math.sqrt on complex, imaginary part, im greater than 0 branch

Percentage Accurate: 41.3% → 89.9%

Time: 13.0s

Alternatives: 7

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 10.5%

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

      1. sub-neg 10.5%

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

      2. sqr-neg 10.5%

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

      3. sub-neg 10.5%

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

      4. sqr-neg 10.5%

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

      5. hypot-define 17.3%

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

   3. Simplified 17.3%

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

   5. Taylor expanded in im around 0 92.6%

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

      1. associate-*r* 92.6%

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

      2. *-commutative 92.6%

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

   7. Simplified 92.6%

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

      1. add-cube-cbrt 91.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \cdot \sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \cdot \sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}} \]

      2. pow3 92.0%

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

      3. associate-*l* 92.0%

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

      4. sqrt-unprod 92.5%

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

      5. metadata-eval 92.5%

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

      6. metadata-eval 92.5%

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

      7. *-rgt-identity 92.5%

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

      8. sqrt-div 92.6%

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

      9. metadata-eval 92.6%

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

      10. un-div-inv 92.3%

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

   9. Applied egg-rr 92.3%

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

       1. rem-cube-cbrt 93.9%

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

       2. clear-num 93.9%

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

       3. un-div-inv 93.9%

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

   11. Applied egg-rr 93.9%

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

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

   1. Initial program 48.9%

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

      1. sub-neg 48.9%

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

      2. sqr-neg 48.9%

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

      3. sub-neg 48.9%

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

      4. sqr-neg 48.9%

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

      5. hypot-define 90.5%

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

   3. Simplified 90.5%

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

      1. add-sqr-sqrt 89.9%

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

      2. sqrt-unprod 90.5%

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

      3. *-commutative 90.5%

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

      4. *-commutative 90.5%

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

      5. swap-sqr 90.5%

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

      6. add-sqr-sqrt 90.5%

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

      7. *-commutative 90.5%

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

      8. metadata-eval 90.5%

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

   6. Applied egg-rr 90.5%

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

      1. associate-*l* 91.0%

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

      2. hypot-undefine 48.9%

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

      3. unpow2 48.9%

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

      4. unpow2 48.9%

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

      5. +-commutative 48.9%

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

      6. unpow2 48.9%

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

      7. unpow2 48.9%

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

      8. hypot-undefine 91.0%

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

      9. metadata-eval 91.0%

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

   8. Simplified 91.0%

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

4. Final simplification 91.5%

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

Alternative 2: 75.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -8 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -2.15 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{-86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{+30}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+58}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- im re)))))
