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

Percentage Accurate: 41.2% → 90.3%

Time: 8.1s

Alternatives: 7

Speedup: 2.1×

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: 41.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

def code(re, im):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = math.sqrt((0.5 * (math.hypot(re, im) - 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(im / sqrt(re)));
	else
		tmp = sqrt(Float64(0.5 * Float64(hypot(re, im) - 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 * (im / sqrt(re));
	else
		tmp = sqrt((0.5 * (hypot(re, im) - 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[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 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 6.0%

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

      1. sqr-neg 6.0%

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

      2. sqr-neg 6.0%

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

      3. hypot-def 15.2%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in re around inf 48.6%

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

      1. unpow2 48.6%

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

      2. associate-/l* 60.6%

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

   6. Simplified 60.6%

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

      1. div-inv 60.7%

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

      2. clear-num 60.7%

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

      3. *-un-lft-identity 60.7%

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

      4. add-sqr-sqrt 60.5%

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

      5. times-frac 60.6%

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

      6. metadata-eval 60.6%

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

      7. sqrt-div 60.6%

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

      8. un-div-inv 60.6%

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

      9. metadata-eval 60.6%

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

      10. sqrt-div 60.5%

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

      11. associate-*l* 60.5%

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

      12. sqrt-unprod 91.2%

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

      13. add-sqr-sqrt 91.7%

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

      14. expm1-log1p-u 91.2%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      15. expm1-udef 23.8%

          \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \sqrt{\frac{1}{re}}\right)} - 1\right)} \]

      16. sqrt-div 23.8%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}\right)} - 1\right) \]

      17. metadata-eval 23.8%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \frac{\color{blue}{1}}{\sqrt{re}}\right)} - 1\right) \]

      18. un-div-inv 23.8%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{im}{\sqrt{re}}}\right)} - 1\right) \]

   8. Applied egg-rr 23.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 91.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 91.8%

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

   10. Simplified 91.8%

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

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

   1. Initial program 42.2%

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

      1. *-commutative 42.2%

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

      2. hypot-udef 90.2%

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

      3. *-commutative 90.2%

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

      4. add-sqr-sqrt 89.4%

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

      5. sqrt-unprod 90.2%

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

      6. *-commutative 90.2%

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

      7. *-commutative 90.2%

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

      8. swap-sqr 90.2%

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

      9. add-sqr-sqrt 90.2%

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

      10. metadata-eval 90.2%

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

   3. Applied egg-rr 90.2%

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

      1. *-commutative 90.2%

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

      2. associate-*r* 90.2%

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

      3. metadata-eval 90.2%

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

   5. Simplified 90.2%

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

4. Final simplification 90.5%

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

Alternative 2: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := \sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -1.8 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -9.2 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -2.9 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 0.016:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (sqrt (* 0.5 (- im re)))))
   (if (<= re -1.8e+152)
     t_0
     (if (<= re -9.2e+103)
       t_1
       (if (<= re -2.9e+41)
         t_0
         (if (<= re 0.016) t_1 (* (pow re -0.5) (* im 0.5))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -1.8e+152) {
		tmp = t_0;
	} else if (re <= -9.2e+103) {
		tmp = t_1;
	} else if (re <= -2.9e+41) {
		tmp = t_0;
	} else if (re <= 0.016) {
		tmp = t_1;
	} else {
		tmp = pow(re, -0.5) * (im * 0.5);
	}
	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((re * (-4.0d0)))
    t_1 = sqrt((0.5d0 * (im - re)))
    if (re <= (-1.8d+152)) then
        tmp = t_0
    else if (re <= (-9.2d+103)) then
        tmp = t_1
    else if (re <= (-2.9d+41)) then
        tmp = t_0
    else if (re <= 0.016d0) then
        tmp = t_1
    else
        tmp = (re ** (-0.5d0)) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = Math.sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -1.8e+152) {
		tmp = t_0;
	} else if (re <= -9.2e+103) {
		tmp = t_1;
	} else if (re <= -2.9e+41) {
		tmp = t_0;
	} else if (re <= 0.016) {
		tmp = t_1;
	} else {
		tmp = Math.pow(re, -0.5) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = math.sqrt((0.5 * (im - re)))
	tmp = 0
	if re <= -1.8e+152:
		tmp = t_0
	elif re <= -9.2e+103:
		tmp = t_1
	elif re <= -2.9e+41:
		tmp = t_0
	elif re <= 0.016:
		tmp = t_1
	else:
		tmp = math.pow(re, -0.5) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = sqrt(Float64(0.5 * Float64(im - re)))
	tmp = 0.0
	if (re <= -1.8e+152)
		tmp = t_0;
	elseif (re <= -9.2e+103)
		tmp = t_1;
	elseif (re <= -2.9e+41)
		tmp = t_0;
	elseif (re <= 0.016)
		tmp = t_1;
	else
		tmp = Float64((re ^ -0.5) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = sqrt((0.5 * (im - re)));
	tmp = 0.0;
	if (re <= -1.8e+152)
		tmp = t_0;
	elseif (re <= -9.2e+103)
		tmp = t_1;
	elseif (re <= -2.9e+41)
		tmp = t_0;
	elseif (re <= 0.016)
		tmp = t_1;
	else
		tmp = (re ^ -0.5) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[re, -1.8e+152], t$95$0, If[LessEqual[re, -9.2e+103], t$95$1, If[LessEqual[re, -2.9e+41], t$95$0, If[LessEqual[re, 0.016], t$95$1, N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.7999999999999999e152 or -9.20000000000000034e103 < re < -2.89999999999999988e41

