math.sqrt on complex, real part

Percentage Accurate: 42.0% → 90.6%

Time: 7.4s

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: 42.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im_m \cdot im_m} \leq 0:\\ \;\;\;\;im_m \cdot \sqrt{\frac{-0.25}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + \mathsf{hypot}\left(re, im_m\right)\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im_m im_m)))) 0.0)
   (* im_m (sqrt (/ -0.25 re)))
   (sqrt (* (+ re (hypot re im_m)) 0.5))))

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 = im_m * sqrt((-0.25 / re));
	} else {
		tmp = sqrt(((re + hypot(re, im_m)) * 0.5));
	}
	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 = im_m * Math.sqrt((-0.25 / re));
	} else {
		tmp = Math.sqrt(((re + Math.hypot(re, im_m)) * 0.5));
	}
	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 = im_m * math.sqrt((-0.25 / re))
	else:
		tmp = math.sqrt(((re + math.hypot(re, im_m)) * 0.5))
	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(im_m * sqrt(Float64(-0.25 / re)));
	else
		tmp = sqrt(Float64(Float64(re + hypot(re, im_m)) * 0.5));
	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 = im_m * sqrt((-0.25 / re));
	else
		tmp = sqrt(((re + hypot(re, im_m)) * 0.5));
	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[(im$95$m * N[Sqrt[N[(-0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * 0.5), $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 9.9%

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

      1. sqr-neg 9.9%

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

      2. +-commutative 9.9%

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

      3. sqr-neg 9.9%

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

      4. +-commutative 9.9%

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

      5. distribute-rgt-in 9.9%

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

      6. cancel-sign-sub 9.9%

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

      7. distribute-rgt-out-- 9.9%

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

      8. sub-neg 9.9%

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

      9. remove-double-neg 9.9%

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

      10. +-commutative 9.9%

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

      11. hypot-def 20.3%

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

   3. Simplified 20.3%

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

   4. Taylor expanded in re around -inf 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-*l/ 56.8%

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

   6. Simplified 56.8%

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

      1. add-sqr-sqrt 56.4%

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

      2. sqrt-unprod 56.8%

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

      3. *-commutative 56.8%

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

      4. *-commutative 56.8%

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

      5. swap-sqr 56.8%

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

      6. add-sqr-sqrt 56.8%

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

      7. associate-*r/ 56.8%

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

      8. *-commutative 56.8%

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

      9. associate-*r* 56.8%

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

      10. metadata-eval 56.8%

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

      11. metadata-eval 56.8%

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

   8. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{\sqrt{\frac{-1 \cdot {im}^{2}}{re} \cdot 0.25}} \]
   9. Step-by-step derivation

      1. associate-/l* 56.8%

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

      2. associate-*l/ 56.8%

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

      3. metadata-eval 56.8%

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

   10. Simplified 56.8%

       \[\leadsto \color{blue}{\sqrt{\frac{-0.25}{\frac{re}{{im}^{2}}}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 56.8%

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

       2. sqrt-prod 56.6%

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

       3. unpow2 56.6%

          \[\leadsto \sqrt{\frac{-0.25}{re}} \cdot \sqrt{\color{blue}{im \cdot im}} \]

       4. sqrt-prod 51.6%

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

       5. add-sqr-sqrt 53.3%

          \[\leadsto \sqrt{\frac{-0.25}{re}} \cdot \color{blue}{im} \]

   12. Applied egg-rr 53.3%

       \[\leadsto \color{blue}{\sqrt{\frac{-0.25}{re}} \cdot im} \]

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

   1. Initial program 47.1%

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

      1. sqr-neg 47.1%

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

      2. +-commutative 47.1%

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

      3. sqr-neg 47.1%

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

      4. +-commutative 47.1%

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

      5. distribute-rgt-in 47.1%

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

      6. cancel-sign-sub 47.1%

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

      7. distribute-rgt-out-- 47.1%

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

      8. sub-neg 47.1%

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

      9. remove-double-neg 47.1%

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

      10. +-commutative 47.1%

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

      11. hypot-def 88.1%

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

   3. Simplified 88.1%

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

      1. add-sqr-sqrt 87.6%

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

      2. sqrt-unprod 88.1%

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

      3. *-commutative 88.1%

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

      4. *-commutative 88.1%

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

      5. swap-sqr 88.1%

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

      6. add-sqr-sqrt 88.1%

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

      7. *-commutative 88.1%

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

      8. metadata-eval 88.1%

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

   5. Applied egg-rr 88.1%

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

      1. associate-*l* 88.1%

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

      2. metadata-eval 88.1%

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

   7. Simplified 88.1%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;im \cdot \sqrt{\frac{-0.25}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5}\\ \end{array} \]

