math.sqrt on complex, real part

Percentage Accurate: 40.9% → 90.0%

Time: 7.9s

Alternatives: 5

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.9% 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: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im_m \cdot im_m}\right)} \leq 0:\\ \;\;\;\;\frac{im_m}{\sqrt{re \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + \mathsf{hypot}\left(re, im_m\right)\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im_m im_m)))))) 0.0)
   (/ im_m (sqrt (* re -4.0)))
   (sqrt (* (+ re (hypot re im_m)) 0.5))))

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 = im_m / sqrt((re * -4.0));
	} else {
		tmp = sqrt(((re + hypot(re, im_m)) * 0.5));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (Math.sqrt((2.0 * (re + Math.sqrt(((re * re) + (im_m * im_m)))))) <= 0.0) {
		tmp = im_m / Math.sqrt((re * -4.0));
	} else {
		tmp = Math.sqrt(((re + Math.hypot(re, im_m)) * 0.5));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if math.sqrt((2.0 * (re + math.sqrt(((re * re) + (im_m * im_m)))))) <= 0.0:
		tmp = im_m / math.sqrt((re * -4.0))
	else:
		tmp = math.sqrt(((re + math.hypot(re, im_m)) * 0.5))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (sqrt(Float64(2.0 * Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))))) <= 0.0)
		tmp = Float64(im_m / sqrt(Float64(re * -4.0)));
	else
		tmp = sqrt(Float64(Float64(re + hypot(re, im_m)) * 0.5));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (sqrt((2.0 * (re + sqrt(((re * re) + (im_m * im_m)))))) <= 0.0)
		tmp = im_m / sqrt((re * -4.0));
	else
		tmp = sqrt(((re + hypot(re, im_m)) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[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[(im$95$m / N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.8%

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

      1. sqr-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. sqr-neg 9.8%

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

      4. +-commutative 9.8%

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

      5. distribute-rgt-in 9.8%

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

      6. cancel-sign-sub 9.8%

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

      7. distribute-rgt-out-- 9.8%

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

      8. sub-neg 9.8%

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

      9. remove-double-neg 9.8%

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

      10. +-commutative 9.8%

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

      11. hypot-def 9.8%

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

   3. Simplified 9.8%

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

      1. *-commutative 9.8%

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

      2. +-commutative 9.8%

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

      3. hypot-udef 9.8%

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

      4. *-commutative 9.8%

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

      5. add-cbrt-cube 9.8%

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

      6. add-sqr-sqrt 9.8%

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

      7. hypot-udef 9.8%

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

      8. +-commutative 9.8%

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

      9. pow1 9.8%

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

      10. pow1/2 9.8%

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

   5. Applied egg-rr 9.8%

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

   6. Taylor expanded in re around -inf 42.0%

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

      1. *-commutative 42.0%

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

      2. associate-*l/ 42.0%

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

   8. Simplified 42.0%

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

      1. add-sqr-sqrt 41.9%

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

      2. sqrt-unprod 42.0%

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

      3. *-commutative 42.0%

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

      4. *-commutative 42.0%

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

      5. swap-sqr 42.0%

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

   10. Applied egg-rr 50.4%

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

       1. associate-*l/ 50.4%

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

       2. associate-*l* 50.4%

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

       3. metadata-eval 50.4%

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

   12. Simplified 50.4%

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

       1. expm1-log1p-u 50.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{im}^{2} \cdot -0.25}{re}}\right)\right)} \]

       2. expm1-udef 11.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{{im}^{2} \cdot -0.25}{re}}\right)} - 1} \]

       3. associate-/l* 11.9%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{im}^{2}}{\frac{re}{-0.25}}}}\right)} - 1 \]

       4. sqrt-div 11.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{im}^{2}}}{\sqrt{\frac{re}{-0.25}}}}\right)} - 1 \]

       5. unpow2 11.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{im \cdot im}}}{\sqrt{\frac{re}{-0.25}}}\right)} - 1 \]

       6. sqrt-prod 5.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}{\sqrt{\frac{re}{-0.25}}}\right)} - 1 \]

       7. add-sqr-sqrt 9.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{im}}{\sqrt{\frac{re}{-0.25}}}\right)} - 1 \]

       8. div-inv 9.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{im}{\sqrt{\color{blue}{re \cdot \frac{1}{-0.25}}}}\right)} - 1 \]

