math.sqrt on complex, real part

Percentage Accurate: 41.5% → 84.0%

Time: 9.4s

Alternatives: 5

Speedup: 1.0×

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: 41.5% 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: 84.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(N[Power[im, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.0%

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

      1. sqr-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. sqr-neg 7.0%

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

      4. +-commutative 7.0%

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

      5. distribute-rgt-in 7.0%

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

      6. cancel-sign-sub 7.0%

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

      7. distribute-rgt-out-- 7.0%

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

      8. sub-neg 7.0%

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

      9. remove-double-neg 7.0%

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

      10. +-commutative 7.0%

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

      11. hypot-define 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in re around -inf 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-*l/ 56.5%

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

   7. Simplified 56.5%

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

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

   1. Initial program 45.4%

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

      1. sqr-neg 45.4%

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

      2. +-commutative 45.4%

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

      3. sqr-neg 45.4%

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

      4. +-commutative 45.4%

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

      5. distribute-rgt-in 45.4%

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

      6. cancel-sign-sub 45.4%

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

      7. distribute-rgt-out-- 45.4%

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

      8. sub-neg 45.4%

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

      9. remove-double-neg 45.4%

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

      10. +-commutative 45.4%

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

      11. hypot-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 86.4%

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

      2. sqrt-unprod 87.1%

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

      3. *-commutative 87.1%

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

      4. *-commutative 87.1%

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

      5. swap-sqr 87.1%

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

      6. add-sqr-sqrt 87.1%

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

      7. metadata-eval 87.1%

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

   6. Applied egg-rr 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*r* 87.1%

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

      3. metadata-eval 87.1%

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

      4. hypot-undefine 45.4%

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

      5. unpow2 45.4%

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

      6. unpow2 45.4%

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

      7. +-commutative 45.4%

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

      8. unpow2 45.4%

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

      9. unpow2 45.4%

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

      10. hypot-undefine 87.1%

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

   8. Simplified 87.1%

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

4. Final simplification 83.4%

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

Alternative 2: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[N[(0.5 * N[(re + N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 40.8%

