math.sqrt on complex, real part

Percentage Accurate: 41.3% → 86.7%

Time: 11.6s

Alternatives: 7

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.3% 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: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.55 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot e^{0.5 \cdot \mathsf{fma}\left(2, \log im\_m, \log \left(\frac{-1}{re}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.55e-14)
   (* 0.5 (exp (* 0.5 (fma 2.0 (log im_m) (log (/ -1.0 re))))))
   (sqrt (* 0.5 (+ re (hypot re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2.55e-14) {
		tmp = 0.5 * exp((0.5 * fma(2.0, log(im_m), log((-1.0 / re)))));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2.55e-14)
		tmp = Float64(0.5 * exp(Float64(0.5 * fma(2.0, log(im_m), log(Float64(-1.0 / re))))));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im_m))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2.55e-14], N[(0.5 * N[Exp[N[(0.5 * N[(2.0 * N[Log[im$95$m], $MachinePrecision] + N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.5499999999999999e-14

   1. Initial program 11.3%

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

      1. sqr-neg 11.3%

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

      2. +-commutative 11.3%

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

      3. sqr-neg 11.3%

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

      4. +-commutative 11.3%

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

      5. distribute-rgt-in 11.3%

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

      6. cancel-sign-sub 11.3%

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

      7. distribute-rgt-out-- 11.3%

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

      8. sub-neg 11.3%

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

      9. remove-double-neg 11.3%

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

      10. +-commutative 11.3%

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

   3. Simplified 35.7%

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

      1. +-commutative 35.7%

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

      2. hypot-udef 11.3%

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

      3. add-cbrt-cube 11.3%

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

      4. add-sqr-sqrt 11.2%

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

      5. hypot-udef 10.0%

         \[\leadsto 0.5 \cdot \sqrt[3]{\left(2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} + re\right)\right) \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \]

      6. +-commutative 10.0%

         \[\leadsto 0.5 \cdot \sqrt[3]{\left(2 \cdot \color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)}\right) \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \]

      7. pow1 10.0%

         \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \]

      8. pow1/2 10.0%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)}^{0.5}}} \]

      9. hypot-udef 26.0%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{1} \cdot {\left(2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} + re\right)\right)}^{0.5}} \]

      10. +-commutative 26.0%

          \[\leadsto 0.5 \cdot \sqrt[3]{{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{1} \cdot {\left(2 \cdot \color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}^{0.5}} \]

   6. Applied egg-rr 26.0%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right)}^{1.5}}} \]

   7. Taylor expanded in re around -inf 51.0%

      \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{e^{1.5 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}}} \]
   8. Step-by-step derivation

      1. exp-prod 51.0%

         \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(e^{1.5}\right)}^{\left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}}} \]

      2. +-commutative 51.0%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(e^{1.5}\right)}^{\color{blue}{\left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}}} \]

      3. log-pow 34.2%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(e^{1.5}\right)}^{\left(\color{blue}{2 \cdot \log im} + \log \left(\frac{-1}{re}\right)\right)}} \]

      4. fma-def 34.2%

         \[\leadsto 0.5 \cdot \sqrt[3]{{\left(e^{1.5}\right)}^{\color{blue}{\left(\mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)\right)}}} \]

   9. Simplified 34.2%

      \[\leadsto 0.5 \cdot \sqrt[3]{\color{blue}{{\left(e^{1.5}\right)}^{\left(\mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)\right)}}} \]

   10. Taylor expanded in im around 0 34.0%

       \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{1.5 \cdot \left(\log \left(\frac{-1}{re}\right) + 2 \cdot \log im\right)}\right)}^{0.3333333333333333}} \]
   11. Step-by-step derivation

       1. +-commutative 34.0%

          \[\leadsto 0.5 \cdot {\left(e^{1.5 \cdot \color{blue}{\left(2 \cdot \log im + \log \left(\frac{-1}{re}\right)\right)}}\right)}^{0.3333333333333333} \]

       2. fma-def 34.0%

          \[\leadsto 0.5 \cdot {\left(e^{1.5 \cdot \color{blue}{\mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)}}\right)}^{0.3333333333333333} \]

       3. exp-prod 46.2%

          \[\leadsto 0.5 \cdot \color{blue}{e^{\left(1.5 \cdot \mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)\right) \cdot 0.3333333333333333}} \]

