math.sqrt on complex, real part

Percentage Accurate: 41.0% → 89.7%

Time: 9.3s

Alternatives: 7

Speedup: 1.8×

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: 41.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.7% accurate, 0.5× speedup

Math

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

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (/ (* im 0.5) (sqrt (- re)))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

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

Java

im = Math.abs(im);
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 = (im * 0.5) / Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

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

Julia

im = abs(im)
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(Float64(im * 0.5) / sqrt(Float64(-re)));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
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[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $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 8.0%

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

      1. +-commutative 8.0%

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

      2. hypot-def 8.0%

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

   3. Simplified 8.0%

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

   4. Taylor expanded in re around -inf 65.5%

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

      1. associate-*r/ 65.5%

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

      2. neg-mul-1 65.5%

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

      3. unpow2 65.5%

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

      4. distribute-rgt-neg-in 65.5%

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

   6. Simplified 65.5%

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

      1. frac-2neg 65.5%

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

      2. sqrt-div 66.4%

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

      3. distribute-rgt-neg-out 66.4%

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

      4. remove-double-neg 66.4%

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

      5. sqrt-unprod 55.5%

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

      6. add-sqr-sqrt 60.4%

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

   8. Applied egg-rr 60.4%

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

      1. associate-*r/ 60.4%

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

   10. Applied egg-rr 60.4%

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

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

   1. Initial program 50.2%

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

      1. +-commutative 50.2%

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

      2. hypot-def 88.6%

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

   3. Simplified 88.6%

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

      1. add-sqr-sqrt 88.0%

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

      2. sqrt-unprod 88.6%

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

      3. *-commutative 88.6%

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

      4. *-commutative 88.6%

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

      5. swap-sqr 88.6%

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

      6. add-sqr-sqrt 88.6%

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

      7. *-commutative 88.6%

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

      8. metadata-eval 88.6%

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

   5. Applied egg-rr 88.6%

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

      1. associate-*l* 88.6%

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

      2. metadata-eval 88.6%

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

   7. Simplified 88.6%

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

4. Final simplification 85.9%

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

Alternative 2: 72.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} t_0 := 2 \cdot \left(re + im\right)\\ t_1 := \sqrt{-re}\\ \mathbf{if}\;re \leq -5.8 \cdot 10^{+142}:\\ \;\;\;\;0.5 \cdot \frac{im}{t_1}\\ \mathbf{elif}\;re \leq -2.9 \cdot 10^{+124}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot \frac{re}{im} + t_0}\\ \mathbf{elif}\;re \leq -2.35 \cdot 10^{+26}:\\ \;\;\;\;\frac{im \cdot 0.5}{t_1}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-76} \lor \neg \left(re \leq 1.76 \cdot 10^{-6}\right) \land re \leq 9 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 2.0 (+ re im))) (t_1 (sqrt (- re))))
   (if (<= re -5.8e+142)
     (* 0.5 (/ im t_1))
     (if (<= re -2.9e+124)
       (* 0.5 (sqrt (+ (* re (/ re im)) t_0)))
       (if (<= re -2.35e+26)
         (/ (* im 0.5) t_1)
         (if (<= re 6.6e-187)
           (* 0.5 (sqrt (* 2.0 im)))
           (if (<= re 1.1e-125)
             (sqrt re)
             (if (or (<= re 1.4e-76) (and (not (<= re 1.76e-6)) (<= re 9e+47)))
               (* 0.5 (sqrt t_0))
               (sqrt re)))))))))