        (t_1 (* 0.5 (sqrt (* 2.0 (* re -2.0))))))
   (if (<= re -8e+34)
     t_1
     (if (<= re -2.6e-40)
       t_0
       (if (<= re -2.15e-70)
         t_1
         (if (<= re 8.5e-86)
           t_0
           (if (<= re 3.4e+30)
             (/ 0.5 (/ (sqrt re) im))
             (if (<= re 1.65e+58) t_0 (/ (* im 0.5) (sqrt re))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (im - re)));
	double t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -8e+34) {
		tmp = t_1;
	} else if (re <= -2.6e-40) {
		tmp = t_0;
	} else if (re <= -2.15e-70) {
		tmp = t_1;
	} else if (re <= 8.5e-86) {
		tmp = t_0;
	} else if (re <= 3.4e+30) {
		tmp = 0.5 / (sqrt(re) / im);
	} else if (re <= 1.65e+58) {
		tmp = t_0;
	} else {
		tmp = (im * 0.5) / sqrt(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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    t_1 = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    if (re <= (-8d+34)) then
        tmp = t_1
    else if (re <= (-2.6d-40)) then
        tmp = t_0
    else if (re <= (-2.15d-70)) then
        tmp = t_1
    else if (re <= 8.5d-86) then
        tmp = t_0
    else if (re <= 3.4d+30) then
        tmp = 0.5d0 / (sqrt(re) / im)
    else if (re <= 1.65d+58) then
        tmp = t_0
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double t_1 = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -8e+34) {
		tmp = t_1;
	} else if (re <= -2.6e-40) {
		tmp = t_0;
	} else if (re <= -2.15e-70) {
		tmp = t_1;
	} else if (re <= 8.5e-86) {
		tmp = t_0;
	} else if (re <= 3.4e+30) {
		tmp = 0.5 / (Math.sqrt(re) / im);
	} else if (re <= 1.65e+58) {
		tmp = t_0;
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (im - re)))
	t_1 = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	tmp = 0
	if re <= -8e+34:
		tmp = t_1
	elif re <= -2.6e-40:
		tmp = t_0
	elif re <= -2.15e-70:
		tmp = t_1
	elif re <= 8.5e-86:
		tmp = t_0
	elif re <= 3.4e+30:
		tmp = 0.5 / (math.sqrt(re) / im)
	elif re <= 1.65e+58:
		tmp = t_0
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))))
	tmp = 0.0
	if (re <= -8e+34)
		tmp = t_1;
	elseif (re <= -2.6e-40)
		tmp = t_0;
	elseif (re <= -2.15e-70)
		tmp = t_1;
	elseif (re <= 8.5e-86)
		tmp = t_0;
	elseif (re <= 3.4e+30)
		tmp = Float64(0.5 / Float64(sqrt(re) / im));
	elseif (re <= 1.65e+58)
		tmp = t_0;
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (im - re)));
	t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	tmp = 0.0;
	if (re <= -8e+34)
		tmp = t_1;
	elseif (re <= -2.6e-40)
		tmp = t_0;
	elseif (re <= -2.15e-70)
		tmp = t_1;
	elseif (re <= 8.5e-86)
		tmp = t_0;
	elseif (re <= 3.4e+30)
		tmp = 0.5 / (sqrt(re) / im);
	elseif (re <= 1.65e+58)
		tmp = t_0;
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -8e+34], t$95$1, If[LessEqual[re, -2.6e-40], t$95$0, If[LessEqual[re, -2.15e-70], t$95$1, If[LessEqual[re, 8.5e-86], t$95$0, If[LessEqual[re, 3.4e+30], N[(0.5 / N[(N[Sqrt[re], $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.65e+58], t$95$0, N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.99999999999999956e34 or -2.6000000000000001e-40 < re < -2.15e-70

   1. Initial program 49.4%

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

   3. Taylor expanded in re around -inf 85.4%

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

      1. *-commutative 85.4%

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

   5. Simplified 85.4%

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

   if -7.99999999999999956e34 < re < -2.6000000000000001e-40 or -2.15e-70 < re < 8.499999999999999e-86 or 3.4000000000000002e30 < re < 1.64999999999999991e58

   1. Initial program 57.1%

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

   3. Taylor expanded in re around 0 81.3%

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

   if 8.499999999999999e-86 < re < 3.4000000000000002e30

   1. Initial program 34.0%

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

      1. sub-neg 34.0%

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

      2. sqr-neg 34.0%

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

      3. sub-neg 34.0%

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

      4. sqr-neg 34.0%

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

      5. hypot-define 45.0%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in im around 0 59.8%

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

      1. associate-*r* 59.8%

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

      2. *-commutative 59.8%

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

   7. Simplified 59.8%

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

      1. add-cube-cbrt 59.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \cdot \sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \cdot \sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}} \]