       4. *-commutative 46.2%

          \[\leadsto 0.5 \cdot e^{\color{blue}{0.3333333333333333 \cdot \left(1.5 \cdot \mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)\right)}} \]

       5. associate-*r* 46.5%

          \[\leadsto 0.5 \cdot e^{\color{blue}{\left(0.3333333333333333 \cdot 1.5\right) \cdot \mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)}} \]

       6. metadata-eval 46.5%

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

   12. Simplified 46.5%

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

   if -2.5499999999999999e-14 < re

   1. Initial program 47.3%

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

      1. sqr-neg 47.3%

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

      2. +-commutative 47.3%

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

      3. sqr-neg 47.3%

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

      4. +-commutative 47.3%

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

      5. distribute-rgt-in 47.3%

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

      6. cancel-sign-sub 47.3%

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

      7. distribute-rgt-out-- 47.3%

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

      8. sub-neg 47.3%

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

      9. remove-double-neg 47.3%

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

      10. +-commutative 47.3%

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

   3. Simplified 94.8%

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

      1. add-sqr-sqrt 94.1%

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

      2. sqrt-unprod 94.8%

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

      3. *-commutative 94.8%

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

      4. *-commutative 94.8%

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

      5. swap-sqr 94.8%

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

      6. add-sqr-sqrt 94.8%

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

      7. *-commutative 94.8%

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

      8. metadata-eval 94.8%

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

   6. Applied egg-rr 94.8%

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

      1. associate-*l* 94.8%

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

      2. metadata-eval 94.8%

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

   8. Simplified 94.8%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.55 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot e^{0.5 \cdot \mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.2 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{{im\_m}^{2}}{re} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.2e-5)
   (sqrt (* 0.5 (* (/ (pow im_m 2.0) re) -0.5)))
   (sqrt (* 0.5 (+ re (hypot re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.2e-5) {
		tmp = sqrt((0.5 * ((pow(im_m, 2.0) / re) * -0.5)));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -1.2e-5) {
		tmp = Math.sqrt((0.5 * ((Math.pow(im_m, 2.0) / re) * -0.5)));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -1.2e-5:
		tmp = math.sqrt((0.5 * ((math.pow(im_m, 2.0) / re) * -0.5)))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.2e-5)
		tmp = sqrt(Float64(0.5 * Float64(Float64((im_m ^ 2.0) / re) * -0.5)));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im_m))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -1.2e-5)
		tmp = sqrt((0.5 * (((im_m ^ 2.0) / re) * -0.5)));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.2e-5], N[Sqrt[N[(0.5 * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] / re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.2e-5