C

im = abs(im);
double code(double re, double im) {
	double t_0 = 2.0 * (re + im);
	double t_1 = sqrt(-re);
	double tmp;
	if (re <= -5.8e+142) {
		tmp = 0.5 * (im / t_1);
	} else if (re <= -2.9e+124) {
		tmp = 0.5 * sqrt(((re * (re / im)) + t_0));
	} else if (re <= -2.35e+26) {
		tmp = (im * 0.5) / t_1;
	} else if (re <= 6.6e-187) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else if (re <= 1.1e-125) {
		tmp = sqrt(re);
	} else if ((re <= 1.4e-76) || (!(re <= 1.76e-6) && (re <= 9e+47))) {
		tmp = 0.5 * sqrt(t_0);
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 * (re + im)
    t_1 = sqrt(-re)
    if (re <= (-5.8d+142)) then
        tmp = 0.5d0 * (im / t_1)
    else if (re <= (-2.9d+124)) then
        tmp = 0.5d0 * sqrt(((re * (re / im)) + t_0))
    else if (re <= (-2.35d+26)) then
        tmp = (im * 0.5d0) / t_1
    else if (re <= 6.6d-187) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else if (re <= 1.1d-125) then
        tmp = sqrt(re)
    else if ((re <= 1.4d-76) .or. (.not. (re <= 1.76d-6)) .and. (re <= 9d+47)) then
        tmp = 0.5d0 * sqrt(t_0)
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double t_0 = 2.0 * (re + im);
	double t_1 = Math.sqrt(-re);
	double tmp;
	if (re <= -5.8e+142) {
		tmp = 0.5 * (im / t_1);
	} else if (re <= -2.9e+124) {
		tmp = 0.5 * Math.sqrt(((re * (re / im)) + t_0));
	} else if (re <= -2.35e+26) {
		tmp = (im * 0.5) / t_1;
	} else if (re <= 6.6e-187) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else if (re <= 1.1e-125) {
		tmp = Math.sqrt(re);
	} else if ((re <= 1.4e-76) || (!(re <= 1.76e-6) && (re <= 9e+47))) {
		tmp = 0.5 * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	t_0 = 2.0 * (re + im)
	t_1 = math.sqrt(-re)
	tmp = 0
	if re <= -5.8e+142:
		tmp = 0.5 * (im / t_1)
	elif re <= -2.9e+124:
		tmp = 0.5 * math.sqrt(((re * (re / im)) + t_0))
	elif re <= -2.35e+26:
		tmp = (im * 0.5) / t_1
	elif re <= 6.6e-187:
		tmp = 0.5 * math.sqrt((2.0 * im))
	elif re <= 1.1e-125:
		tmp = math.sqrt(re)
	elif (re <= 1.4e-76) or (not (re <= 1.76e-6) and (re <= 9e+47)):
		tmp = 0.5 * math.sqrt(t_0)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im = abs(im)
function code(re, im)
	t_0 = Float64(2.0 * Float64(re + im))
	t_1 = sqrt(Float64(-re))
	tmp = 0.0
	if (re <= -5.8e+142)
		tmp = Float64(0.5 * Float64(im / t_1));
	elseif (re <= -2.9e+124)
		tmp = Float64(0.5 * sqrt(Float64(Float64(re * Float64(re / im)) + t_0)));
	elseif (re <= -2.35e+26)
		tmp = Float64(Float64(im * 0.5) / t_1);
	elseif (re <= 6.6e-187)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	elseif (re <= 1.1e-125)
		tmp = sqrt(re);
	elseif ((re <= 1.4e-76) || (!(re <= 1.76e-6) && (re <= 9e+47)))
		tmp = Float64(0.5 * sqrt(t_0));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	t_0 = 2.0 * (re + im);
	t_1 = sqrt(-re);
	tmp = 0.0;
	if (re <= -5.8e+142)
		tmp = 0.5 * (im / t_1);
	elseif (re <= -2.9e+124)
		tmp = 0.5 * sqrt(((re * (re / im)) + t_0));
	elseif (re <= -2.35e+26)
		tmp = (im * 0.5) / t_1;
	elseif (re <= 6.6e-187)
		tmp = 0.5 * sqrt((2.0 * im));
	elseif (re <= 1.1e-125)
		tmp = sqrt(re);
	elseif ((re <= 1.4e-76) || (~((re <= 1.76e-6)) && (re <= 9e+47)))
		tmp = 0.5 * sqrt(t_0);
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := Block[{t$95$0 = N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-re)], $MachinePrecision]}, If[LessEqual[re, -5.8e+142], N[(0.5 * N[(im / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -2.9e+124], N[(0.5 * N[Sqrt[N[(N[(re * N[(re / im), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -2.35e+26], N[(N[(im * 0.5), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[re, 6.6e-187], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.1e-125], N[Sqrt[re], $MachinePrecision], If[Or[LessEqual[re, 1.4e-76], And[N[Not[LessEqual[re, 1.76e-6]], $MachinePrecision], LessEqual[re, 9e+47]]], N[(0.5 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if re < -5.80000000000000027e142

