math.sqrt on complex, real part

Percentage Accurate: 41.6% → 90.3%

Time: 6.0s

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: 41.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im_m \cdot im_m}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \left(im_m \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im_m im_m)))))) 0.0)
   (* 0.5 (* im_m (sqrt (/ -1.0 re))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (sqrt((2.0 * (re + sqrt(((re * re) + (im_m * im_m)))))) <= 0.0) {
		tmp = 0.5 * (im_m * sqrt((-1.0 / re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (Math.sqrt((2.0 * (re + Math.sqrt(((re * re) + (im_m * im_m)))))) <= 0.0) {
		tmp = 0.5 * (im_m * Math.sqrt((-1.0 / re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if math.sqrt((2.0 * (re + math.sqrt(((re * re) + (im_m * im_m)))))) <= 0.0:
		tmp = 0.5 * (im_m * math.sqrt((-1.0 / re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im_m))))
	return tmp

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[(im$95$m * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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 7.4%

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

      1. sqr-neg 7.4%

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

      2. +-commutative 7.4%

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

      3. sqr-neg 7.4%

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

      4. +-commutative 7.4%

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

      5. distribute-rgt-in 7.4%

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

      6. cancel-sign-sub 7.4%

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

      7. distribute-rgt-out-- 7.4%

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

      8. sub-neg 7.4%

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

      9. remove-double-neg 7.4%

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

      10. +-commutative 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in re around -inf 62.5%

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

      1. associate-*r/ 62.4%

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

      2. associate-/l* 62.5%

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

   7. Simplified 62.5%

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

      1. *-un-lft-identity 62.5%

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

      2. unpow2 62.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{-0.5}{\frac{1 \cdot re}{\color{blue}{im \cdot im}}}} \]

      3. times-frac 71.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{-0.5}{\color{blue}{\frac{1}{im} \cdot \frac{re}{im}}}} \]

   9. Applied egg-rr 71.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{-0.5}{\color{blue}{\frac{1}{im} \cdot \frac{re}{im}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 71.2%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2 \cdot -0.5}{\frac{1}{im} \cdot \frac{re}{im}}}} \]

       2. metadata-eval 71.2%

          \[\leadsto 0.5 \cdot \sqrt{\frac{\color{blue}{-1}}{\frac{1}{im} \cdot \frac{re}{im}}} \]

       3. frac-times 62.5%

          \[\leadsto 0.5 \cdot \sqrt{\frac{-1}{\color{blue}{\frac{1 \cdot re}{im \cdot im}}}} \]

       4. *-un-lft-identity 62.5%

          \[\leadsto 0.5 \cdot \sqrt{\frac{-1}{\frac{\color{blue}{re}}{im \cdot im}}} \]