Alternative 2: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := im_m \cdot \sqrt{\frac{-0.25}{re}}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re + im_m\right)}\\ \mathbf{if}\;re \leq -5.4 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -6400:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -4 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* im_m (sqrt (/ -0.25 re))))
        (t_1 (* 0.5 (sqrt (* 2.0 (+ re im_m))))))
   (if (<= re -5.4e+50)
     t_0
     (if (<= re -6400.0)
       t_1
       (if (<= re -4e-59) t_0 (if (<= re 1.85e+30) t_1 (sqrt re)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = im_m * sqrt((-0.25 / re));
	double t_1 = 0.5 * sqrt((2.0 * (re + im_m)));
	double tmp;
	if (re <= -5.4e+50) {
		tmp = t_0;
	} else if (re <= -6400.0) {
		tmp = t_1;
	} else if (re <= -4e-59) {
		tmp = t_0;
	} else if (re <= 1.85e+30) {
		tmp = t_1;
	} else {
		tmp = 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 = im_m * sqrt(((-0.25d0) / re))
    t_1 = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    if (re <= (-5.4d+50)) then
        tmp = t_0
    else if (re <= (-6400.0d0)) then
        tmp = t_1
    else if (re <= (-4d-59)) then
        tmp = t_0
    else if (re <= 1.85d+30) then
        tmp = t_1
    else
        tmp = 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 = im_m * Math.sqrt((-0.25 / re));
	double t_1 = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	double tmp;
	if (re <= -5.4e+50) {
		tmp = t_0;
	} else if (re <= -6400.0) {
		tmp = t_1;
	} else if (re <= -4e-59) {
		tmp = t_0;
	} else if (re <= 1.85e+30) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = im_m * math.sqrt((-0.25 / re))
	t_1 = 0.5 * math.sqrt((2.0 * (re + im_m)))
	tmp = 0
	if re <= -5.4e+50:
		tmp = t_0
	elif re <= -6400.0:
		tmp = t_1
	elif re <= -4e-59:
		tmp = t_0
	elif re <= 1.85e+30:
		tmp = t_1
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(im_m * sqrt(Float64(-0.25 / re)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))))
	tmp = 0.0
	if (re <= -5.4e+50)
		tmp = t_0;
	elseif (re <= -6400.0)
		tmp = t_1;
	elseif (re <= -4e-59)
		tmp = t_0;
	elseif (re <= 1.85e+30)
		tmp = t_1;
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = im_m * sqrt((-0.25 / re));
	t_1 = 0.5 * sqrt((2.0 * (re + im_m)));
	tmp = 0.0;
	if (re <= -5.4e+50)
		tmp = t_0;
	elseif (re <= -6400.0)
		tmp = t_1;
	elseif (re <= -4e-59)
		tmp = t_0;
	elseif (re <= 1.85e+30)
		tmp = t_1;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[Sqrt[N[(-0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -5.4e+50], t$95$0, If[LessEqual[re, -6400.0], t$95$1, If[LessEqual[re, -4e-59], t$95$0, If[LessEqual[re, 1.85e+30], t$95$1, N[Sqrt[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.4e50 or -6400 < re < -4.0000000000000001e-59

   1. Initial program 12.5%

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

      1. sqr-neg 12.5%

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

      2. +-commutative 12.5%

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

      3. sqr-neg 12.5%

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

      4. +-commutative 12.5%

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

      5. distribute-rgt-in 12.5%

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

      6. cancel-sign-sub 12.5%

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

      7. distribute-rgt-out-- 12.5%

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

      8. sub-neg 12.5%

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

      9. remove-double-neg 12.5%

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

      10. +-commutative 12.5%

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

      11. hypot-def 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in re around -inf 51.2%