   1. Initial program 2.8%

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

      1. +-commutative 2.8%

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

      2. hypot-def 35.7%

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

   3. Simplified 35.7%

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

   4. Taylor expanded in re around -inf 59.9%

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

      1. associate-*r/ 59.9%

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

      2. neg-mul-1 59.9%

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

      3. unpow2 59.9%

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

      4. distribute-rgt-neg-in 59.9%

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

   6. Simplified 59.9%

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

      1. frac-2neg 59.9%

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

      2. sqrt-div 69.6%

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

      3. distribute-rgt-neg-out 69.6%

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

      4. remove-double-neg 69.6%

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

      5. sqrt-unprod 46.4%

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

      6. add-sqr-sqrt 60.5%

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

   8. Applied egg-rr 60.5%

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

   if -5.80000000000000027e142 < re < -2.90000000000000021e124

   1. Initial program 35.5%

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

      1. +-commutative 35.5%

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

      2. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in re around 0 80.4%

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

      1. unpow2 80.4%

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

      2. distribute-lft-out 80.4%

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

   6. Simplified 80.4%

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

      1. associate-/l* 80.4%

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

      2. associate-/r/ 80.4%

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

   8. Applied egg-rr 80.4%

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

   if -2.90000000000000021e124 < re < -2.3499999999999999e26

   1. Initial program 7.6%

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

      1. +-commutative 7.6%

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

      2. hypot-def 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in re around -inf 62.3%

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

      1. associate-*r/ 62.3%

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

      2. neg-mul-1 62.3%

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

      3. unpow2 62.3%

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

      4. distribute-rgt-neg-in 62.3%

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

   6. Simplified 62.3%

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

      1. frac-2neg 62.3%

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

      2. sqrt-div 63.9%

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

      3. distribute-rgt-neg-out 63.9%

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

      4. remove-double-neg 63.9%

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

      5. sqrt-unprod 50.0%

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

      6. add-sqr-sqrt 54.4%

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

   8. Applied egg-rr 54.4%

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

      1. associate-*r/ 54.4%

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

   10. Applied egg-rr 54.4%

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

   if -2.3499999999999999e26 < re < 6.6e-187

   1. Initial program 52.3%

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

      1. +-commutative 52.3%

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

      2. hypot-def 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in re around 0 46.4%

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

      1. *-commutative 46.4%

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

   6. Simplified 46.4%

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

   if 6.6e-187 < re < 1.09999999999999997e-125 or 1.40000000000000005e-76 < re < 1.7600000000000001e-6 or 8.99999999999999958e47 < re

   1. Initial program 61.4%

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

      1. +-commutative 61.4%

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

      2. 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 77.4%

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

      1. associate-*r* 77.4%

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

      2. unpow2 77.4%

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

      3. rem-square-sqrt 78.9%

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

      4. metadata-eval 78.9%

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

      5. *-lft-identity 78.9%

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

   6. Simplified 78.9%

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

   if 1.09999999999999997e-125 < re < 1.40000000000000005e-76 or 1.7600000000000001e-6 < re < 8.99999999999999958e47