       5. unpow2 62.5%

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

       6. associate-/r/ 62.4%

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

       7. metadata-eval 62.4%

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

       8. associate-*r/ 62.4%

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

       9. add-sqr-sqrt 62.3%

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

       10. sqrt-unprod 62.2%

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

       11. sqrt-unprod 62.2%

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

       12. unpow2 62.2%

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

       13. swap-sqr 71.0%

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

       14. *-commutative 71.0%

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

       15. *-commutative 71.0%

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

       16. sqrt-unprod 55.1%

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

       17. add-sqr-sqrt 56.8%

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

       18. *-commutative 56.8%

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

   11. Applied egg-rr 56.9%

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

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

   1. Initial program 47.4%

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

      1. sqr-neg 47.4%

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

      2. +-commutative 47.4%

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

      3. sqr-neg 47.4%

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

      4. +-commutative 47.4%

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

      5. distribute-rgt-in 47.4%

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

      6. cancel-sign-sub 47.4%

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

      7. distribute-rgt-out-- 47.4%

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

      8. sub-neg 47.4%

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

      9. remove-double-neg 47.4%

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

      10. +-commutative 47.4%

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

   3. Simplified 89.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \left(im_m \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.2e+42)
   (* 0.5 (* im_m (sqrt (/ -1.0 re))))
   (if (<= re 1.9e+96)
     (* 0.5 (sqrt (* 2.0 (+ re im_m))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2.2e+42) {
		tmp = 0.5 * (im_m * sqrt((-1.0 / re)));
	} else if (re <= 1.9e+96) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} else {
		tmp = 0.5 * (2.0 * 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 <= (-2.2d+42)) then
        tmp = 0.5d0 * (im_m * sqrt(((-1.0d0) / re)))
    else if (re <= 1.9d+96) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    else
        tmp = 0.5d0 * (2.0d0 * 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 <= -2.2e+42) {
		tmp = 0.5 * (im_m * Math.sqrt((-1.0 / re)));
	} else if (re <= 1.9e+96) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -2.2e+42:
		tmp = 0.5 * (im_m * math.sqrt((-1.0 / re)))
	elif re <= 1.9e+96:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2.2e+42)
		tmp = Float64(0.5 * Float64(im_m * sqrt(Float64(-1.0 / re))));
	elseif (re <= 1.9e+96)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -2.2e+42)
		tmp = 0.5 * (im_m * sqrt((-1.0 / re)));
	elseif (re <= 1.9e+96)
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2.2e+42], N[(0.5 * N[(im$95$m * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+96], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.2000000000000001e42

   1. Initial program 9.4%

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

      1. sqr-neg 9.4%

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

      2. +-commutative 9.4%

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

      3. sqr-neg 9.4%

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

      4. +-commutative 9.4%

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

      5. distribute-rgt-in 9.4%

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

      6. cancel-sign-sub 9.4%

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

      7. distribute-rgt-out-- 9.4%

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

      8. sub-neg 9.4%

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

      9. remove-double-neg 9.4%

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

      10. +-commutative 9.4%

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

   3. Simplified 39.0%

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

   5. Taylor expanded in re around -inf 46.4%

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

      1. associate-*r/ 46.4%

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

      2. associate-/l* 45.2%

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

   7. Simplified 45.2%

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

      1. *-un-lft-identity 45.2%

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

      2. unpow2 45.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{-0.5}{\frac{1 \cdot re}{\color{blue}{im \cdot im}}}} \]

      3. times-frac 51.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{-0.5}{\color{blue}{\frac{1}{im} \cdot \frac{re}{im}}}} \]

   9. Applied egg-rr 51.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{-0.5}{\color{blue}{\frac{1}{im} \cdot \frac{re}{im}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 51.9%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2 \cdot -0.5}{\frac{1}{im} \cdot \frac{re}{im}}}} \]

       2. metadata-eval 51.9%

          \[\leadsto 0.5 \cdot \sqrt{\frac{\color{blue}{-1}}{\frac{1}{im} \cdot \frac{re}{im}}} \]

       3. frac-times 45.2%

          \[\leadsto 0.5 \cdot \sqrt{\frac{-1}{\color{blue}{\frac{1 \cdot re}{im \cdot im}}}} \]

       4. *-un-lft-identity 45.2%

          \[\leadsto 0.5 \cdot \sqrt{\frac{-1}{\frac{\color{blue}{re}}{im \cdot im}}} \]

       5. unpow2 45.2%

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

       6. associate-/r/ 46.3%

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

       7. metadata-eval 46.3%

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

       8. associate-*r/ 46.3%

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

       9. add-sqr-sqrt 46.3%

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

       10. sqrt-unprod 46.2%

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

       11. sqrt-unprod 46.2%

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

       12. unpow2 46.2%

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

       13. swap-sqr 53.1%

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

       14. *-commutative 53.1%

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

       15. *-commutative 53.1%

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

       16. sqrt-unprod 41.5%

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

       17. add-sqr-sqrt 44.0%

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

       18. *-commutative 44.0%

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

   11. Applied egg-rr 44.0%

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

   if -2.2000000000000001e42 < re < 1.9000000000000001e96

   1. Initial program 59.4%

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

      1. sqr-neg 59.4%

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

      2. +-commutative 59.4%

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

      3. sqr-neg 59.4%

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

      4. +-commutative 59.4%

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

      5. distribute-rgt-in 59.4%

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

      6. cancel-sign-sub 59.4%

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

      7. distribute-rgt-out-- 59.4%

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

      8. sub-neg 59.4%

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

      9. remove-double-neg 59.4%

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

      10. +-commutative 59.4%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in re around 0 44.8%

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

   if 1.9000000000000001e96 < re

   1. Initial program 24.0%

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

      1. sqr-neg 24.0%

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

      2. +-commutative 24.0%

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

      3. sqr-neg 24.0%

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

      4. +-commutative 24.0%

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

      5. distribute-rgt-in 24.0%

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

      6. cancel-sign-sub 24.0%

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

      7. distribute-rgt-out-- 24.0%

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

      8. sub-neg 24.0%

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

      9. remove-double-neg 24.0%

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

      10. +-commutative 24.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. rem-square-sqrt 79.0%