      2. pow3 59.3%

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

      3. associate-*l* 59.3%

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

      4. sqrt-unprod 59.7%

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

      5. metadata-eval 59.7%

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

      6. metadata-eval 59.7%

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

      7. *-rgt-identity 59.7%

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

      8. sqrt-div 59.7%

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

      9. metadata-eval 59.7%

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

      10. un-div-inv 59.7%

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

   9. Applied egg-rr 59.7%

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

       1. rem-cube-cbrt 60.7%

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

       2. clear-num 60.8%

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

       3. un-div-inv 60.8%

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

   11. Applied egg-rr 60.8%

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

   if 1.64999999999999991e58 < re

   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. sub-neg 9.4%

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

      2. sqr-neg 9.4%

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

      3. sub-neg 9.4%

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

      4. sqr-neg 9.4%

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

      5. hypot-define 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in im around 0 87.5%

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

      1. associate-*r* 87.6%

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

      2. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. sqrt-div 87.7%

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

      2. metadata-eval 87.7%

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

      3. un-div-inv 87.6%

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

      4. *-commutative 87.6%

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

      5. sqrt-unprod 88.5%

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

      6. metadata-eval 88.5%

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

      7. metadata-eval 88.5%

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

      8. *-rgt-identity 88.5%

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

   9. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -2.15 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{+30}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -1.6 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -1.52 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -1.45 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{\frac{1}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- im re)))))
        (t_1 (* 0.5 (sqrt (* 2.0 (* re -2.0))))))
   (if (<= re -1.6e+35)
     t_1
     (if (<= re -1.52e-43)
       t_0
       (if (<= re -1.45e-70)
         t_1
         (if (<= re 8e-86)
           t_0
           (if (<= re 3.6e+30)
             (/ 0.5 (/ (sqrt re) im))
             (if (<= re 1.5e+58) t_0 (* (* im 0.5) (sqrt (/ 1.0 re)))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (im - re)));
	double t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -1.6e+35) {
		tmp = t_1;
	} else if (re <= -1.52e-43) {
		tmp = t_0;
	} else if (re <= -1.45e-70) {
		tmp = t_1;
	} else if (re <= 8e-86) {
		tmp = t_0;
	} else if (re <= 3.6e+30) {
		tmp = 0.5 / (sqrt(re) / im);
	} else if (re <= 1.5e+58) {
		tmp = t_0;
	} else {
		tmp = (im * 0.5) * sqrt((1.0 / 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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    t_1 = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    if (re <= (-1.6d+35)) then
        tmp = t_1
    else if (re <= (-1.52d-43)) then
        tmp = t_0
    else if (re <= (-1.45d-70)) then
        tmp = t_1
    else if (re <= 8d-86) then
        tmp = t_0
    else if (re <= 3.6d+30) then
        tmp = 0.5d0 / (sqrt(re) / im)
    else if (re <= 1.5d+58) then
        tmp = t_0
    else
        tmp = (im * 0.5d0) * sqrt((1.0d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double t_1 = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -1.6e+35) {
		tmp = t_1;
	} else if (re <= -1.52e-43) {
		tmp = t_0;
	} else if (re <= -1.45e-70) {
		tmp = t_1;
	} else if (re <= 8e-86) {
		tmp = t_0;
	} else if (re <= 3.6e+30) {
		tmp = 0.5 / (Math.sqrt(re) / im);
	} else if (re <= 1.5e+58) {
		tmp = t_0;
	} else {
		tmp = (im * 0.5) * Math.sqrt((1.0 / re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (im - re)))
	t_1 = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	tmp = 0
	if re <= -1.6e+35:
		tmp = t_1
	elif re <= -1.52e-43:
		tmp = t_0
	elif re <= -1.45e-70:
		tmp = t_1
	elif re <= 8e-86:
		tmp = t_0
	elif re <= 3.6e+30:
		tmp = 0.5 / (math.sqrt(re) / im)
	elif re <= 1.5e+58:
		tmp = t_0
	else:
		tmp = (im * 0.5) * math.sqrt((1.0 / re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))))
	tmp = 0.0
	if (re <= -1.6e+35)
		tmp = t_1;
	elseif (re <= -1.52e-43)
		tmp = t_0;
	elseif (re <= -1.45e-70)
		tmp = t_1;
	elseif (re <= 8e-86)
		tmp = t_0;
	elseif (re <= 3.6e+30)
		tmp = Float64(0.5 / Float64(sqrt(re) / im));
	elseif (re <= 1.5e+58)
		tmp = t_0;
	else
		tmp = Float64(Float64(im * 0.5) * sqrt(Float64(1.0 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (im - re)));
	t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	tmp = 0.0;
	if (re <= -1.6e+35)
		tmp = t_1;
	elseif (re <= -1.52e-43)
		tmp = t_0;
	elseif (re <= -1.45e-70)
		tmp = t_1;
	elseif (re <= 8e-86)
		tmp = t_0;
	elseif (re <= 3.6e+30)
		tmp = 0.5 / (sqrt(re) / im);
	elseif (re <= 1.5e+58)
		tmp = t_0;
	else
		tmp = (im * 0.5) * sqrt((1.0 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.6e+35], t$95$1, If[LessEqual[re, -1.52e-43], t$95$0, If[LessEqual[re, -1.45e-70], t$95$1, If[LessEqual[re, 8e-86], t$95$0, If[LessEqual[re, 3.6e+30], N[(0.5 / N[(N[Sqrt[re], $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.5e+58], t$95$0, N[(N[(im * 0.5), $MachinePrecision] * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.59999999999999991e35 or -1.52e-43 < re < -1.44999999999999986e-70