   1. Initial program 61.7%

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

      1. +-commutative 61.7%

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

      2. 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 re around 0 39.2%

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

      1. distribute-lft-out 39.2%

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

      2. +-commutative 39.2%

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

      3. *-commutative 39.2%

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

      4. +-commutative 39.2%

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

   6. Simplified 39.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right) \cdot 2}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{+142}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq -2.9 \cdot 10^{+124}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot \frac{re}{im} + 2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq -2.35 \cdot 10^{+26}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-76} \lor \neg \left(re \leq 1.76 \cdot 10^{-6}\right) \land re \leq 9 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 3: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-77} \lor \neg \left(re \leq 4.2 \cdot 10^{-5}\right) \land re \leq 5.2 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -4.8e+25)
   (/ (* im 0.5) (sqrt (- re)))
   (if (<= re 6.6e-187)
     (* 0.5 (sqrt (* 2.0 im)))
     (if (<= re 1.1e-125)
       (sqrt re)
       (if (or (<= re 1.6e-77) (and (not (<= re 4.2e-5)) (<= re 5.2e+47)))
         (* 0.5 (sqrt (* 2.0 (+ re im))))
         (sqrt re))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -4.8e+25) {
		tmp = (im * 0.5) / sqrt(-re);
	} else if (re <= 6.6e-187) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else if (re <= 1.1e-125) {
		tmp = sqrt(re);
	} else if ((re <= 1.6e-77) || (!(re <= 4.2e-5) && (re <= 5.2e+47))) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.8d+25)) then
        tmp = (im * 0.5d0) / sqrt(-re)
    else if (re <= 6.6d-187) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else if (re <= 1.1d-125) then
        tmp = sqrt(re)
    else if ((re <= 1.6d-77) .or. (.not. (re <= 4.2d-5)) .and. (re <= 5.2d+47)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -4.8e+25) {
		tmp = (im * 0.5) / Math.sqrt(-re);
	} else if (re <= 6.6e-187) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else if (re <= 1.1e-125) {
		tmp = Math.sqrt(re);
	} else if ((re <= 1.6e-77) || (!(re <= 4.2e-5) && (re <= 5.2e+47))) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -4.8e+25:
		tmp = (im * 0.5) / math.sqrt(-re)
	elif re <= 6.6e-187:
		tmp = 0.5 * math.sqrt((2.0 * im))
	elif re <= 1.1e-125:
		tmp = math.sqrt(re)
	elif (re <= 1.6e-77) or (not (re <= 4.2e-5) and (re <= 5.2e+47)):
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -4.8e+25)
		tmp = Float64(Float64(im * 0.5) / sqrt(Float64(-re)));
	elseif (re <= 6.6e-187)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	elseif (re <= 1.1e-125)
		tmp = sqrt(re);
	elseif ((re <= 1.6e-77) || (!(re <= 4.2e-5) && (re <= 5.2e+47)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.8e+25)
		tmp = (im * 0.5) / sqrt(-re);
	elseif (re <= 6.6e-187)
		tmp = 0.5 * sqrt((2.0 * im));
	elseif (re <= 1.1e-125)
		tmp = sqrt(re);
	elseif ((re <= 1.6e-77) || (~((re <= 4.2e-5)) && (re <= 5.2e+47)))
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -4.8e+25], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.6e-187], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.1e-125], N[Sqrt[re], $MachinePrecision], If[Or[LessEqual[re, 1.6e-77], And[N[Not[LessEqual[re, 4.2e-5]], $MachinePrecision], LessEqual[re, 5.2e+47]]], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.79999999999999992e25

   1. Initial program 7.3%

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

      1. +-commutative 7.3%

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

      2. hypot-def 38.8%

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

   3. Simplified 38.8%

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

   4. Taylor expanded in re around -inf 55.2%

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

      1. associate-*r/ 55.2%

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

      2. neg-mul-1 55.2%

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

      3. unpow2 55.2%

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

      4. distribute-rgt-neg-in 55.2%

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

   6. Simplified 55.2%

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

      1. frac-2neg 55.2%

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

      2. sqrt-div 62.0%

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

      3. distribute-rgt-neg-out 62.0%

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

      4. remove-double-neg 62.0%

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

      5. sqrt-unprod 43.7%

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

      6. add-sqr-sqrt 54.0%

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

   8. Applied egg-rr 54.0%

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

      1. associate-*r/ 54.0%

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

   10. Applied egg-rr 54.0%

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

   if -4.79999999999999992e25 < re < 6.6e-187

   1. Initial program 52.3%

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

      1. +-commutative 52.3%

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

      2. hypot-def 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in re around 0 46.4%

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

      1. *-commutative 46.4%

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

   6. Simplified 46.4%

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

   if 6.6e-187 < re < 1.09999999999999997e-125 or 1.6e-77 < re < 4.19999999999999977e-5 or 5.20000000000000007e47 < re

   1. Initial program 61.4%

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

      1. +-commutative 61.4%

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

      2. 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 77.4%

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

      1. associate-*r* 77.4%

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

      2. unpow2 77.4%

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

      3. rem-square-sqrt 78.9%

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

      4. metadata-eval 78.9%

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

      5. *-lft-identity 78.9%

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

   6. Simplified 78.9%

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

   if 1.09999999999999997e-125 < re < 1.6e-77 or 4.19999999999999977e-5 < re < 5.20000000000000007e47