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

   1. sqr-neg 40.8%

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

   2. +-commutative 40.8%

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

   3. sqr-neg 40.8%

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

   4. +-commutative 40.8%

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

   5. distribute-rgt-in 40.8%

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

   6. cancel-sign-sub 40.8%

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

   7. distribute-rgt-out-- 40.8%

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

   8. sub-neg 40.8%

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

   9. remove-double-neg 40.8%

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

   10. +-commutative 40.8%

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

   11. hypot-define 77.4%

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

3. Simplified 77.4%

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

   1. add-sqr-sqrt 76.8%

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

   2. sqrt-unprod 77.4%

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

   3. *-commutative 77.4%

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

   4. *-commutative 77.4%

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

   5. swap-sqr 77.4%

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

   6. add-sqr-sqrt 77.4%

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

   7. metadata-eval 77.4%

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

6. Applied egg-rr 77.4%

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

   1. *-commutative 77.4%

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

   2. associate-*r* 77.4%

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

   3. metadata-eval 77.4%

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

   4. hypot-undefine 40.8%

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

   5. unpow2 40.8%

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

   6. unpow2 40.8%

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

   7. +-commutative 40.8%

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

   8. unpow2 40.8%

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

   9. unpow2 40.8%

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

   10. hypot-undefine 77.4%

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

8. Simplified 77.4%

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

9. Final simplification 77.4%

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

Alternative 3: 40.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 13000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re \cdot \left(1 + 0.5 \cdot \frac{re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 13000000.0)
   (* 0.5 (sqrt (* 2.0 (+ im (* re (+ 1.0 (* 0.5 (/ re im))))))))
   (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 13000000.0) {
		tmp = 0.5 * sqrt((2.0 * (im + (re * (1.0 + (0.5 * (re / im)))))));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 13000000.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im + (re * (1.0d0 + (0.5d0 * (re / im)))))))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 13000000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (im + (re * (1.0 + (0.5 * (re / im)))))));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 13000000.0:
		tmp = 0.5 * math.sqrt((2.0 * (im + (re * (1.0 + (0.5 * (re / im)))))))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 13000000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im + Float64(re * Float64(1.0 + Float64(0.5 * Float64(re / im))))))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 13000000.0)
		tmp = 0.5 * sqrt((2.0 * (im + (re * (1.0 + (0.5 * (re / im)))))));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 13000000.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(im + N[(re * N[(1.0 + N[(0.5 * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.3e7

   1. Initial program 42.0%

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

      1. sqr-neg 42.0%

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

      2. +-commutative 42.0%

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

      3. sqr-neg 42.0%

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

      4. +-commutative 42.0%

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

      5. distribute-rgt-in 42.0%

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

      6. cancel-sign-sub 42.0%

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

      7. distribute-rgt-out-- 42.0%

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

      8. sub-neg 42.0%

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

      9. remove-double-neg 42.0%

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

      10. +-commutative 42.0%

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

      11. hypot-define 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in re around 0 28.1%

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

   if 1.3e7 < re

   1. Initial program 37.2%

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

      1. add-sqr-sqrt 37.1%

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

      2. pow2 37.1%

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

      3. hypot-define 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. pow1/2 99.9%

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

      2. unpow2 99.9%

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

      3. add-sqr-sqrt 100.0%

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

      4. +-commutative 100.0%

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

      5. exp-to-pow 92.2%

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

      6. *-commutative 92.2%

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

      7. exp-prod 90.8%

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

   6. Applied egg-rr 90.8%

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

   7. Taylor expanded in im around 0 81.3%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 13000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re \cdot \left(1 + 0.5 \cdot \frac{re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 40.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 18500000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 18500000.0) (* 0.5 (sqrt (* 2.0 im))) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 18500000.0) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 18500000.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 18500000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 18500000.0:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 18500000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 18500000.0)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 18500000.0], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.85e7

   1. Initial program 42.0%

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

      1. sqr-neg 42.0%

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

      2. +-commutative 42.0%

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

      3. sqr-neg 42.0%

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

      4. +-commutative 42.0%

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

      5. distribute-rgt-in 42.0%

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

      6. cancel-sign-sub 42.0%

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

      7. distribute-rgt-out-- 42.0%

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

      8. sub-neg 42.0%

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

      9. remove-double-neg 42.0%

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

      10. +-commutative 42.0%

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

      11. hypot-define 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in re around 0 27.6%

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

   if 1.85e7 < re

   1. Initial program 37.2%

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

      1. add-sqr-sqrt 37.1%

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

      2. pow2 37.1%

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

      3. hypot-define 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. pow1/2 99.9%

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

      2. unpow2 99.9%

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

      3. add-sqr-sqrt 100.0%

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

      4. +-commutative 100.0%

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

      5. exp-to-pow 92.2%

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

      6. *-commutative 92.2%

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

      7. exp-prod 90.8%

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

   6. Applied egg-rr 90.8%

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

   7. Taylor expanded in im around 0 81.3%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 18500000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 26.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt re))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return sqrt(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 40.8%

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

   1. add-sqr-sqrt 40.6%

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

   2. pow2 40.6%

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

   3. hypot-define 76.2%

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

4. Applied egg-rr 76.2%

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

   1. pow1/2 76.2%

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

   2. unpow2 76.2%

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

   3. add-sqr-sqrt 77.4%

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

   4. +-commutative 77.4%

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

   5. exp-to-pow 72.0%

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

   6. *-commutative 72.0%

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

   7. exp-prod 71.4%

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

6. Applied egg-rr 71.4%

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

7. Taylor expanded in im around 0 25.9%

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

8. Final simplification 25.9%

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

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 2024073 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))