       9. metadata-eval 9.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re \cdot \color{blue}{-4}}}\right)} - 1 \]

   14. Applied egg-rr 9.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re \cdot -4}}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 62.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re \cdot -4}}\right)\right)} \]

       2. expm1-log1p 62.9%

          \[\leadsto \color{blue}{\frac{im}{\sqrt{re \cdot -4}}} \]

   16. Simplified 62.9%

       \[\leadsto \color{blue}{\frac{im}{\sqrt{re \cdot -4}}} \]

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

   1. Initial program 42.7%

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

      1. sqr-neg 42.7%

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

      2. +-commutative 42.7%

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

      3. sqr-neg 42.7%

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

      4. +-commutative 42.7%

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

      5. distribute-rgt-in 42.7%

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

      6. cancel-sign-sub 42.7%

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

      7. distribute-rgt-out-- 42.7%

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

      8. sub-neg 42.7%

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

      9. remove-double-neg 42.7%

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

      10. +-commutative 42.7%

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

      11. hypot-def 84.8%

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

   3. Simplified 84.8%

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

      1. add-sqr-sqrt 84.1%

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

      2. sqrt-unprod 84.8%

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

      3. *-commutative 84.8%

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

      4. *-commutative 84.8%

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

      5. swap-sqr 84.8%

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

      6. add-sqr-sqrt 84.8%

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

      7. *-commutative 84.8%

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

      8. metadata-eval 84.8%

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

   5. Applied egg-rr 84.8%

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

      1. associate-*l* 85.3%

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

      2. metadata-eval 85.3%

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

   7. Simplified 85.3%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;\frac{im}{\sqrt{re \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5}\\ \end{array} \]

Alternative 2: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \frac{im_m}{\sqrt{re \cdot -4}}\\ t_1 := \sqrt{im_m \cdot 0.5}\\ \mathbf{if}\;re \leq -65000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -3 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1.2 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.45 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -3.5 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 95000000000000:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (/ im_m (sqrt (* re -4.0)))) (t_1 (sqrt (* im_m 0.5))))
   (if (<= re -65000000000.0)
     t_0
     (if (<= re -3e-22)
       t_1
       (if (<= re -1.2e-48)
         t_0
         (if (<= re -1.45e-105)
           t_1
           (if (<= re -3.5e-153)
             t_0
             (if (<= re 95000000000000.0)
               (sqrt (* 0.5 (+ re im_m)))
               (sqrt re)))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = im_m / sqrt((re * -4.0));
	double t_1 = sqrt((im_m * 0.5));
	double tmp;
	if (re <= -65000000000.0) {
		tmp = t_0;
	} else if (re <= -3e-22) {
		tmp = t_1;
	} else if (re <= -1.2e-48) {
		tmp = t_0;
	} else if (re <= -1.45e-105) {
		tmp = t_1;
	} else if (re <= -3.5e-153) {
		tmp = t_0;
	} else if (re <= 95000000000000.0) {
		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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im_m / sqrt((re * (-4.0d0)))
    t_1 = sqrt((im_m * 0.5d0))
    if (re <= (-65000000000.0d0)) then
        tmp = t_0
    else if (re <= (-3d-22)) then
        tmp = t_1
    else if (re <= (-1.2d-48)) then
        tmp = t_0
    else if (re <= (-1.45d-105)) then
        tmp = t_1
    else if (re <= (-3.5d-153)) then
        tmp = t_0
    else if (re <= 95000000000000.0d0) 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 t_0 = im_m / Math.sqrt((re * -4.0));
	double t_1 = Math.sqrt((im_m * 0.5));
	double tmp;
	if (re <= -65000000000.0) {
		tmp = t_0;
	} else if (re <= -3e-22) {
		tmp = t_1;
	} else if (re <= -1.2e-48) {
		tmp = t_0;
	} else if (re <= -1.45e-105) {
		tmp = t_1;
	} else if (re <= -3.5e-153) {
		tmp = t_0;
	} else if (re <= 95000000000000.0) {
		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):
	t_0 = im_m / math.sqrt((re * -4.0))
	t_1 = math.sqrt((im_m * 0.5))
	tmp = 0
	if re <= -65000000000.0:
		tmp = t_0
	elif re <= -3e-22:
		tmp = t_1
	elif re <= -1.2e-48:
		tmp = t_0
	elif re <= -1.45e-105:
		tmp = t_1
	elif re <= -3.5e-153:
		tmp = t_0
	elif re <= 95000000000000.0:
		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)
	t_0 = Float64(im_m / sqrt(Float64(re * -4.0)))
	t_1 = sqrt(Float64(im_m * 0.5))
	tmp = 0.0
	if (re <= -65000000000.0)
		tmp = t_0;
	elseif (re <= -3e-22)
		tmp = t_1;
	elseif (re <= -1.2e-48)
		tmp = t_0;
	elseif (re <= -1.45e-105)
		tmp = t_1;
	elseif (re <= -3.5e-153)
		tmp = t_0;
	elseif (re <= 95000000000000.0)
		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)
	t_0 = im_m / sqrt((re * -4.0));
	t_1 = sqrt((im_m * 0.5));
	tmp = 0.0;
	if (re <= -65000000000.0)
		tmp = t_0;
	elseif (re <= -3e-22)
		tmp = t_1;
	elseif (re <= -1.2e-48)
		tmp = t_0;
	elseif (re <= -1.45e-105)
		tmp = t_1;
	elseif (re <= -3.5e-153)
		tmp = t_0;
	elseif (re <= 95000000000000.0)
		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_] := Block[{t$95$0 = N[(im$95$m / N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(im$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[re, -65000000000.0], t$95$0, If[LessEqual[re, -3e-22], t$95$1, If[LessEqual[re, -1.2e-48], t$95$0, If[LessEqual[re, -1.45e-105], t$95$1, If[LessEqual[re, -3.5e-153], t$95$0, If[LessEqual[re, 95000000000000.0], N[Sqrt[N[(0.5 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -6.5e10 or -2.9999999999999999e-22 < re < -1.2e-48 or -1.45000000000000002e-105 < re < -3.49999999999999981e-153