   1. Initial program 49.4%

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

   3. Taylor expanded in re around -inf 85.4%

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

      1. *-commutative 85.4%

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

   5. Simplified 85.4%

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

   if -1.59999999999999991e35 < re < -1.52e-43 or -1.44999999999999986e-70 < re < 8.00000000000000068e-86 or 3.6000000000000002e30 < re < 1.5000000000000001e58

   1. Initial program 57.1%

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

   3. Taylor expanded in re around 0 81.3%

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

   if 8.00000000000000068e-86 < re < 3.6000000000000002e30

   1. Initial program 34.0%

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

      1. sub-neg 34.0%

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

      2. sqr-neg 34.0%

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

      3. sub-neg 34.0%

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

      4. sqr-neg 34.0%

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

      5. hypot-define 45.0%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in im around 0 59.8%

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

      1. associate-*r* 59.8%

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

      2. *-commutative 59.8%

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

   7. Simplified 59.8%

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

      1. add-cube-cbrt 59.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \cdot \sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \cdot \sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}} \]

      2. pow3 59.3%

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

      3. associate-*l* 59.3%

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

      4. sqrt-unprod 59.7%

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

      5. metadata-eval 59.7%

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

      6. metadata-eval 59.7%

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

      7. *-rgt-identity 59.7%

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

      8. sqrt-div 59.7%

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

      9. metadata-eval 59.7%

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

      10. un-div-inv 59.7%

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

   9. Applied egg-rr 59.7%

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

       1. rem-cube-cbrt 60.7%

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

       2. clear-num 60.8%

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

       3. un-div-inv 60.8%

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

   11. Applied egg-rr 60.8%

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

   if 1.5000000000000001e58 < re

   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. sub-neg 9.4%

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

      2. sqr-neg 9.4%

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

      3. sub-neg 9.4%

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

      4. sqr-neg 9.4%

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

      5. hypot-define 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in im around 0 87.5%

