math.sqrt on complex, real part

Percentage Accurate: 42.0% → 90.3%

Time: 11.5s

Alternatives: 6

Speedup: 1.9×

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: 90.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 13.6%

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

      1. sqr-neg 13.6%

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

      2. +-commutative 13.6%

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

      3. sqr-neg 13.6%

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

      4. +-commutative 13.6%

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

      5. distribute-rgt-in 13.6%

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

      6. cancel-sign-sub 13.6%

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

      7. distribute-rgt-out-- 13.6%

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

      8. sub-neg 13.6%

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

      9. remove-double-neg 13.6%

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

      10. +-commutative 13.6%

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

      11. hypot-define 13.6%

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

   3. Simplified 13.6%

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

      1. pow1/2 13.6%

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

      2. hypot-define 13.6%

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

      3. +-commutative 13.6%

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

      4. pow-to-exp 13.6%

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

      5. +-commutative 13.6%

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

      6. hypot-define 13.6%

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

   6. Applied egg-rr 13.6%

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

   7. Taylor expanded in re around -inf 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-*l/ 42.1%

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

   9. Simplified 42.1%

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

       1. exp-to-pow 44.0%

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

       2. pow1/2 44.0%

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

       3. *-commutative 44.0%

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

       4. sqrt-prod 43.9%

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

       5. associate-/l* 43.9%

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

       6. sqrt-prod 48.5%

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

       7. sqrt-pow1 62.1%

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

       8. metadata-eval 62.1%

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

       9. pow1 62.1%

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

   11. Applied egg-rr 62.1%

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

       1. associate-*l* 62.0%

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

   13. Simplified 62.0%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 44.6%

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

      1. sqr-neg 44.6%

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

      2. +-commutative 44.6%

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

      3. sqr-neg 44.6%

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

      4. +-commutative 44.6%

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

      5. distribute-rgt-in 44.6%

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

      6. cancel-sign-sub 44.6%

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

      7. distribute-rgt-out-- 44.6%

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

      8. sub-neg 44.6%

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

      9. remove-double-neg 44.6%

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

      10. +-commutative 44.6%

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

      11. hypot-define 91.0%

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

   3. Simplified 91.0%

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

      1. *-commutative 91.0%

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

      2. hypot-define 44.6%

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

      3. +-commutative 44.6%

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

      4. *-commutative 44.6%

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

      5. add-sqr-sqrt 44.3%

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

      6. sqrt-unprod 44.6%

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

      7. *-commutative 44.6%

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

      8. *-commutative 44.6%

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

      9. swap-sqr 44.6%

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

   6. Applied egg-rr 91.0%

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

      1. *-commutative 91.0%

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

      2. associate-*r* 91.4%

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

      3. metadata-eval 91.4%

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

      4. hypot-undefine 44.6%

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

      5. unpow2 44.6%

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

      6. unpow2 44.6%

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

      7. +-commutative 44.6%

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

      8. unpow2 44.6%

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

      9. unpow2 44.6%

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

      10. hypot-undefine 91.4%

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

   8. Simplified 91.4%

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

4. Final simplification 86.8%

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

Alternative 2: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(im\_m, re\right)\right)} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