   1. Initial program 11.0%

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

      1. sqr-neg 11.0%

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

      2. +-commutative 11.0%

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

      3. sqr-neg 11.0%

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

      4. +-commutative 11.0%

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

      5. distribute-rgt-in 11.0%

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

      6. cancel-sign-sub 11.0%

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

      7. distribute-rgt-out-- 11.0%

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

      8. sub-neg 11.0%

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

      9. remove-double-neg 11.0%

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

      10. +-commutative 11.0%

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

      11. hypot-def 33.4%

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

   3. Simplified 33.4%

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

      1. *-commutative 33.4%

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

      2. +-commutative 33.4%

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

      3. hypot-udef 11.0%

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

      4. *-commutative 11.0%

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

      5. add-cbrt-cube 10.9%

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

      6. add-sqr-sqrt 11.0%

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

      7. hypot-udef 10.0%

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

      8. +-commutative 10.0%

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

      9. pow1 10.0%

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

      10. pow1/2 10.0%

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

   5. Applied egg-rr 21.7%

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

   6. Taylor expanded in re around -inf 38.5%

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

      1. *-commutative 38.5%

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

      2. associate-*l/ 38.5%

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

   8. Simplified 38.5%

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

      1. add-sqr-sqrt 38.5%

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

      2. sqrt-unprod 38.5%

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

      3. *-commutative 38.5%

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

      4. *-commutative 38.5%

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

      5. swap-sqr 38.5%

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

   10. Applied egg-rr 47.1%

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

       1. associate-*l/ 47.1%

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

       2. associate-*l* 47.1%

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

       3. metadata-eval 47.1%

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

   12. Simplified 47.1%

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

       1. expm1-log1p-u 46.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{im}^{2} \cdot -0.25}{re}}\right)\right)} \]

       2. expm1-udef 20.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{{im}^{2} \cdot -0.25}{re}}\right)} - 1} \]

       3. associate-/l* 20.4%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{im}^{2}}{\frac{re}{-0.25}}}}\right)} - 1 \]

       4. sqrt-div 20.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{im}^{2}}}{\sqrt{\frac{re}{-0.25}}}}\right)} - 1 \]

       5. unpow2 20.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{im \cdot im}}}{\sqrt{\frac{re}{-0.25}}}\right)} - 1 \]

       6. sqrt-prod 9.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}{\sqrt{\frac{re}{-0.25}}}\right)} - 1 \]

