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

Percentage Accurate: 41.4% → 90.0%

Time: 13.0s

Alternatives: 6

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: 41.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))) <= 0.0)
		tmp = Float64((re ^ -0.5) * Float64(im * 0.5));
	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((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0)
		tmp = (re ^ -0.5) * (im * 0.5);
	else
		tmp = sqrt((0.5 * (hypot(im, re) - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $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 (sqrt.f64 (*.f64 2 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 14.6%

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

      1. sub-neg 14.6%

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

      2. sqr-neg 14.6%

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

      3. sub-neg 14.6%

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

      4. sqr-neg 14.6%

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

      5. hypot-define 14.6%

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

   3. Simplified 14.6%

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

   5. Taylor expanded in im around 0 98.2%

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

      1. associate-*r* 98.2%

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

      2. *-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. add0 98.2%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}} + 0} \]

      2. *-commutative 98.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right)} + 0 \]

      3. fma-define 98.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{re}}, 0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right), 0\right)} \]

      4. inv-pow 98.2%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{{re}^{-1}}}, 0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right), 0\right) \]

      5. sqrt-pow1 98.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{\left(\frac{-1}{2}\right)}}, 0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right), 0\right) \]

      6. metadata-eval 98.2%

         \[\leadsto \mathsf{fma}\left({re}^{\color{blue}{-0.5}}, 0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right), 0\right) \]

      7. *-commutative 98.2%

         \[\leadsto \mathsf{fma}\left({re}^{-0.5}, \color{blue}{\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot 0.5}, 0\right) \]

      8. sqrt-unprod 99.5%

         \[\leadsto \mathsf{fma}\left({re}^{-0.5}, \left(im \cdot \color{blue}{\sqrt{2 \cdot 0.5}}\right) \cdot 0.5, 0\right) \]

      9. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left({re}^{-0.5}, \left(im \cdot \sqrt{\color{blue}{1}}\right) \cdot 0.5, 0\right) \]

      10. metadata-eval 99.5%

          \[\leadsto \mathsf{fma}\left({re}^{-0.5}, \left(im \cdot \color{blue}{1}\right) \cdot 0.5, 0\right) \]

      11. *-rgt-identity 99.5%

          \[\leadsto \mathsf{fma}\left({re}^{-0.5}, \color{blue}{im} \cdot 0.5, 0\right) \]

   9. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{-0.5}, im \cdot 0.5, 0\right)} \]
   10. Step-by-step derivation

       1. fma-undefine 99.5%

          \[\leadsto \color{blue}{{re}^{-0.5} \cdot \left(im \cdot 0.5\right) + 0} \]

       2. +-rgt-identity 99.5%

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

       3. *-commutative 99.5%

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

   11. Simplified 99.5%

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

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

   1. Initial program 46.4%

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

      1. sub-neg 46.4%

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

      2. sqr-neg 46.4%

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

      3. sub-neg 46.4%

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

      4. sqr-neg 46.4%

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

      5. hypot-define 90.8%

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

   3. Simplified 90.8%

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

      1. add-sqr-sqrt 90.0%

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

      2. sqrt-unprod 90.8%

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

      3. *-commutative 90.8%

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

      4. *-commutative 90.8%

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

      5. swap-sqr 90.8%

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

      6. add-sqr-sqrt 90.8%

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

      7. *-commutative 90.8%

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

      8. metadata-eval 90.8%

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

   6. Applied egg-rr 90.8%

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

      1. associate-*l* 90.8%

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

      2. hypot-undefine 46.4%

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

      3. unpow2 46.4%

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

      4. unpow2 46.4%

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

      5. +-commutative 46.4%

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

      6. unpow2 46.4%

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

      7. unpow2 46.4%

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

      8. hypot-undefine 90.8%

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

      9. metadata-eval 90.8%

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

   8. Simplified 90.8%

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

4. Final simplification 92.0%

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

Alternative 2: 75.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{if}\;re \leq -1.6 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq -9.5 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -1.05 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