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

      1. *-commutative 51.2%

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

      2. associate-*l/ 51.2%

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

   6. Simplified 51.2%

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

      1. add-sqr-sqrt 50.9%

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

      2. sqrt-unprod 51.1%

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

      3. *-commutative 51.1%

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

      4. *-commutative 51.1%

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

      5. swap-sqr 51.1%

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

      6. add-sqr-sqrt 51.1%

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

      7. associate-*r/ 51.1%

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

      8. *-commutative 51.1%

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

      9. associate-*r* 51.1%

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

      10. metadata-eval 51.1%

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

      11. metadata-eval 51.1%

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

   8. Applied egg-rr 51.1%

      \[\leadsto \color{blue}{\sqrt{\frac{-1 \cdot {im}^{2}}{re} \cdot 0.25}} \]
   9. Step-by-step derivation

      1. associate-/l* 49.8%

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

      2. associate-*l/ 49.8%

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

      3. metadata-eval 49.8%

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

   10. Simplified 49.8%

       \[\leadsto \color{blue}{\sqrt{\frac{-0.25}{\frac{re}{{im}^{2}}}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 51.1%

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

       2. sqrt-prod 64.7%

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

       3. unpow2 64.7%

          \[\leadsto \sqrt{\frac{-0.25}{re}} \cdot \sqrt{\color{blue}{im \cdot im}} \]

       4. sqrt-prod 49.4%

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

       5. add-sqr-sqrt 53.4%

          \[\leadsto \sqrt{\frac{-0.25}{re}} \cdot \color{blue}{im} \]

   12. Applied egg-rr 53.4%

       \[\leadsto \color{blue}{\sqrt{\frac{-0.25}{re}} \cdot im} \]

   if -5.4e50 < re < -6400 or -4.0000000000000001e-59 < re < 1.85000000000000008e30

   1. Initial program 59.4%

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

      1. sqr-neg 59.4%

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

      2. +-commutative 59.4%

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

      3. sqr-neg 59.4%

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

      4. +-commutative 59.4%

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

      5. distribute-rgt-in 59.4%

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

      6. cancel-sign-sub 59.4%

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

      7. distribute-rgt-out-- 59.4%

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

      8. sub-neg 59.4%

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

      9. remove-double-neg 59.4%

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

      10. +-commutative 59.4%

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

      11. hypot-def 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in re around 0 48.6%

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

   if 1.85000000000000008e30 < re

   1. Initial program 37.7%

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

      1. sqr-neg 37.7%

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

      2. +-commutative 37.7%

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

      3. sqr-neg 37.7%

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

      4. +-commutative 37.7%

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

      5. distribute-rgt-in 37.7%

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

      6. cancel-sign-sub 37.7%

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

      7. distribute-rgt-out-- 37.7%

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

      8. sub-neg 37.7%

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

      9. remove-double-neg 37.7%

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

      10. +-commutative 37.7%

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

      11. hypot-def 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. Taylor expanded in im around 0 81.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   5. 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 83.3%