       7. add-sqr-sqrt 15.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{im}}{\sqrt{\frac{re}{-0.25}}}\right)} - 1 \]

       8. div-inv 15.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{im}{\sqrt{\color{blue}{re \cdot \frac{1}{-0.25}}}}\right)} - 1 \]

       9. metadata-eval 15.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re \cdot \color{blue}{-4}}}\right)} - 1 \]

   14. Applied egg-rr 15.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re \cdot -4}}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 49.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re \cdot -4}}\right)\right)} \]

       2. expm1-log1p 49.7%

          \[\leadsto \color{blue}{\frac{im}{\sqrt{re \cdot -4}}} \]

   16. Simplified 49.7%

       \[\leadsto \color{blue}{\frac{im}{\sqrt{re \cdot -4}}} \]

   if -6.5e10 < re < -2.9999999999999999e-22 or -1.2e-48 < re < -1.45000000000000002e-105

   1. Initial program 61.0%

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

      1. sqr-neg 61.0%

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

      2. +-commutative 61.0%

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

      3. sqr-neg 61.0%

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

      4. +-commutative 61.0%

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

      5. distribute-rgt-in 61.0%

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

      6. cancel-sign-sub 61.0%

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

      7. distribute-rgt-out-- 61.0%

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

      8. sub-neg 61.0%

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

      9. remove-double-neg 61.0%

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

      10. +-commutative 61.0%

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

      11. hypot-def 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in re around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-*l* 50.6%

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

   6. Simplified 50.6%

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

      1. expm1-log1p-u 47.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right)\right)} \]

      2. expm1-udef 46.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right)} - 1} \]

      3. add-sqr-sqrt 46.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)} \cdot \sqrt{\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)}}\right)} - 1 \]

      4. sqrt-unprod 46.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right) \cdot \left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right)}}\right)} - 1 \]

      5. swap-sqr 46.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right) \cdot \left(\left(\sqrt{2} \cdot 0.5\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\right)}}\right)} - 1 \]

      6. add-sqr-sqrt 46.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{im} \cdot \left(\left(\sqrt{2} \cdot 0.5\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\right)}\right)} - 1 \]

      7. swap-sqr 46.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(0.5 \cdot 0.5\right)\right)}}\right)} - 1 \]

      8. rem-square-sqrt 46.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \left(\color{blue}{2} \cdot \left(0.5 \cdot 0.5\right)\right)}\right)} - 1 \]

      9. metadata-eval 46.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \left(2 \cdot \color{blue}{0.25}\right)}\right)} - 1 \]

      10. metadata-eval 46.3%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \color{blue}{0.5}}\right)} - 1 \]

   8. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{im \cdot 0.5}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 47.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{im \cdot 0.5}\right)\right)} \]

      2. expm1-log1p 50.9%

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

   10. Simplified 50.9%

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

   if -3.49999999999999981e-153 < re < 9.5e13

   1. Initial program 56.2%

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

      1. sqr-neg 56.2%

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

      2. +-commutative 56.2%

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

      3. sqr-neg 56.2%

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

      4. +-commutative 56.2%

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

      5. distribute-rgt-in 56.2%

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

      6. cancel-sign-sub 56.2%

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

      7. distribute-rgt-out-- 56.2%

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

      8. sub-neg 56.2%

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

      9. remove-double-neg 56.2%

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

      10. +-commutative 56.2%

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

      11. hypot-def 95.6%

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

   3. Simplified 95.6%

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

      1. add-sqr-sqrt 94.8%

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

      2. sqrt-unprod 95.6%

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

      3. *-commutative 95.6%

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

      4. *-commutative 95.6%

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

      5. swap-sqr 95.6%

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

      6. add-sqr-sqrt 95.6%

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

      7. *-commutative 95.6%

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

      8. metadata-eval 95.6%

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

   5. Applied egg-rr 95.6%

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

      1. associate-*l* 96.7%

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

      2. metadata-eval 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in re around 0 35.2%