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

      1. associate-*r* 87.6%

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

      2. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. sqrt-unprod 88.6%

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

      2. metadata-eval 88.6%

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

      3. metadata-eval 88.6%

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

   9. Applied egg-rr 88.6%

      \[\leadsto \left(0.5 \cdot \left(im \cdot \color{blue}{1}\right)\right) \cdot \sqrt{\frac{1}{re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{+35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -1.52 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -1.45 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{\frac{1}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -3.6 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -3.8 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -1.6 \cdot 10^{-89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* im 2.0))))
        (t_1 (* 0.5 (sqrt (* 2.0 (* re -2.0))))))
   (if (<= re -3.6e+35)
     t_1
     (if (<= re -3.8e-44)
       t_0
       (if (<= re -1.6e-89)
         t_1
         (if (<= re 1.5e+58) t_0 (/ (* im 0.5) (sqrt re))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((im * 2.0));
	double t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -3.6e+35) {
		tmp = t_1;
	} else if (re <= -3.8e-44) {
		tmp = t_0;
	} else if (re <= -1.6e-89) {
		tmp = t_1;
	} else if (re <= 1.5e+58) {
		tmp = t_0;
	} else {
		tmp = (im * 0.5) / sqrt(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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((im * 2.0d0))
    t_1 = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    if (re <= (-3.6d+35)) then
        tmp = t_1
    else if (re <= (-3.8d-44)) then
        tmp = t_0
    else if (re <= (-1.6d-89)) then
        tmp = t_1
    else if (re <= 1.5d+58) then
        tmp = t_0
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((im * 2.0));
	double t_1 = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	double tmp;
	if (re <= -3.6e+35) {
		tmp = t_1;
	} else if (re <= -3.8e-44) {
		tmp = t_0;
	} else if (re <= -1.6e-89) {
		tmp = t_1;
	} else if (re <= 1.5e+58) {
		tmp = t_0;
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((im * 2.0))
	t_1 = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	tmp = 0
	if re <= -3.6e+35:
		tmp = t_1
	elif re <= -3.8e-44:
		tmp = t_0
	elif re <= -1.6e-89:
		tmp = t_1
	elif re <= 1.5e+58:
		tmp = t_0
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))))
	tmp = 0.0
	if (re <= -3.6e+35)
		tmp = t_1;
	elseif (re <= -3.8e-44)
		tmp = t_0;
	elseif (re <= -1.6e-89)
		tmp = t_1;
	elseif (re <= 1.5e+58)
		tmp = t_0;
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((im * 2.0));
	t_1 = 0.5 * sqrt((2.0 * (re * -2.0)));
	tmp = 0.0;
	if (re <= -3.6e+35)
		tmp = t_1;
	elseif (re <= -3.8e-44)
		tmp = t_0;
	elseif (re <= -1.6e-89)
		tmp = t_1;
	elseif (re <= 1.5e+58)
		tmp = t_0;
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3.6e+35], t$95$1, If[LessEqual[re, -3.8e-44], t$95$0, If[LessEqual[re, -1.6e-89], t$95$1, If[LessEqual[re, 1.5e+58], t$95$0, N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.6e35 or -3.8000000000000001e-44 < re < -1.59999999999999999e-89

   1. Initial program 50.8%

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

   3. Taylor expanded in re around -inf 84.5%

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

      1. *-commutative 84.5%

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

   5. Simplified 84.5%

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

   if -3.6e35 < re < -3.8000000000000001e-44 or -1.59999999999999999e-89 < re < 1.5000000000000001e58

   1. Initial program 51.7%

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

   3. Taylor expanded in re around 0 73.3%

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

      1. pow1 73.3%

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

      2. sqrt-unprod 73.0%

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

   5. Applied egg-rr 73.0%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{im \cdot 2}\right)}^{1}} \]
   6. Step-by-step derivation

      1. unpow1 73.0%

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

   7. Simplified 73.0%

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

   if 1.5000000000000001e58 < re

   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. sub-neg 9.4%

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

      2. sqr-neg 9.4%

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

      3. sub-neg 9.4%

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

      4. sqr-neg 9.4%

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

      5. hypot-define 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in im around 0 87.5%