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

      4. associate-*r* 83.3%

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

      5. metadata-eval 83.3%

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      6. *-lft-identity 83.3%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   6. Simplified 83.3%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.4 \cdot 10^{+50}:\\ \;\;\;\;im \cdot \sqrt{\frac{-0.25}{re}}\\ \mathbf{elif}\;re \leq -6400:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq -4 \cdot 10^{-59}:\\ \;\;\;\;im \cdot \sqrt{\frac{-0.25}{re}}\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 3: 75.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := im_m \cdot \sqrt{\frac{-0.25}{re}}\\ t_1 := 0.5 \cdot \sqrt{im_m \cdot 2}\\ \mathbf{if}\;re \leq -7.2 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -940:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -3.9 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.56 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* im_m (sqrt (/ -0.25 re)))) (t_1 (* 0.5 (sqrt (* im_m 2.0)))))
   (if (<= re -7.2e+48)
     t_0
     (if (<= re -940.0)
       t_1
       (if (<= re -3.9e-59) t_0 (if (<= re 1.56e+29) t_1 (sqrt re)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = im_m * sqrt((-0.25 / re));
	double t_1 = 0.5 * sqrt((im_m * 2.0));
	double tmp;
	if (re <= -7.2e+48) {
		tmp = t_0;
	} else if (re <= -940.0) {
		tmp = t_1;
	} else if (re <= -3.9e-59) {
		tmp = t_0;
	} else if (re <= 1.56e+29) {
		tmp = t_1;
	} else {
		tmp = 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 = im_m * sqrt(((-0.25d0) / re))
    t_1 = 0.5d0 * sqrt((im_m * 2.0d0))
    if (re <= (-7.2d+48)) then
        tmp = t_0
    else if (re <= (-940.0d0)) then
        tmp = t_1
    else if (re <= (-3.9d-59)) then
        tmp = t_0
    else if (re <= 1.56d+29) then
        tmp = t_1
    else
        tmp = 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 = im_m * Math.sqrt((-0.25 / re));
	double t_1 = 0.5 * Math.sqrt((im_m * 2.0));
	double tmp;
	if (re <= -7.2e+48) {
		tmp = t_0;
	} else if (re <= -940.0) {
		tmp = t_1;
	} else if (re <= -3.9e-59) {
		tmp = t_0;
	} else if (re <= 1.56e+29) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = im_m * math.sqrt((-0.25 / re))
	t_1 = 0.5 * math.sqrt((im_m * 2.0))
	tmp = 0
	if re <= -7.2e+48:
		tmp = t_0
	elif re <= -940.0:
		tmp = t_1
	elif re <= -3.9e-59:
		tmp = t_0
	elif re <= 1.56e+29:
		tmp = t_1
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(im_m * sqrt(Float64(-0.25 / re)))
	t_1 = Float64(0.5 * sqrt(Float64(im_m * 2.0)))
	tmp = 0.0
	if (re <= -7.2e+48)
		tmp = t_0;
	elseif (re <= -940.0)
		tmp = t_1;
	elseif (re <= -3.9e-59)
		tmp = t_0;
	elseif (re <= 1.56e+29)
		tmp = t_1;
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = im_m * sqrt((-0.25 / re));
	t_1 = 0.5 * sqrt((im_m * 2.0));
	tmp = 0.0;
	if (re <= -7.2e+48)
		tmp = t_0;
	elseif (re <= -940.0)
		tmp = t_1;
	elseif (re <= -3.9e-59)
		tmp = t_0;
	elseif (re <= 1.56e+29)
		tmp = t_1;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[Sqrt[N[(-0.25 / 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, -7.2e+48], t$95$0, If[LessEqual[re, -940.0], t$95$1, If[LessEqual[re, -3.9e-59], t$95$0, If[LessEqual[re, 1.56e+29], t$95$1, N[Sqrt[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.19999999999999967e48 or -940 < re < -3.90000000000000019e-59

   1. Initial program 12.5%

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

      1. sqr-neg 12.5%

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

      2. +-commutative 12.5%

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

      3. sqr-neg 12.5%

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

      4. +-commutative 12.5%

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

      5. distribute-rgt-in 12.5%

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

      6. cancel-sign-sub 12.5%

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

      7. distribute-rgt-out-- 12.5%

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

      8. sub-neg 12.5%

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

      9. remove-double-neg 12.5%

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

      10. +-commutative 12.5%

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

      11. hypot-def 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in re around -inf 51.2%

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

      1. *-commutative 51.2%

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

      2. associate-*l/ 51.2%

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

   6. Simplified 51.2%

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

      1. add-sqr-sqrt 50.9%

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

      2. sqrt-unprod 51.1%

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

      3. *-commutative 51.1%

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

      4. *-commutative 51.1%

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

      5. swap-sqr 51.1%

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

      6. add-sqr-sqrt 51.1%

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

      7. associate-*r/ 51.1%

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

      8. *-commutative 51.1%

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

      9. associate-*r* 51.1%

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

      10. metadata-eval 51.1%

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

      11. metadata-eval 51.1%

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

   8. Applied egg-rr 51.1%

      \[\leadsto \color{blue}{\sqrt{\frac{-1 \cdot {im}^{2}}{re} \cdot 0.25}} \]
   9. Step-by-step derivation

      1. associate-/l* 49.8%

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

      2. associate-*l/ 49.8%

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

      3. metadata-eval 49.8%

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

   10. Simplified 49.8%

       \[\leadsto \color{blue}{\sqrt{\frac{-0.25}{\frac{re}{{im}^{2}}}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 51.1%

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

       2. sqrt-prod 64.7%

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

       3. unpow2 64.7%

          \[\leadsto \sqrt{\frac{-0.25}{re}} \cdot \sqrt{\color{blue}{im \cdot im}} \]

       4. sqrt-prod 49.4%

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

       5. add-sqr-sqrt 53.4%

          \[\leadsto \sqrt{\frac{-0.25}{re}} \cdot \color{blue}{im} \]

   12. Applied egg-rr 53.4%

       \[\leadsto \color{blue}{\sqrt{\frac{-0.25}{re}} \cdot im} \]

   if -7.19999999999999967e48 < re < -940 or -3.90000000000000019e-59 < re < 1.5599999999999999e29