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

   7. Simplified 79.0%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 1.56 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 1.56e+97)
   (* 0.5 (sqrt (* 2.0 (+ re im_m))))
   (* 0.5 (* 2.0 (sqrt re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 1.56e+97) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} else {
		tmp = 0.5 * (2.0 * 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 <= 1.56d+97) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    else
        tmp = 0.5d0 * (2.0d0 * 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 <= 1.56e+97) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 1.56e+97:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 1.56e+97)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 1.56e+97)
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 1.56e+97], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.56e97

   1. Initial program 44.5%

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

      1. sqr-neg 44.5%

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

      2. +-commutative 44.5%

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

      3. sqr-neg 44.5%

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

      4. +-commutative 44.5%

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

      5. distribute-rgt-in 44.5%

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

      6. cancel-sign-sub 44.5%

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

      7. distribute-rgt-out-- 44.5%

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

      8. sub-neg 44.5%

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

      9. remove-double-neg 44.5%

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

      10. +-commutative 44.5%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in re around 0 35.1%

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

   if 1.56e97 < re

   1. Initial program 24.0%

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

      1. sqr-neg 24.0%

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

      2. +-commutative 24.0%

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

      3. sqr-neg 24.0%

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

      4. +-commutative 24.0%

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

      5. distribute-rgt-in 24.0%

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

      6. cancel-sign-sub 24.0%

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

      7. distribute-rgt-out-- 24.0%

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

      8. sub-neg 24.0%

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

      9. remove-double-neg 24.0%

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

      10. +-commutative 24.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. rem-square-sqrt 79.0%

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

   7. Simplified 79.0%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.56 \cdot 10^{+97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 1.8 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 1.8e+96) (* 0.5 (sqrt (* 2.0 im_m))) (* 0.5 (* 2.0 (sqrt re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 1.8e+96) {
		tmp = 0.5 * sqrt((2.0 * im_m));
	} else {
		tmp = 0.5 * (2.0 * 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 <= 1.8d+96) then
        tmp = 0.5d0 * sqrt((2.0d0 * im_m))
    else
        tmp = 0.5d0 * (2.0d0 * 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 <= 1.8e+96) {
		tmp = 0.5 * Math.sqrt((2.0 * im_m));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 1.8e+96:
		tmp = 0.5 * math.sqrt((2.0 * im_m))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 1.8e+96)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im_m)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 1.8e+96)
		tmp = 0.5 * sqrt((2.0 * im_m));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 1.8e+96], N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.80000000000000007e96

   1. Initial program 44.5%

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

      1. sqr-neg 44.5%

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

      2. +-commutative 44.5%

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

      3. sqr-neg 44.5%

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

      4. +-commutative 44.5%

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

      5. distribute-rgt-in 44.5%

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

      6. cancel-sign-sub 44.5%

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

      7. distribute-rgt-out-- 44.5%

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

      8. sub-neg 44.5%

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

      9. remove-double-neg 44.5%

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

      10. +-commutative 44.5%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in re around 0 34.2%

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

   if 1.80000000000000007e96 < re

   1. Initial program 24.0%

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

      1. sqr-neg 24.0%

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

      2. +-commutative 24.0%

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

      3. sqr-neg 24.0%

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

      4. +-commutative 24.0%

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

      5. distribute-rgt-in 24.0%

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

      6. cancel-sign-sub 24.0%

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

      7. distribute-rgt-out-- 24.0%

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

      8. sub-neg 24.0%

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

      9. remove-double-neg 24.0%

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

      10. +-commutative 24.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. rem-square-sqrt 79.0%

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

   7. Simplified 79.0%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.8 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.5 \cdot \sqrt{2 \cdot im_m} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* 0.5 (sqrt (* 2.0 im_m))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.5 * sqrt((2.0 * im_m));
}

Fortran

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

Java

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

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.5 * math.sqrt((2.0 * im_m))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(0.5 * sqrt(Float64(2.0 * im_m)))
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.8%

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

   1. sqr-neg 41.8%

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

   2. +-commutative 41.8%

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

   3. sqr-neg 41.8%

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

   4. +-commutative 41.8%

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

   5. distribute-rgt-in 41.8%

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

   6. cancel-sign-sub 41.8%

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

   7. distribute-rgt-out-- 41.8%

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

   8. sub-neg 41.8%

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

   9. remove-double-neg 41.8%

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

   10. +-commutative 41.8%

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

3. Simplified 78.3%

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

5. Taylor expanded in re around 0 30.7%

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

6. Final simplification 30.7%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
7. Add Preprocessing

Developer target: 48.5% 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 2024019 
(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)))))