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

   if 9.5e13 < re

   1. Initial program 43.9%

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

      1. sqr-neg 43.9%

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

      2. +-commutative 43.9%

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

      3. sqr-neg 43.9%

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

      4. +-commutative 43.9%

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

      5. distribute-rgt-in 43.9%

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

      6. cancel-sign-sub 43.9%

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

      7. distribute-rgt-out-- 43.9%

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

      8. sub-neg 43.9%

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

      9. remove-double-neg 43.9%

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

      10. +-commutative 43.9%

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

      11. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 80.3%

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

      1. *-commutative 80.3%

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

      2. unpow2 80.3%

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

      3. rem-square-sqrt 82.0%

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

      4. associate-*r* 82.0%

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

      5. metadata-eval 82.0%

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

      6. *-lft-identity 82.0%

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

   6. Simplified 82.0%

      \[\leadsto \color{blue}{\sqrt{re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -65000000000:\\ \;\;\;\;\frac{im}{\sqrt{re \cdot -4}}\\ \mathbf{elif}\;re \leq -3 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{elif}\;re \leq -1.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{im}{\sqrt{re \cdot -4}}\\ \mathbf{elif}\;re \leq -1.45 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{elif}\;re \leq -3.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{im}{\sqrt{re \cdot -4}}\\ \mathbf{elif}\;re \leq 95000000000000:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 3: 63.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{im_m \cdot 0.5}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{-71} \lor \neg \left(re \leq 1.2 \cdot 10^{+15}\right):\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im_m\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 1.6e-119)
   (sqrt (* im_m 0.5))
   (if (or (<= re 1.65e-71) (not (<= re 1.2e+15)))
     (sqrt re)
     (sqrt (* 0.5 (+ re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 1.6e-119) {
		tmp = sqrt((im_m * 0.5));
	} else if ((re <= 1.65e-71) || !(re <= 1.2e+15)) {
		tmp = sqrt(re);
	} else {
		tmp = sqrt((0.5 * (re + im_m)));
	}
	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 <= 1.6d-119) then
        tmp = sqrt((im_m * 0.5d0))
    else if ((re <= 1.65d-71) .or. (.not. (re <= 1.2d+15))) then
        tmp = sqrt(re)
    else
        tmp = sqrt((0.5d0 * (re + im_m)))
    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 <= 1.6e-119) {
		tmp = Math.sqrt((im_m * 0.5));
	} else if ((re <= 1.65e-71) || !(re <= 1.2e+15)) {
		tmp = Math.sqrt(re);
	} else {
		tmp = Math.sqrt((0.5 * (re + im_m)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 1.6e-119:
		tmp = math.sqrt((im_m * 0.5))
	elif (re <= 1.65e-71) or not (re <= 1.2e+15):
		tmp = math.sqrt(re)
	else:
		tmp = math.sqrt((0.5 * (re + im_m)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 1.6e-119)
		tmp = sqrt(Float64(im_m * 0.5));
	elseif ((re <= 1.65e-71) || !(re <= 1.2e+15))
		tmp = sqrt(re);
	else
		tmp = sqrt(Float64(0.5 * Float64(re + im_m)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 1.6e-119)
		tmp = sqrt((im_m * 0.5));
	elseif ((re <= 1.65e-71) || ~((re <= 1.2e+15)))
		tmp = sqrt(re);
	else
		tmp = sqrt((0.5 * (re + im_m)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 1.6e-119], N[Sqrt[N[(im$95$m * 0.5), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[re, 1.65e-71], N[Not[LessEqual[re, 1.2e+15]], $MachinePrecision]], N[Sqrt[re], $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.59999999999999997e-119

   1. Initial program 31.7%

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

      1. sqr-neg 31.7%

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

      2. +-commutative 31.7%

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

      3. sqr-neg 31.7%

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

      4. +-commutative 31.7%

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

      5. distribute-rgt-in 31.7%

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

      6. cancel-sign-sub 31.7%

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

      7. distribute-rgt-out-- 31.7%

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

      8. sub-neg 31.7%

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

      9. remove-double-neg 31.7%

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

      10. +-commutative 31.7%

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

      11. hypot-def 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in re around 0 26.4%

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

      1. *-commutative 26.4%

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

      2. associate-*l* 26.4%

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

   6. Simplified 26.4%

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

      1. expm1-log1p-u 25.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right)\right)} \]

      2. expm1-udef 22.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right)} - 1} \]

      3. add-sqr-sqrt 22.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)} \cdot \sqrt{\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)}}\right)} - 1 \]