   1. Initial program 11.4%

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

      1. sqr-neg 11.4%

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

      2. +-commutative 11.4%

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

      3. sqr-neg 11.4%

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

      4. +-commutative 11.4%

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

      5. distribute-rgt-in 11.4%

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

      6. cancel-sign-sub 11.4%

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

      7. distribute-rgt-out-- 11.4%

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

      8. sub-neg 11.4%

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

      9. remove-double-neg 11.4%

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

      10. +-commutative 11.4%

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

   3. Simplified 36.1%

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

      1. add-sqr-sqrt 36.0%

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

      2. sqrt-unprod 36.1%

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

      3. *-commutative 36.1%

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

      4. *-commutative 36.1%

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

      5. swap-sqr 36.1%

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

      6. add-sqr-sqrt 36.1%

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

      7. *-commutative 36.1%

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

      8. metadata-eval 36.1%

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

   6. Applied egg-rr 36.1%

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

      1. associate-*l* 36.1%

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

      2. metadata-eval 36.1%

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

   8. Simplified 36.1%

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

   9. Taylor expanded in re around -inf 60.7%

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

       1. *-commutative 60.7%

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

   11. Simplified 60.7%

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

   if -1.2e-5 < 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%

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

   3. Simplified 94.4%

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

      1. add-sqr-sqrt 93.6%

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

      2. sqrt-unprod 94.4%

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

      3. *-commutative 94.4%

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

      4. *-commutative 94.4%

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

      5. swap-sqr 94.4%

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

      6. add-sqr-sqrt 94.4%

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

      7. *-commutative 94.4%

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

      8. metadata-eval 94.4%

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

   6. Applied egg-rr 94.4%

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

      1. associate-*l* 94.4%

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

      2. metadata-eval 94.4%

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

   8. Simplified 94.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.2 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{{im}^{2}}{re} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{+162}:\\ \;\;\;\;\sqrt[3]{{\left(0.5 \cdot im\_m\right)}^{1.5}}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -8.5e+162)
   (cbrt (pow (* 0.5 im_m) 1.5))
   (if (<= re 4.5e+103) (sqrt (* 0.5 (+ re im_m))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -8.5e+162) {
		tmp = cbrt(pow((0.5 * im_m), 1.5));
	} else if (re <= 4.5e+103) {
		tmp = sqrt((0.5 * (re + im_m)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -8.5e+162) {
		tmp = Math.cbrt(Math.pow((0.5 * im_m), 1.5));
	} else if (re <= 4.5e+103) {
		tmp = Math.sqrt((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 <= -8.5e+162)
		tmp = cbrt((Float64(0.5 * im_m) ^ 1.5));
	elseif (re <= 4.5e+103)
		tmp = sqrt(Float64(0.5 * Float64(re + im_m)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -8.5e+162], N[Power[N[Power[N[(0.5 * im$95$m), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[re, 4.5e+103], N[Sqrt[N[(0.5 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.5000000000000003e162

   1. Initial program 2.5%

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

      1. sqr-neg 2.5%

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

      2. +-commutative 2.5%

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

      3. sqr-neg 2.5%

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

      4. +-commutative 2.5%

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

      5. distribute-rgt-in 2.5%

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

      6. cancel-sign-sub 2.5%

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

      7. distribute-rgt-out-- 2.5%

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

      8. sub-neg 2.5%

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

      9. remove-double-neg 2.5%

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

      10. +-commutative 2.5%

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

   3. Simplified 33.5%

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

      1. add-sqr-sqrt 33.4%

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

      2. sqrt-unprod 33.5%

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

      3. *-commutative 33.5%

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

      4. *-commutative 33.5%

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

      5. swap-sqr 33.5%

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

      6. add-sqr-sqrt 33.5%

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

      7. *-commutative 33.5%

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

      8. metadata-eval 33.5%

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

   6. Applied egg-rr 33.5%

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

      1. associate-*l* 33.5%

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

      2. metadata-eval 33.5%

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

   8. Simplified 33.5%

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

      1. add-cbrt-cube 26.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5} \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5}\right) \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5}}} \]

      2. add-sqr-sqrt 26.1%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5\right)} \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5}} \]

      3. pow1 26.1%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5\right)}^{1}} \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5}} \]

      4. pow1/2 26.1%

         \[\leadsto \sqrt[3]{{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5\right)}^{1} \cdot \color{blue}{{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5\right)}^{0.5}}} \]

      5. pow-prod-up 26.1%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5\right)}^{\left(1 + 0.5\right)}}} \]

      6. metadata-eval 26.1%

         \[\leadsto \sqrt[3]{{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5\right)}^{\color{blue}{1.5}}} \]

   10. Applied egg-rr 26.1%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5\right)}^{1.5}}} \]

   11. Taylor expanded in re around 0 10.6%

       \[\leadsto \sqrt[3]{{\left(\color{blue}{im} \cdot 0.5\right)}^{1.5}} \]

   if -8.5000000000000003e162 < re < 4.50000000000000001e103

   1. Initial program 49.6%

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

      1. sqr-neg 49.6%

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

      2. +-commutative 49.6%

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

      3. sqr-neg 49.6%

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

      4. +-commutative 49.6%

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

      5. distribute-rgt-in 49.6%

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

      6. cancel-sign-sub 49.6%

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

      7. distribute-rgt-out-- 49.6%

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

      8. sub-neg 49.6%

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

      9. remove-double-neg 49.6%

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

      10. +-commutative 49.6%

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

   3. Simplified 83.8%

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

      1. add-sqr-sqrt 83.1%

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

      2. sqrt-unprod 83.8%

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

      3. *-commutative 83.8%

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

      4. *-commutative 83.8%

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

      5. swap-sqr 83.8%

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

      6. add-sqr-sqrt 83.8%

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

      7. *-commutative 83.8%

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

      8. metadata-eval 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. associate-*l* 83.8%