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 -1.6e+96)
     t_1
     (if (<= re -9.5e+32)
       t_0
       (if (<= re -1.05e-51)
         t_1
         (if (<= re 1.12e+15) 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 <= -1.6e+96) {
		tmp = t_1;
	} else if (re <= -9.5e+32) {
		tmp = t_0;
	} else if (re <= -1.05e-51) {
		tmp = t_1;
	} else if (re <= 1.12e+15) {
		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 <= (-1.6d+96)) then
        tmp = t_1
    else if (re <= (-9.5d+32)) then
        tmp = t_0
    else if (re <= (-1.05d-51)) then
        tmp = t_1
    else if (re <= 1.12d+15) 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 <= -1.6e+96) {
		tmp = t_1;
	} else if (re <= -9.5e+32) {
		tmp = t_0;
	} else if (re <= -1.05e-51) {
		tmp = t_1;
	} else if (re <= 1.12e+15) {
		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 <= -1.6e+96:
		tmp = t_1
	elif re <= -9.5e+32:
		tmp = t_0
	elif re <= -1.05e-51:
		tmp = t_1
	elif re <= 1.12e+15:
		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 <= -1.6e+96)
		tmp = t_1;
	elseif (re <= -9.5e+32)
		tmp = t_0;
	elseif (re <= -1.05e-51)
		tmp = t_1;
	elseif (re <= 1.12e+15)
		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 <= -1.6e+96)
		tmp = t_1;
	elseif (re <= -9.5e+32)
		tmp = t_0;
	elseif (re <= -1.05e-51)
		tmp = t_1;
	elseif (re <= 1.12e+15)
		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, -1.6e+96], t$95$1, If[LessEqual[re, -9.5e+32], t$95$0, If[LessEqual[re, -1.05e-51], t$95$1, If[LessEqual[re, 1.12e+15], t$95$0, N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.60000000000000003e96 or -9.50000000000000006e32 < re < -1.05000000000000001e-51

   1. Initial program 45.7%

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

   3. Taylor expanded in re around -inf 79.0%

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

      1. *-commutative 79.0%

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

   5. Simplified 79.0%

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

   if -1.60000000000000003e96 < re < -9.50000000000000006e32 or -1.05000000000000001e-51 < re < 1.12e15

   1. Initial program 54.9%

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

   3. Taylor expanded in re around 0 77.9%

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

   if 1.12e15 < re

   1. Initial program 10.6%

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

      1. sub-neg 10.6%

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

      2. sqr-neg 10.6%

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

      3. sub-neg 10.6%

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

      4. sqr-neg 10.6%

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

      5. hypot-define 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in im around 0 80.6%

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

      1. associate-*r* 80.6%

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

      2. *-commutative 80.6%

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

   7. Simplified 80.6%

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

      1. sqrt-div 80.6%

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

      2. metadata-eval 80.6%

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

      3. un-div-inv 80.5%

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

      4. *-commutative 80.5%

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

      5. sqrt-unprod 81.4%

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

      6. metadata-eval 81.4%

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

      7. metadata-eval 81.4%

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

      8. *-rgt-identity 81.4%

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

   9. Applied egg-rr 81.4%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq -9.5 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -1.05 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.6 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.6e-51)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 4.5e+14) (* 0.5 (sqrt (* 2.0 im))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.6e-51) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 4.5e+14) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= (-9.6d-51)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 4.5d+14) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= -9.6e-51) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 4.5e+14) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.6e-51:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 4.5e+14:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.6e-51)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 4.5e+14)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= -9.6e-51)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 4.5e+14)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.6e-51], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.5e+14], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -9.6000000000000001e-51

   1. Initial program 47.2%

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

   3. Taylor expanded in re around -inf 72.4%

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

      1. *-commutative 72.4%

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

   5. Simplified 72.4%

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

   if -9.6000000000000001e-51 < re < 4.5e14

   1. Initial program 54.8%

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

   3. Taylor expanded in re around 0 76.8%

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

      1. add0 76.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im} \cdot \sqrt{2} + 0\right)} \]