      4. sqrt-unprod 22.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right) \cdot \left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right)}}\right)} - 1 \]

      5. swap-sqr 22.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right) \cdot \left(\left(\sqrt{2} \cdot 0.5\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\right)}}\right)} - 1 \]

      6. add-sqr-sqrt 22.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{im} \cdot \left(\left(\sqrt{2} \cdot 0.5\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\right)}\right)} - 1 \]

      7. swap-sqr 22.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(0.5 \cdot 0.5\right)\right)}}\right)} - 1 \]

      8. rem-square-sqrt 22.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \left(\color{blue}{2} \cdot \left(0.5 \cdot 0.5\right)\right)}\right)} - 1 \]

      9. metadata-eval 22.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \left(2 \cdot \color{blue}{0.25}\right)}\right)} - 1 \]

      10. metadata-eval 22.7%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \color{blue}{0.5}}\right)} - 1 \]

   8. Applied egg-rr 22.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{im \cdot 0.5}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 25.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{im \cdot 0.5}\right)\right)} \]

      2. expm1-log1p 26.6%

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

   10. Simplified 26.6%

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

   if 1.59999999999999997e-119 < re < 1.6500000000000001e-71 or 1.2e15 < re

   1. Initial program 48.5%

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

      1. sqr-neg 48.5%

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

      2. +-commutative 48.5%

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

      3. sqr-neg 48.5%

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

      4. +-commutative 48.5%

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

      5. distribute-rgt-in 48.5%

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

      6. cancel-sign-sub 48.5%

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

      7. distribute-rgt-out-- 48.5%

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

      8. sub-neg 48.5%

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

      9. remove-double-neg 48.5%

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

      10. +-commutative 48.5%

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

      11. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 80.8%

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

      1. *-commutative 80.8%

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

      2. unpow2 80.8%

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

      3. rem-square-sqrt 82.4%

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

      4. associate-*r* 82.4%

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

      5. metadata-eval 82.4%

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

      6. *-lft-identity 82.4%

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

   6. Simplified 82.4%

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

   if 1.6500000000000001e-71 < re < 1.2e15

   1. Initial program 70.5%

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

      1. sqr-neg 70.5%

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

      2. +-commutative 70.5%

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

      3. sqr-neg 70.5%

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

      4. +-commutative 70.5%

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

      5. distribute-rgt-in 70.5%

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

      6. cancel-sign-sub 70.5%

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

      7. distribute-rgt-out-- 70.5%

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

      8. sub-neg 70.5%

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

      9. remove-double-neg 70.5%

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

      10. +-commutative 70.5%

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

      11. hypot-def 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. swap-sqr 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. *-commutative 100.0%

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

      8. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-*l* 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 21.6%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{-71} \lor \neg \left(re \leq 1.2 \cdot 10^{+15}\right):\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4: 62.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{-119} \lor \neg \left(re \leq 5.3 \cdot 10^{-71}\right) \land re \leq 460:\\ \;\;\;\;\sqrt{im_m \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (or (<= re 1.6e-119) (and (not (<= re 5.3e-71)) (<= re 460.0)))
   (sqrt (* im_m 0.5))
   (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re <= 1.6e-119) || (!(re <= 5.3e-71) && (re <= 460.0))) {
		tmp = sqrt((im_m * 0.5));
	} 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 <= 1.6d-119) .or. (.not. (re <= 5.3d-71)) .and. (re <= 460.0d0)) then
        tmp = sqrt((im_m * 0.5d0))
    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 <= 1.6e-119) || (!(re <= 5.3e-71) && (re <= 460.0))) {
		tmp = Math.sqrt((im_m * 0.5));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if (re <= 1.6e-119) or (not (re <= 5.3e-71) and (re <= 460.0)):
		tmp = math.sqrt((im_m * 0.5))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if ((re <= 1.6e-119) || (!(re <= 5.3e-71) && (re <= 460.0)))
		tmp = sqrt(Float64(im_m * 0.5));
	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 <= 1.6e-119) || (~((re <= 5.3e-71)) && (re <= 460.0)))
		tmp = sqrt((im_m * 0.5));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[Or[LessEqual[re, 1.6e-119], And[N[Not[LessEqual[re, 5.3e-71]], $MachinePrecision], LessEqual[re, 460.0]]], N[Sqrt[N[(im$95$m * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.59999999999999997e-119 or 5.29999999999999999e-71 < re < 460