   1. Initial program 59.4%

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

      1. sqr-neg 59.4%

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

      2. +-commutative 59.4%

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

      3. sqr-neg 59.4%

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

      4. +-commutative 59.4%

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

      5. distribute-rgt-in 59.4%

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

      6. cancel-sign-sub 59.4%

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

      7. distribute-rgt-out-- 59.4%

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

      8. sub-neg 59.4%

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

      9. remove-double-neg 59.4%

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

      10. +-commutative 59.4%

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

      11. hypot-def 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in re around 0 47.2%

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

   if 1.5599999999999999e29 < re

   1. Initial program 37.7%

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

      1. sqr-neg 37.7%

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

      2. +-commutative 37.7%

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

      3. sqr-neg 37.7%

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

      4. +-commutative 37.7%

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

      5. distribute-rgt-in 37.7%

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

      6. cancel-sign-sub 37.7%

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

      7. distribute-rgt-out-- 37.7%

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

      8. sub-neg 37.7%

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

      9. remove-double-neg 37.7%

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

      10. +-commutative 37.7%

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

      11. hypot-def 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. Taylor expanded in im around 0 81.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   5. 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 83.3%

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

      4. associate-*r* 83.3%

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

      5. metadata-eval 83.3%

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      6. *-lft-identity 83.3%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   6. Simplified 83.3%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+48}:\\ \;\;\;\;im \cdot \sqrt{\frac{-0.25}{re}}\\ \mathbf{elif}\;re \leq -940:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq -3.9 \cdot 10^{-59}:\\ \;\;\;\;im \cdot \sqrt{\frac{-0.25}{re}}\\ \mathbf{elif}\;re \leq 1.56 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 4: 64.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 8.6 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \sqrt{im_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 8.6e+28) (* 0.5 (sqrt (* im_m 2.0))) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 8.6e+28) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else {
		tmp = 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 <= 8.6d+28) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else
        tmp = 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 <= 8.6e+28) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 8.6e+28:
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 8.6e+28)
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 8.6e+28)
		tmp = 0.5 * sqrt((im_m * 2.0));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 8.6e+28], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.59999999999999951e28

   1. Initial program 42.1%

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

      1. sqr-neg 42.1%

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

      2. +-commutative 42.1%

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

      3. sqr-neg 42.1%

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

      4. +-commutative 42.1%

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

      5. distribute-rgt-in 42.1%

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

      6. cancel-sign-sub 42.1%

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

      7. distribute-rgt-out-- 42.1%

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

      8. sub-neg 42.1%

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

      9. remove-double-neg 42.1%

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

      10. +-commutative 42.1%

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

      11. hypot-def 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in re around 0 34.3%

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

   if 8.59999999999999951e28 < re

   1. Initial program 37.7%

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

      1. sqr-neg 37.7%

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

      2. +-commutative 37.7%

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

      3. sqr-neg 37.7%

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

      4. +-commutative 37.7%

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

      5. distribute-rgt-in 37.7%

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

      6. cancel-sign-sub 37.7%

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

      7. distribute-rgt-out-- 37.7%

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

      8. sub-neg 37.7%

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

      9. remove-double-neg 37.7%

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

      10. +-commutative 37.7%

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

      11. hypot-def 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. Taylor expanded in im around 0 81.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   5. 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 83.3%

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

      4. associate-*r* 83.3%

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

      5. metadata-eval 83.3%

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      6. *-lft-identity 83.3%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   6. Simplified 83.3%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8.6 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 5: 26.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sqrt{re} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

im_m = abs(im)
function code(re, im_m)
	return sqrt(re)
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[Sqrt[re], $MachinePrecision]

Derivation

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%

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

   11. hypot-def 77.3%

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

3. Simplified 77.3%

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

4. Taylor expanded in im around 0 24.6%

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

   1. *-commutative 24.6%

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

   2. unpow2 24.6%

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

   3. rem-square-sqrt 25.1%

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

   4. associate-*r* 25.1%

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

   5. metadata-eval 25.1%

      \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

   6. *-lft-identity 25.1%

      \[\leadsto \color{blue}{\sqrt{re}} \]

6. Simplified 25.1%

   \[\leadsto \color{blue}{\sqrt{re}} \]

7. Final simplification 25.1%

   \[\leadsto \sqrt{re} \]

Developer target: 49.0% 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 2023318 
(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)))))