   1. sqr-neg 39.8%

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

   2. +-commutative 39.8%

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

   3. sqr-neg 39.8%

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

   4. +-commutative 39.8%

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

   5. distribute-rgt-in 39.8%

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

   6. cancel-sign-sub 39.8%

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

   7. distribute-rgt-out-- 39.8%

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

   8. sub-neg 39.8%

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

   9. remove-double-neg 39.8%

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

   10. +-commutative 39.8%

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

   11. hypot-define 78.9%

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

3. Simplified 78.9%

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

   1. *-commutative 78.9%

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

   2. hypot-define 39.8%

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

   3. +-commutative 39.8%

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

   4. *-commutative 39.8%

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

   5. add-sqr-sqrt 39.5%

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

   6. sqrt-unprod 39.8%

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

   7. *-commutative 39.8%

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

   8. *-commutative 39.8%

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

   9. swap-sqr 39.8%

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

6. Applied egg-rr 78.9%

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

   1. *-commutative 78.9%

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

   2. associate-*r* 79.3%

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

   3. metadata-eval 79.3%

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

   4. hypot-undefine 39.8%

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

   5. unpow2 39.8%

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

   6. unpow2 39.8%

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

   7. +-commutative 39.8%

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

   8. unpow2 39.8%

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

   9. unpow2 39.8%

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

   10. hypot-undefine 79.3%

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

8. Simplified 79.3%

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

9. Final simplification 79.3%

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

Alternative 3: 64.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im\_m + re \cdot \left(1 + 0.5 \cdot \frac{re}{im\_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -4.5e+118)
   (* 0.5 (sqrt (* 2.0 (- re re))))
   (if (<= re 2.9e-19)
     (* 0.5 (sqrt (* 2.0 (+ im_m (* re (+ 1.0 (* 0.5 (/ re im_m))))))))
     (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -4.5e+118) {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	} else if (re <= 2.9e-19) {
		tmp = 0.5 * sqrt((2.0 * (im_m + (re * (1.0 + (0.5 * (re / im_m)))))));
	} 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 <= (-4.5d+118)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    else if (re <= 2.9d-19) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im_m + (re * (1.0d0 + (0.5d0 * (re / im_m)))))))
    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 <= -4.5e+118) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	} else if (re <= 2.9e-19) {
		tmp = 0.5 * Math.sqrt((2.0 * (im_m + (re * (1.0 + (0.5 * (re / im_m)))))));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -4.5e+118:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	elif re <= 2.9e-19:
		tmp = 0.5 * math.sqrt((2.0 * (im_m + (re * (1.0 + (0.5 * (re / im_m)))))))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -4.5e+118)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	elseif (re <= 2.9e-19)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im_m + Float64(re * Float64(1.0 + Float64(0.5 * Float64(re / im_m))))))));
	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 <= -4.5e+118)
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	elseif (re <= 2.9e-19)
		tmp = 0.5 * sqrt((2.0 * (im_m + (re * (1.0 + (0.5 * (re / im_m)))))));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -4.5e+118], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.9e-19], N[(0.5 * N[Sqrt[N[(2.0 * N[(im$95$m + N[(re * N[(1.0 + N[(0.5 * N[(re / im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.50000000000000002e118

   1. Initial program 9.3%

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

   3. Taylor expanded in re around -inf 30.4%

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

      1. mul-1-neg 30.4%

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

   5. Simplified 30.4%

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

   if -4.50000000000000002e118 < re < 2.9e-19

   1. Initial program 46.1%

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

      1. sqr-neg 46.1%

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

      2. +-commutative 46.1%

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

      3. sqr-neg 46.1%

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

      4. +-commutative 46.1%

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

      5. distribute-rgt-in 46.1%

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

      6. cancel-sign-sub 46.1%

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

      7. distribute-rgt-out-- 46.1%

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

      8. sub-neg 46.1%

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

      9. remove-double-neg 46.1%

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

      10. +-commutative 46.1%

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

      11. hypot-define 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in re around 0 33.9%

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

   if 2.9e-19 < re

   1. Initial program 43.5%

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

      1. sqr-neg 43.5%

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

      2. +-commutative 43.5%

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

      3. sqr-neg 43.5%

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

      4. +-commutative 43.5%

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

      5. distribute-rgt-in 43.5%

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

      6. cancel-sign-sub 43.5%

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

      7. distribute-rgt-out-- 43.5%

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

      8. sub-neg 43.5%

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

      9. remove-double-neg 43.5%

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

      10. +-commutative 43.5%

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

      11. hypot-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around inf 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. rem-square-sqrt 79.1%

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

      4. associate-*r* 79.1%

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

      5. metadata-eval 79.1%

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

      6. *-lft-identity 79.1%

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

   7. Simplified 79.1%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re \cdot \left(1 + 0.5 \cdot \frac{re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.42 \cdot 10^{+116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.42e+116)
   (* 0.5 (sqrt (* 2.0 (- re re))))
   (if (<= re 5.5e-20) (* 0.5 (sqrt (* 2.0 (+ re im_m)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.42e+116) {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	} else if (re <= 5.5e-20) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} 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 <= (-1.42d+116)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    else if (re <= 5.5d-20) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    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 <= -1.42e+116) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	} else if (re <= 5.5e-20) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -1.42e+116:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	elif re <= 5.5e-20:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.42e+116)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	elseif (re <= 5.5e-20)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	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 <= -1.42e+116)
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	elseif (re <= 5.5e-20)
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.42e+116], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.5e-20], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.4199999999999999e116