   1. Initial program 33.6%

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

      1. sqr-neg 33.6%

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

      2. +-commutative 33.6%

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

      3. sqr-neg 33.6%

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

      4. +-commutative 33.6%

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

      5. distribute-rgt-in 33.6%

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

      6. cancel-sign-sub 33.6%

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

      7. distribute-rgt-out-- 33.6%

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

      8. sub-neg 33.6%

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

      9. remove-double-neg 33.6%

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

      10. +-commutative 33.6%

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

      11. hypot-def 64.8%

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

   3. Simplified 64.8%

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

   4. Taylor expanded in re around 0 25.5%

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

      1. *-commutative 25.5%

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

      2. associate-*l* 25.5%

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

   6. Simplified 25.5%

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

      1. expm1-log1p-u 24.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right)\right)} \]

      2. expm1-udef 21.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right)} - 1} \]

      3. add-sqr-sqrt 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)} \cdot \sqrt{\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)}}\right)} - 1 \]

      4. sqrt-unprod 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right) \cdot \left(\sqrt{im} \cdot \left(\sqrt{2} \cdot 0.5\right)\right)}}\right)} - 1 \]

      5. swap-sqr 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right) \cdot \left(\left(\sqrt{2} \cdot 0.5\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\right)}}\right)} - 1 \]

      6. add-sqr-sqrt 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{im} \cdot \left(\left(\sqrt{2} \cdot 0.5\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\right)}\right)} - 1 \]

      7. swap-sqr 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(0.5 \cdot 0.5\right)\right)}}\right)} - 1 \]

      8. rem-square-sqrt 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \left(\color{blue}{2} \cdot \left(0.5 \cdot 0.5\right)\right)}\right)} - 1 \]

      9. metadata-eval 21.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \left(2 \cdot \color{blue}{0.25}\right)}\right)} - 1 \]

      10. metadata-eval 21.9%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{im \cdot \color{blue}{0.5}}\right)} - 1 \]

   8. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{im \cdot 0.5}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 24.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{im \cdot 0.5}\right)\right)} \]

      2. expm1-log1p 25.6%

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

   10. Simplified 25.6%

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

   if 1.59999999999999997e-119 < re < 5.29999999999999999e-71 or 460 < re

   1. Initial program 50.0%

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

      1. sqr-neg 50.0%

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

      2. +-commutative 50.0%

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

      3. sqr-neg 50.0%

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

      4. +-commutative 50.0%

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

      5. distribute-rgt-in 50.0%

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

      6. cancel-sign-sub 50.0%

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

      7. distribute-rgt-out-- 50.0%

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

      8. sub-neg 50.0%

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

      9. remove-double-neg 50.0%

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

      10. +-commutative 50.0%

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

      11. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 80.0%

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

      1. *-commutative 80.0%

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

      2. unpow2 80.0%

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

      3. rem-square-sqrt 81.6%

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

      4. associate-*r* 81.6%

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

      5. metadata-eval 81.6%

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

      6. *-lft-identity 81.6%

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

   6. Simplified 81.6%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{-119} \lor \neg \left(re \leq 5.3 \cdot 10^{-71}\right) \land re \leq 460:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 5: 26.0% 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 37.9%

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

   1. sqr-neg 37.9%

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

   2. +-commutative 37.9%

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

   3. sqr-neg 37.9%

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

   4. +-commutative 37.9%

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

   5. distribute-rgt-in 37.9%

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

   6. cancel-sign-sub 37.9%

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

   7. distribute-rgt-out-- 37.9%

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

   8. sub-neg 37.9%

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

   9. remove-double-neg 37.9%

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

   10. +-commutative 37.9%

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

   11. hypot-def 74.0%

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

3. Simplified 74.0%

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

4. Taylor expanded in im around 0 23.9%

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

   1. *-commutative 23.9%

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

   2. unpow2 23.9%

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

   3. rem-square-sqrt 24.3%

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

   4. associate-*r* 24.3%

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

   5. metadata-eval 24.3%

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

   6. *-lft-identity 24.3%

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

6. Simplified 24.3%

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

7. Final simplification 24.3%

   \[\leadsto \sqrt{re} \]

Developer target: 48.2% 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 2023319 
(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)))))