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

      2. metadata-eval 83.8%

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

   8. Simplified 83.8%

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

   9. Taylor expanded in re around 0 36.1%

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

   if 4.50000000000000001e103 < re

   1. Initial program 21.6%

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

      1. sqr-neg 21.6%

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

      2. +-commutative 21.6%

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

      3. sqr-neg 21.6%

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

      4. +-commutative 21.6%

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

      5. distribute-rgt-in 21.6%

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

      6. cancel-sign-sub 21.6%

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

      7. distribute-rgt-out-- 21.6%

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

      8. sub-neg 21.6%

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

      9. remove-double-neg 21.6%

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

      10. +-commutative 21.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 77.9%

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

      1. *-commutative 77.9%

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

      2. unpow2 77.9%

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

      3. rem-square-sqrt 79.5%

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

      4. associate-*r* 79.5%

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

      5. metadata-eval 79.5%

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

      6. *-lft-identity 79.5%

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

   7. Simplified 79.5%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{+162}:\\ \;\;\;\;\sqrt[3]{{\left(0.5 \cdot im\right)}^{1.5}}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

   1. sqr-neg 37.6%

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

   2. +-commutative 37.6%

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

   3. sqr-neg 37.6%

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

   4. +-commutative 37.6%

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

   5. distribute-rgt-in 37.6%

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

   6. cancel-sign-sub 37.6%

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

   7. distribute-rgt-out-- 37.6%

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

   8. sub-neg 37.6%

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

   9. remove-double-neg 37.6%

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

   10. +-commutative 37.6%

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

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. add-sqr-sqrt 78.3%

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

   2. sqrt-unprod 78.9%

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

   3. *-commutative 78.9%

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

   4. *-commutative 78.9%

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

   5. swap-sqr 78.9%

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

   6. add-sqr-sqrt 78.9%

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

   7. *-commutative 78.9%

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

   8. metadata-eval 78.9%

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

6. Applied egg-rr 78.9%

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

   1. associate-*l* 78.9%

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

   2. metadata-eval 78.9%

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

8. Simplified 78.9%

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

9. Final simplification 78.9%

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

Alternative 5: 63.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+164}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.5e+164)
   (* 0.5 (sqrt (* 2.0 (- re re))))
   (if (<= re 6.5e+97) (sqrt (* 0.5 (+ re im_m))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.5e+164) {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	} else if (re <= 6.5e+97) {
		tmp = sqrt((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 <= (-1.5d+164)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    else if (re <= 6.5d+97) then
        tmp = sqrt((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 <= -1.5e+164) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	} else if (re <= 6.5e+97) {
		tmp = Math.sqrt((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 <= -1.5e+164:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	elif re <= 6.5e+97:
		tmp = math.sqrt((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 <= -1.5e+164)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	elseif (re <= 6.5e+97)
		tmp = sqrt(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 <= -1.5e+164)
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	elseif (re <= 6.5e+97)
		tmp = sqrt((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, -1.5e+164], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e+97], N[Sqrt[N[(0.5 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.5e164

   1. Initial program 2.5%

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

   3. Taylor expanded in re around -inf 25.9%

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

      1. mul-1-neg 25.9%

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

   5. Simplified 25.9%

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

   if -1.5e164 < re < 6.4999999999999999e97

   1. Initial program 49.6%

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

      1. sqr-neg 49.6%

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

      2. +-commutative 49.6%

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

      3. sqr-neg 49.6%

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

      4. +-commutative 49.6%

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

      5. distribute-rgt-in 49.6%

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

      6. cancel-sign-sub 49.6%

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

      7. distribute-rgt-out-- 49.6%

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

      8. sub-neg 49.6%

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

      9. remove-double-neg 49.6%

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

      10. +-commutative 49.6%

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

   3. Simplified 83.8%

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

      1. add-sqr-sqrt 83.1%

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

      2. sqrt-unprod 83.8%

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

      3. *-commutative 83.8%

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

      4. *-commutative 83.8%

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

      5. swap-sqr 83.8%

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

      6. add-sqr-sqrt 83.8%

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

      7. *-commutative 83.8%

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

      8. metadata-eval 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. associate-*l* 83.8%