   1. Initial program 61.7%

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

      1. +-commutative 61.7%

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

      2. 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 re around 0 39.2%

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

      1. distribute-lft-out 39.2%

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

      2. +-commutative 39.2%

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

      3. *-commutative 39.2%

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

      4. +-commutative 39.2%

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

   6. Simplified 39.2%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-77} \lor \neg \left(re \leq 4.2 \cdot 10^{-5}\right) \land re \leq 5.2 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 4: 72.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-187} \lor \neg \left(re \leq 1.1 \cdot 10^{-125}\right) \land \left(re \leq 1.95 \cdot 10^{-78} \lor \neg \left(re \leq 0.0031\right) \land re \leq 1.8 \cdot 10^{+53}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -3.3e+28)
   (* 0.5 (/ im (sqrt (- re))))
   (if (or (<= re 6.6e-187)
           (and (not (<= re 1.1e-125))
                (or (<= re 1.95e-78)
                    (and (not (<= re 0.0031)) (<= re 1.8e+53)))))
     (* 0.5 (sqrt (* 2.0 im)))
     (sqrt re))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -3.3e+28) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if ((re <= 6.6e-187) || (!(re <= 1.1e-125) && ((re <= 1.95e-78) || (!(re <= 0.0031) && (re <= 1.8e+53))))) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.3d+28)) then
        tmp = 0.5d0 * (im / sqrt(-re))
    else if ((re <= 6.6d-187) .or. (.not. (re <= 1.1d-125)) .and. (re <= 1.95d-78) .or. (.not. (re <= 0.0031d0)) .and. (re <= 1.8d+53)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -3.3e+28) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if ((re <= 6.6e-187) || (!(re <= 1.1e-125) && ((re <= 1.95e-78) || (!(re <= 0.0031) && (re <= 1.8e+53))))) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -3.3e+28:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif (re <= 6.6e-187) or (not (re <= 1.1e-125) and ((re <= 1.95e-78) or (not (re <= 0.0031) and (re <= 1.8e+53)))):
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -3.3e+28)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif ((re <= 6.6e-187) || (!(re <= 1.1e-125) && ((re <= 1.95e-78) || (!(re <= 0.0031) && (re <= 1.8e+53)))))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.3e+28)
		tmp = 0.5 * (im / sqrt(-re));
	elseif ((re <= 6.6e-187) || (~((re <= 1.1e-125)) && ((re <= 1.95e-78) || (~((re <= 0.0031)) && (re <= 1.8e+53)))))
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -3.3e+28], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 6.6e-187], And[N[Not[LessEqual[re, 1.1e-125]], $MachinePrecision], Or[LessEqual[re, 1.95e-78], And[N[Not[LessEqual[re, 0.0031]], $MachinePrecision], LessEqual[re, 1.8e+53]]]]], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.3e28

   1. Initial program 7.3%

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

      1. +-commutative 7.3%

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

      2. hypot-def 38.8%

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

   3. Simplified 38.8%

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

   4. Taylor expanded in re around -inf 55.2%

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

      1. associate-*r/ 55.2%

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

      2. neg-mul-1 55.2%

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

      3. unpow2 55.2%

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

      4. distribute-rgt-neg-in 55.2%

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

   6. Simplified 55.2%

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

      1. frac-2neg 55.2%

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

      2. sqrt-div 62.0%

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

      3. distribute-rgt-neg-out 62.0%

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

      4. remove-double-neg 62.0%

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

      5. sqrt-unprod 43.7%

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

      6. add-sqr-sqrt 54.0%

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

   8. Applied egg-rr 54.0%

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

   if -3.3e28 < re < 6.6e-187 or 1.09999999999999997e-125 < re < 1.9500000000000001e-78 or 0.00309999999999999989 < re < 1.8e53

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

      2. hypot-def 87.5%

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

   3. Simplified 87.5%

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

   4. Taylor expanded in re around 0 44.2%

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

      1. *-commutative 44.2%

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

   6. Simplified 44.2%

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

   if 6.6e-187 < re < 1.09999999999999997e-125 or 1.9500000000000001e-78 < re < 0.00309999999999999989 or 1.8e53 < re