      2. sqrt-unprod 77.4%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{im \cdot 2}} + 0\right) \]

   5. Applied egg-rr 77.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im \cdot 2} + 0\right)} \]
   6. Step-by-step derivation

      1. add0 77.4%

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

   7. Simplified 77.4%

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

   if 4.5e14 < re

   1. Initial program 10.6%

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

      1. sub-neg 10.6%

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

      2. sqr-neg 10.6%

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

      3. sub-neg 10.6%

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

      4. sqr-neg 10.6%

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

      5. hypot-define 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in im around 0 80.6%

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

      1. associate-*r* 80.6%

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

      2. *-commutative 80.6%

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

   7. Simplified 80.6%

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

      1. sqrt-div 80.6%

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

      2. metadata-eval 80.6%

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

      3. un-div-inv 80.5%

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

      4. *-commutative 80.5%

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

      5. sqrt-unprod 81.4%

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

      6. metadata-eval 81.4%

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

      7. metadata-eval 81.4%

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

      8. *-rgt-identity 81.4%

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

   9. Applied egg-rr 81.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.6 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8.8 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8.8e+14) (* 0.5 (sqrt (* 2.0 im))) (* im (/ 0.5 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8.8e+14) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= 8.8d+14) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= 8.8e+14) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8.8e+14:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8.8e+14)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 8.8e+14)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8.8e+14], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.8e14

   1. Initial program 52.1%

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

   3. Taylor expanded in re around 0 60.7%

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

      1. add0 60.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im} \cdot \sqrt{2} + 0\right)} \]

      2. sqrt-unprod 61.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{im \cdot 2}} + 0\right) \]

   5. Applied egg-rr 61.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im \cdot 2} + 0\right)} \]
   6. Step-by-step derivation

      1. add0 61.1%

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

   7. Simplified 61.1%

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

   if 8.8e14 < re

   1. Initial program 10.6%

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

      1. sub-neg 10.6%

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

      2. sqr-neg 10.6%

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

      3. sub-neg 10.6%

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

      4. sqr-neg 10.6%

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

      5. hypot-define 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in im around 0 80.6%

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

      1. associate-*r* 80.6%

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

      2. *-commutative 80.6%

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

   7. Simplified 80.6%

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

      1. add0 80.6%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}} + 0} \]

      2. *-commutative 80.6%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right)\right)} + 0 \]

      3. fma-define 80.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{re}}, 0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right), 0\right)} \]

      4. inv-pow 80.6%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{{re}^{-1}}}, 0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right), 0\right) \]

      5. sqrt-pow1 80.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{\left(\frac{-1}{2}\right)}}, 0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right), 0\right) \]

      6. metadata-eval 80.5%

         \[\leadsto \mathsf{fma}\left({re}^{\color{blue}{-0.5}}, 0.5 \cdot \left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right), 0\right) \]

      7. *-commutative 80.5%

         \[\leadsto \mathsf{fma}\left({re}^{-0.5}, \color{blue}{\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot 0.5}, 0\right) \]

      8. sqrt-unprod 81.4%

         \[\leadsto \mathsf{fma}\left({re}^{-0.5}, \left(im \cdot \color{blue}{\sqrt{2 \cdot 0.5}}\right) \cdot 0.5, 0\right) \]

      9. metadata-eval 81.4%

         \[\leadsto \mathsf{fma}\left({re}^{-0.5}, \left(im \cdot \sqrt{\color{blue}{1}}\right) \cdot 0.5, 0\right) \]

      10. metadata-eval 81.4%

          \[\leadsto \mathsf{fma}\left({re}^{-0.5}, \left(im \cdot \color{blue}{1}\right) \cdot 0.5, 0\right) \]

      11. *-rgt-identity 81.4%

          \[\leadsto \mathsf{fma}\left({re}^{-0.5}, \color{blue}{im} \cdot 0.5, 0\right) \]

   9. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{-0.5}, im \cdot 0.5, 0\right)} \]
   10. Step-by-step derivation