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

      2. metadata-eval 83.8%

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

   8. Simplified 83.8%

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

   9. Taylor expanded in re around 0 36.1%

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

   if 6.4999999999999999e97 < re

   1. Initial program 21.6%

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

      1. sqr-neg 21.6%

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

      2. +-commutative 21.6%

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

      3. sqr-neg 21.6%

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

      4. +-commutative 21.6%

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

      5. distribute-rgt-in 21.6%

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

      6. cancel-sign-sub 21.6%

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

      7. distribute-rgt-out-- 21.6%

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

      8. sub-neg 21.6%

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

      9. remove-double-neg 21.6%

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

      10. +-commutative 21.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 77.9%

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

      1. *-commutative 77.9%

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

      2. unpow2 77.9%

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

      3. rem-square-sqrt 79.5%

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

      4. associate-*r* 79.5%

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

      5. metadata-eval 79.5%

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

      6. *-lft-identity 79.5%

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

   7. Simplified 79.5%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+164}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{0.5 \cdot im\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 6.5e+97) (sqrt (* 0.5 im_m)) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 6.5e+97) {
		tmp = sqrt((0.5 * 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 <= 6.5d+97) then
        tmp = sqrt((0.5d0 * 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 <= 6.5e+97) {
		tmp = Math.sqrt((0.5 * im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 6.5e+97:
		tmp = math.sqrt((0.5 * im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 6.5e+97)
		tmp = sqrt(Float64(0.5 * 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 <= 6.5e+97)
		tmp = sqrt((0.5 * 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, 6.5e+97], N[Sqrt[N[(0.5 * im$95$m), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.4999999999999999e97

   1. Initial program 40.9%

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

      1. sqr-neg 40.9%

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

      2. +-commutative 40.9%

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

      3. sqr-neg 40.9%

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

      4. +-commutative 40.9%

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

      5. distribute-rgt-in 40.9%

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

      6. cancel-sign-sub 40.9%

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

      7. distribute-rgt-out-- 40.9%

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

      8. sub-neg 40.9%

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

      9. remove-double-neg 40.9%

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

      10. +-commutative 40.9%

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

   3. Simplified 74.5%

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

      1. add-sqr-sqrt 73.9%

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

      2. sqrt-unprod 74.5%

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

      3. *-commutative 74.5%

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

      4. *-commutative 74.5%

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

      5. swap-sqr 74.5%

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

      6. add-sqr-sqrt 74.5%

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

      7. *-commutative 74.5%

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

      8. metadata-eval 74.5%

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

   6. Applied egg-rr 74.5%

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

      1. associate-*l* 74.5%

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

      2. metadata-eval 74.5%

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

   8. Simplified 74.5%

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

   9. Taylor expanded in re around 0 29.8%

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

   if 6.4999999999999999e97 < re

   1. Initial program 21.6%

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

      1. sqr-neg 21.6%

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

      2. +-commutative 21.6%

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

      3. sqr-neg 21.6%

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

      4. +-commutative 21.6%

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

      5. distribute-rgt-in 21.6%

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

      6. cancel-sign-sub 21.6%

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

      7. distribute-rgt-out-- 21.6%

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

      8. sub-neg 21.6%

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

      9. remove-double-neg 21.6%

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

      10. +-commutative 21.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 77.9%

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

      1. *-commutative 77.9%

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

      2. unpow2 77.9%

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

      3. rem-square-sqrt 79.5%

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

      4. associate-*r* 79.5%

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

      5. metadata-eval 79.5%

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

      6. *-lft-identity 79.5%

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

   7. Simplified 79.5%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.1% 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 37.6%

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

   1. sqr-neg 37.6%

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

   2. +-commutative 37.6%

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

   3. sqr-neg 37.6%

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

   4. +-commutative 37.6%

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

   5. distribute-rgt-in 37.6%

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

   6. cancel-sign-sub 37.6%

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

   7. distribute-rgt-out-- 37.6%

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

   8. sub-neg 37.6%

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

   9. remove-double-neg 37.6%

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

   10. +-commutative 37.6%

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

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 im around 0 24.4%

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

   1. *-commutative 24.4%

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

   2. unpow2 24.4%

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

   3. rem-square-sqrt 24.8%

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

   4. associate-*r* 24.8%

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

   5. metadata-eval 24.8%

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

   6. *-lft-identity 24.8%

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

7. Simplified 24.8%

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

8. Final simplification 24.8%

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

Developer target: 48.4% 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 2024031 
(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)))))