%

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

      2. unpow2 74.5%

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

      3. rem-square-sqrt 76.0%

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

   7. Simplified 76.0%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.05 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq -7.8 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq -2.05 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 8.2 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{im_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 8.2e-43) (* 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 <= 8.2e-43) {
		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 <= 8.2d-43) 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 <= 8.2e-43) {
		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 <= 8.2e-43:
		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 <= 8.2e-43)
		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 <= 8.2e-43)
		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, 8.2e-43], 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 < 8.1999999999999996e-43

   1. Initial program 41.1%

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

      1. sqr-neg 41.1%

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

      2. +-commutative 41.1%

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

      3. sqr-neg 41.1%

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

      4. +-commutative 41.1%

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

      5. distribute-rgt-in 41.1%

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

      6. cancel-sign-sub 41.1%

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

      7. distribute-rgt-out-- 41.1%

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

      8. sub-neg 41.1%

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

      9. remove-double-neg 41.1%

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

      10. +-commutative 41.1%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in re around 0 28.3%

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

   if 8.1999999999999996e-43 < re

   1. Initial program 42.8%

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

      1. sqr-neg 42.8%

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

      2. +-commutative 42.8%

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

      3. sqr-neg 42.8%

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

      4. +-commutative 42.8%

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

      5. distribute-rgt-in 42.8%

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

      6. cancel-sign-sub 42.8%

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

      7. distribute-rgt-out-- 42.8%

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

      8. sub-neg 42.8%

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

      9. remove-double-neg 42.8%

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

      10. +-commutative 42.8%

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

   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 74.5%

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

      1. *-commutative 74.5%

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

      2. unpow2 74.5%

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

      3. rem-square-sqrt 76.0%

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

   7. Simplified 76.0%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8.2 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.9% 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 41.5%

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

   1. sqr-neg 41.5%

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

   2. +-commutative 41.5%

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

   3. sqr-neg 41.5%

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

   4. +-commutative 41.5%

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

   5. distribute-rgt-in 41.5%

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

   6. cancel-sign-sub 41.5%

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

   7. distribute-rgt-out-- 41.5%

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

   8. sub-neg 41.5%

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

   9. remove-double-neg 41.5%

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

   10. +-commutative 41.5%

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

3. Simplified 77.9%

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

5. Taylor expanded in re around 0 24.1%

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

6. Final simplification 24.1%

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

Developer target: 47.6% 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 2024017 
(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)))))