   1. Initial program 9.1%

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

   3. Taylor expanded in re around -inf 29.8%

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

      1. mul-1-neg 29.8%

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

   5. Simplified 29.8%

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

   if -1.4199999999999999e116 < re < 5.4999999999999996e-20

   1. Initial program 46.4%

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

      1. sqr-neg 46.4%

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

      2. +-commutative 46.4%

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

      3. sqr-neg 46.4%

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

      4. +-commutative 46.4%

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

      5. distribute-rgt-in 46.4%

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

      6. cancel-sign-sub 46.4%

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

      7. distribute-rgt-out-- 46.4%

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

      8. sub-neg 46.4%

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

      9. remove-double-neg 46.4%

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

      10. +-commutative 46.4%

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

      11. hypot-define 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in re around 0 35.0%

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

   if 5.4999999999999996e-20 < re

   1. Initial program 43.5%

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

      1. sqr-neg 43.5%

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

      2. +-commutative 43.5%

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

      3. sqr-neg 43.5%

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

      4. +-commutative 43.5%

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

      5. distribute-rgt-in 43.5%

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

      6. cancel-sign-sub 43.5%

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

      7. distribute-rgt-out-- 43.5%

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

      8. sub-neg 43.5%

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

      9. remove-double-neg 43.5%

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

      10. +-commutative 43.5%

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

      11. hypot-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around inf 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. rem-square-sqrt 79.1%

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

      4. associate-*r* 79.1%

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

      5. metadata-eval 79.1%

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

      6. *-lft-identity 79.1%

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

   7. Simplified 79.1%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.42 \cdot 10^{+116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 6.2e-13) (* 0.5 (sqrt (* 2.0 im_m))) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 6.2e-13) {
		tmp = 0.5 * sqrt((2.0 * im_m));
	} 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 <= 6.2d-13) then
        tmp = 0.5d0 * sqrt((2.0d0 * im_m))
    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 <= 6.2e-13) {
		tmp = 0.5 * Math.sqrt((2.0 * im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 6.2e-13:
		tmp = 0.5 * math.sqrt((2.0 * im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 6.2e-13)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im_m)));
	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 <= 6.2e-13)
		tmp = 0.5 * sqrt((2.0 * im_m));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 6.2e-13], N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.1999999999999998e-13

   1. Initial program 38.6%

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

      1. sqr-neg 38.6%

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

      2. +-commutative 38.6%

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

      3. sqr-neg 38.6%

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

      4. +-commutative 38.6%

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

      5. distribute-rgt-in 38.6%

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

      6. cancel-sign-sub 38.6%

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

      7. distribute-rgt-out-- 38.6%

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

      8. sub-neg 38.6%

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

      9. remove-double-neg 38.6%

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

      10. +-commutative 38.6%

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

      11. hypot-define 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in re around 0 27.7%

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

   if 6.1999999999999998e-13 < re

   1. Initial program 43.5%

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

      1. sqr-neg 43.5%

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

      2. +-commutative 43.5%

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

      3. sqr-neg 43.5%

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

      4. +-commutative 43.5%

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

      5. distribute-rgt-in 43.5%

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

      6. cancel-sign-sub 43.5%

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

      7. distribute-rgt-out-- 43.5%

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

      8. sub-neg 43.5%

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

      9. remove-double-neg 43.5%

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

      10. +-commutative 43.5%

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

      11. hypot-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around inf 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. rem-square-sqrt 79.1%

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

      4. associate-*r* 79.1%

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

      5. metadata-eval 79.1%

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

      6. *-lft-identity 79.1%

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

   7. Simplified 79.1%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 26.6% 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 39.8%

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

   1. sqr-neg 39.8%

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

   2. +-commutative 39.8%

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

   3. sqr-neg 39.8%

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

   4. +-commutative 39.8%

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

   5. distribute-rgt-in 39.8%

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

   6. cancel-sign-sub 39.8%

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

   7. distribute-rgt-out-- 39.8%

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

   8. sub-neg 39.8%

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

   9. remove-double-neg 39.8%

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

   10. +-commutative 39.8%

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

   11. hypot-define 78.9%

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

3. Simplified 78.9%

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

5. Taylor expanded in re around inf 24.7%

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

   1. *-commutative 24.7%

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

   2. unpow2 24.7%

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

   3. rem-square-sqrt 25.2%

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

   4. associate-*r* 25.2%

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

   5. metadata-eval 25.2%

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

   6. *-lft-identity 25.2%

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

7. Simplified 25.2%

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

8. Final simplification 25.2%

   \[\leadsto \sqrt{re} \]
9. Add Preprocessing

Developer target: 49.2% 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 2024080 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (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)))))