   1. Initial program 61.4%

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

      1. +-commutative 61.4%

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

      2. 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 77.4%

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

      1. associate-*r* 77.4%

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

      2. unpow2 77.4%

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

      3. rem-square-sqrt 78.9%

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

      4. metadata-eval 78.9%

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

      5. *-lft-identity 78.9%

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

   6. Simplified 78.9%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-187} \lor \neg \left(re \leq 1.1 \cdot 10^{-125}\right) \land \left(re \leq 1.95 \cdot 10^{-78} \lor \neg \left(re \leq 0.0031\right) \land re \leq 1.8 \cdot 10^{+53}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 5: 72.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-187} \lor \neg \left(re \leq 1.1 \cdot 10^{-125}\right) \land \left(re \leq 1.25 \cdot 10^{-76} \lor \neg \left(re \leq 8.5 \cdot 10^{-5}\right) \land re \leq 3.5 \cdot 10^{+45}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -1.5e+25)
   (/ (* im 0.5) (sqrt (- re)))
   (if (or (<= re 6.6e-187)
           (and (not (<= re 1.1e-125))
                (or (<= re 1.25e-76)
                    (and (not (<= re 8.5e-5)) (<= re 3.5e+45)))))
     (* 0.5 (sqrt (* 2.0 im)))
     (sqrt re))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -1.5e+25) {
		tmp = (im * 0.5) / sqrt(-re);
	} else if ((re <= 6.6e-187) || (!(re <= 1.1e-125) && ((re <= 1.25e-76) || (!(re <= 8.5e-5) && (re <= 3.5e+45))))) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.5d+25)) then
        tmp = (im * 0.5d0) / sqrt(-re)
    else if ((re <= 6.6d-187) .or. (.not. (re <= 1.1d-125)) .and. (re <= 1.25d-76) .or. (.not. (re <= 8.5d-5)) .and. (re <= 3.5d+45)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -1.5e+25) {
		tmp = (im * 0.5) / Math.sqrt(-re);
	} else if ((re <= 6.6e-187) || (!(re <= 1.1e-125) && ((re <= 1.25e-76) || (!(re <= 8.5e-5) && (re <= 3.5e+45))))) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -1.5e+25:
		tmp = (im * 0.5) / math.sqrt(-re)
	elif (re <= 6.6e-187) or (not (re <= 1.1e-125) and ((re <= 1.25e-76) or (not (re <= 8.5e-5) and (re <= 3.5e+45)))):
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -1.5e+25)
		tmp = Float64(Float64(im * 0.5) / sqrt(Float64(-re)));
	elseif ((re <= 6.6e-187) || (!(re <= 1.1e-125) && ((re <= 1.25e-76) || (!(re <= 8.5e-5) && (re <= 3.5e+45)))))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.5e+25)
		tmp = (im * 0.5) / sqrt(-re);
	elseif ((re <= 6.6e-187) || (~((re <= 1.1e-125)) && ((re <= 1.25e-76) || (~((re <= 8.5e-5)) && (re <= 3.5e+45)))))
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -1.5e+25], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 6.6e-187], And[N[Not[LessEqual[re, 1.1e-125]], $MachinePrecision], Or[LessEqual[re, 1.25e-76], And[N[Not[LessEqual[re, 8.5e-5]], $MachinePrecision], LessEqual[re, 3.5e+45]]]]], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.50000000000000003e25

   1. Initial program 7.3%

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

      1. +-commutative 7.3%

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

      2. hypot-def 38.8%

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

   3. Simplified 38.8%

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

   4. Taylor expanded in re around -inf 55.2%

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

      1. associate-*r/ 55.2%

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

      2. neg-mul-1 55.2%

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

      3. unpow2 55.2%

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

      4. distribute-rgt-neg-in 55.2%

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

   6. Simplified 55.2%

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

      1. frac-2neg 55.2%

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

      2. sqrt-div 62.0%

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

      3. distribute-rgt-neg-out 62.0%

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

      4. remove-double-neg 62.0%

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

      5. sqrt-unprod 43.7%

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

      6. add-sqr-sqrt 54.0%

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

   8. Applied egg-rr 54.0%

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

      1. associate-*r/ 54.0%

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

   10. Applied egg-rr 54.0%

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

   if -1.50000000000000003e25 < re < 6.6e-187 or 1.09999999999999997e-125 < re < 1.2499999999999999e-76 or 8.500000000000001e-5 < re < 3.50000000000000023e45

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

      2. hypot-def 87.5%

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

   3. Simplified 87.5%

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

   4. Taylor expanded in re around 0 44.2%

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

      1. *-commutative 44.2%

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

   6. Simplified 44.2%

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

   if 6.6e-187 < re < 1.09999999999999997e-125 or 1.2499999999999999e-76 < re < 8.500000000000001e-5 or 3.50000000000000023e45 < re