       1. fma-undefine 81.4%

          \[\leadsto \color{blue}{{re}^{-0.5} \cdot \left(im \cdot 0.5\right) + 0} \]

       2. +-rgt-identity 81.4%

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

       3. *-commutative 81.4%

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

   11. Simplified 81.4%

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

       1. add0 81.4%

          \[\leadsto \color{blue}{{re}^{-0.5} \cdot \left(0.5 \cdot im\right) + 0} \]

       2. *-commutative 81.4%

          \[\leadsto \color{blue}{\left(0.5 \cdot im\right) \cdot {re}^{-0.5}} + 0 \]

       3. metadata-eval 81.4%

          \[\leadsto \left(0.5 \cdot im\right) \cdot {re}^{\color{blue}{\left(-0.5\right)}} + 0 \]

       4. pow-flip 81.3%

          \[\leadsto \left(0.5 \cdot im\right) \cdot \color{blue}{\frac{1}{{re}^{0.5}}} + 0 \]

       5. pow1/2 81.3%

          \[\leadsto \left(0.5 \cdot im\right) \cdot \frac{1}{\color{blue}{\sqrt{re}}} + 0 \]

       6. div-inv 81.4%

          \[\leadsto \color{blue}{\frac{0.5 \cdot im}{\sqrt{re}}} + 0 \]

       7. *-commutative 81.4%

          \[\leadsto \frac{\color{blue}{im \cdot 0.5}}{\sqrt{re}} + 0 \]

       8. associate-/l* 81.3%

          \[\leadsto \color{blue}{im \cdot \frac{0.5}{\sqrt{re}}} + 0 \]

   13. Applied egg-rr 81.3%

       \[\leadsto \color{blue}{im \cdot \frac{0.5}{\sqrt{re}} + 0} \]
   14. Step-by-step derivation

       1. add0 81.3%

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

   15. Simplified 81.3%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8.8 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8.5 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8.5e+14) (* 0.5 (sqrt (* 2.0 im))) (/ (* im 0.5) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8.5e+14) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= 8.5d+14) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= 8.5e+14) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8.5e+14:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8.5e+14)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= 8.5e+14)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8.5e+14], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.5e14

   1. Initial program 52.1%

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

   3. Taylor expanded in re around 0 60.7%

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

      1. add0 60.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im} \cdot \sqrt{2} + 0\right)} \]

      2. sqrt-unprod 61.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{im \cdot 2}} + 0\right) \]

   5. Applied egg-rr 61.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im \cdot 2} + 0\right)} \]
   6. Step-by-step derivation

      1. add0 61.1%

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

   7. Simplified 61.1%

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

   if 8.5e14 < re

   1. Initial program 10.6%

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

      1. sub-neg 10.6%

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

      2. sqr-neg 10.6%

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

      3. sub-neg 10.6%

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

      4. sqr-neg 10.6%

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

      5. hypot-define 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in im around 0 80.6%

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

      1. associate-*r* 80.6%

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

      2. *-commutative 80.6%

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

   7. Simplified 80.6%

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

      1. sqrt-div 80.6%

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

      2. metadata-eval 80.6%

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

      3. un-div-inv 80.5%

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

      4. *-commutative 80.5%

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

      5. sqrt-unprod 81.4%

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

      6. metadata-eval 81.4%

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

      7. metadata-eval 81.4%

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

      8. *-rgt-identity 81.4%

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

   9. Applied egg-rr 81.4%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8.5 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.0%

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

3. Taylor expanded in re around 0 51.5%

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

   1. add0 51.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im} \cdot \sqrt{2} + 0\right)} \]

   2. sqrt-unprod 51.9%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{im \cdot 2}} + 0\right) \]

5. Applied egg-rr 51.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{im \cdot 2} + 0\right)} \]
6. Step-by-step derivation

   1. add0 51.9%

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

7. Simplified 51.9%

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

8. Final simplification 51.9%

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

Reproduce

herbie shell --seed 2024046 
(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)))))