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

      1. associate-*r* 87.6%

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

      2. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. sqrt-div 87.7%

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

      2. metadata-eval 87.7%

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

      3. un-div-inv 87.6%

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

      4. *-commutative 87.6%

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

      5. sqrt-unprod 88.5%

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

      6. metadata-eval 88.5%

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

      7. metadata-eval 88.5%

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

      8. *-rgt-identity 88.5%

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

   9. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -3.8 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq -1.6 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.5e+58) (* 0.5 (sqrt (* im 2.0))) (/ 0.5 (/ (sqrt re) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+58) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 / (sqrt(re) / im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.5d+58) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 / (sqrt(re) / im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+58) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 / (Math.sqrt(re) / im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.5e+58:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 / (math.sqrt(re) / im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.5e+58)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 / Float64(sqrt(re) / im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.5e+58)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 / (sqrt(re) / im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.5e+58], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[Sqrt[re], $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.5000000000000001e58

   1. Initial program 51.4%

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

   3. Taylor expanded in re around 0 54.2%

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

      1. pow1 54.2%

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

      2. sqrt-unprod 54.0%

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

   5. Applied egg-rr 54.0%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{im \cdot 2}\right)}^{1}} \]
   6. Step-by-step derivation

      1. unpow1 54.0%

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

   7. Simplified 54.0%

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

   if 1.5000000000000001e58 < re

   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. sub-neg 9.4%

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

      2. sqr-neg 9.4%

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

      3. sub-neg 9.4%

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

      4. sqr-neg 9.4%

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

      5. hypot-define 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in im around 0 87.5%

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

      1. associate-*r* 87.6%

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

      2. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. add-cube-cbrt 87.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \cdot \sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \cdot \sqrt[3]{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}} \]

      2. pow3 87.2%

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

      3. associate-*l* 87.2%

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

      4. sqrt-unprod 87.4%

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

      5. metadata-eval 87.4%

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

      6. metadata-eval 87.4%

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

      7. *-rgt-identity 87.4%

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

      8. sqrt-div 87.4%

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

      9. metadata-eval 87.4%

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

      10. un-div-inv 87.4%

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

   9. Applied egg-rr 87.4%

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

       1. rem-cube-cbrt 88.5%

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

       2. clear-num 86.9%

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

       3. un-div-inv 86.9%

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

   11. Applied egg-rr 86.9%

       \[\leadsto \color{blue}{\frac{0.5}{\frac{\sqrt{re}}{im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\sqrt{re}}{im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.5e+58) (* 0.5 (sqrt (* im 2.0))) (/ (* im 0.5) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+58) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.5d+58) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+58) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.5e+58:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.5e+58)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.5e+58)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.5e+58], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.5000000000000001e58

   1. Initial program 51.4%

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

   3. Taylor expanded in re around 0 54.2%

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

      1. pow1 54.2%

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

      2. sqrt-unprod 54.0%

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

   5. Applied egg-rr 54.0%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{im \cdot 2}\right)}^{1}} \]
   6. Step-by-step derivation

      1. unpow1 54.0%

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

   7. Simplified 54.0%

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

   if 1.5000000000000001e58 < re

   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. sub-neg 9.4%

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

      2. sqr-neg 9.4%

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

      3. sub-neg 9.4%

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

      4. sqr-neg 9.4%

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

      5. hypot-define 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in im around 0 87.5%

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

      1. associate-*r* 87.6%

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

      2. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. sqrt-div 87.7%

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

      2. metadata-eval 87.7%

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

      3. un-div-inv 87.6%

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

      4. *-commutative 87.6%

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

      5. sqrt-unprod 88.5%

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

      6. metadata-eval 88.5%

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

      7. metadata-eval 88.5%

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

      8. *-rgt-identity 88.5%

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

   9. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{im \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.9%

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

3. Taylor expanded in re around 0 45.7%

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

   1. pow1 45.7%

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

   2. sqrt-unprod 45.6%

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

5. Applied egg-rr 45.6%

   \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{im \cdot 2}\right)}^{1}} \]
6. Step-by-step derivation

   1. unpow1 45.6%

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

7. Simplified 45.6%

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

8. Final simplification 45.6%

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

Reproduce

herbie shell --seed 2024040 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))