   1. Initial program 61.4%

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

      1. +-commutative 61.4%

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

      2. 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 77.4%

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

      1. associate-*r* 77.4%

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

      2. unpow2 77.4%

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

      3. rem-square-sqrt 78.9%

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

      4. metadata-eval 78.9%

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

      5. *-lft-identity 78.9%

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

   6. Simplified 78.9%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-187} \lor \neg \left(re \leq 1.1 \cdot 10^{-125}\right) \land \left(re \leq 1.25 \cdot 10^{-76} \lor \neg \left(re \leq 8.5 \cdot 10^{-5}\right) \land re \leq 3.5 \cdot 10^{+45}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 6: 60.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 6.6 \cdot 10^{-187} \lor \neg \left(re \leq 1.1 \cdot 10^{-125}\right) \land \left(re \leq 1.85 \cdot 10^{-78} \lor \neg \left(re \leq 2.85 \cdot 10^{-6}\right) \land re \leq 8.8 \cdot 10^{+46}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (or (<= re 6.6e-187)
         (and (not (<= re 1.1e-125))
              (or (<= re 1.85e-78)
                  (and (not (<= re 2.85e-6)) (<= re 8.8e+46)))))
   (* 0.5 (sqrt (* 2.0 im)))
   (sqrt re)))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if ((re <= 6.6e-187) || (!(re <= 1.1e-125) && ((re <= 1.85e-78) || (!(re <= 2.85e-6) && (re <= 8.8e+46))))) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= 6.6d-187) .or. (.not. (re <= 1.1d-125)) .and. (re <= 1.85d-78) .or. (.not. (re <= 2.85d-6)) .and. (re <= 8.8d+46)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if ((re <= 6.6e-187) || (!(re <= 1.1e-125) && ((re <= 1.85e-78) || (!(re <= 2.85e-6) && (re <= 8.8e+46))))) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if (re <= 6.6e-187) or (not (re <= 1.1e-125) and ((re <= 1.85e-78) or (not (re <= 2.85e-6) and (re <= 8.8e+46)))):
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if ((re <= 6.6e-187) || (!(re <= 1.1e-125) && ((re <= 1.85e-78) || (!(re <= 2.85e-6) && (re <= 8.8e+46)))))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 6.6e-187) || (~((re <= 1.1e-125)) && ((re <= 1.85e-78) || (~((re <= 2.85e-6)) && (re <= 8.8e+46)))))
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[Or[LessEqual[re, 6.6e-187], And[N[Not[LessEqual[re, 1.1e-125]], $MachinePrecision], Or[LessEqual[re, 1.85e-78], And[N[Not[LessEqual[re, 2.85e-6]], $MachinePrecision], LessEqual[re, 8.8e+46]]]]], 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 < 6.6e-187 or 1.09999999999999997e-125 < re < 1.85000000000000003e-78 or 2.8499999999999998e-6 < re < 8.8000000000000001e46

   1. Initial program 38.2%

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

      1. +-commutative 38.2%

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

      2. hypot-def 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in re around 0 33.8%

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

      1. *-commutative 33.8%

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

   6. Simplified 33.8%

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

   if 6.6e-187 < re < 1.09999999999999997e-125 or 1.85000000000000003e-78 < re < 2.8499999999999998e-6 or 8.8000000000000001e46 < re

   1. Initial program 61.4%

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

      1. +-commutative 61.4%

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

      2. 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 77.4%

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

      1. associate-*r* 77.4%

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

      2. unpow2 77.4%

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

      3. rem-square-sqrt 78.9%

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

      4. metadata-eval 78.9%

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

      5. *-lft-identity 78.9%

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

   6. Simplified 78.9%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.6 \cdot 10^{-187} \lor \neg \left(re \leq 1.1 \cdot 10^{-125}\right) \land \left(re \leq 1.85 \cdot 10^{-78} \lor \neg \left(re \leq 2.85 \cdot 10^{-6}\right) \land re \leq 8.8 \cdot 10^{+46}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 7: 26.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \sqrt{re} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (sqrt re))

C

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

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 46.1%

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

   1. +-commutative 46.1%

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

   2. hypot-def 80.7%

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

3. Simplified 80.7%

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

4. Taylor expanded in im around 0 29.3%

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

   1. associate-*r* 29.3%

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

   2. unpow2 29.3%

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

   3. rem-square-sqrt 29.8%

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

   4. metadata-eval 29.8%

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

   5. *-lft-identity 29.8%

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

6. Simplified 29.8%

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

7. Final simplification 29.8%

   \[\leadsto \sqrt{re} \]

Developer target: 47.6% 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 2023258 
(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)))))