   1. Initial program 23.4%

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

      1. sqr-neg 23.4%

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

      2. sqr-neg 23.4%

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

      3. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around -inf 91.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   5. Step-by-step derivation

      1. *-commutative 91.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   6. Simplified 91.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.7999999999999999e152 < re < -9.20000000000000034e103 or -2.89999999999999988e41 < re < 0.016

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

      2. hypot-udef 89.7%

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

      3. *-commutative 89.7%

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

      4. add-sqr-sqrt 88.8%

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

      5. sqrt-unprod 89.7%

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

      6. *-commutative 89.7%

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

      7. *-commutative 89.7%

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

      8. swap-sqr 89.7%

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

      9. add-sqr-sqrt 89.7%

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

      10. metadata-eval 89.7%

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

   3. Applied egg-rr 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*r* 89.7%

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

      3. metadata-eval 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in re around 0 75.9%

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

      1. neg-mul-1 75.9%

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

      2. sub-neg 75.9%

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

   8. Simplified 75.9%

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

   if 0.016 < re

   1. Initial program 13.9%

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

      1. sqr-neg 13.9%

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

      2. sqr-neg 13.9%

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

      3. hypot-def 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in re around inf 48.9%

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

      1. unpow2 48.9%

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

      2. associate-/l* 59.3%

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

   6. Simplified 59.3%

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

      1. add-sqr-sqrt 59.1%

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

      2. pow2 59.1%

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

   8. Applied egg-rr 76.0%

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

      1. unpow2 76.0%

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

      2. add-sqr-sqrt 76.3%

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

      3. associate-*r/ 76.3%

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

      4. *-commutative 76.3%

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

      5. *-un-lft-identity 76.3%

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

      6. associate-*l/ 76.2%

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

      7. pow1/2 76.2%

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

      8. pow-flip 76.3%

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

      9. metadata-eval 76.3%

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

   10. Applied egg-rr 76.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -9.2 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -2.9 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 0.016:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := \sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -1.8 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.45 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -2.9 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (sqrt (* 0.5 (- im re)))))
   (if (<= re -1.8e+152)
     t_0
     (if (<= re -1.45e+103)
       t_1
       (if (<= re -2.9e+41)
         t_0
         (if (<= re 8e-5) t_1 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -1.8e+152) {
		tmp = t_0;
	} else if (re <= -1.45e+103) {
		tmp = t_1;
	} else if (re <= -2.9e+41) {
		tmp = t_0;
	} else if (re <= 8e-5) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / 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((re * (-4.0d0)))
    t_1 = sqrt((0.5d0 * (im - re)))
    if (re <= (-1.8d+152)) then
        tmp = t_0
    else if (re <= (-1.45d+103)) then
        tmp = t_1
    else if (re <= (-2.9d+41)) then
        tmp = t_0
    else if (re <= 8d-5) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = Math.sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -1.8e+152) {
		tmp = t_0;
	} else if (re <= -1.45e+103) {
		tmp = t_1;
	} else if (re <= -2.9e+41) {
		tmp = t_0;
	} else if (re <= 8e-5) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = math.sqrt((0.5 * (im - re)))
	tmp = 0
	if re <= -1.8e+152:
		tmp = t_0
	elif re <= -1.45e+103:
		tmp = t_1
	elif re <= -2.9e+41:
		tmp = t_0
	elif re <= 8e-5:
		tmp = t_1
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = sqrt(Float64(0.5 * Float64(im - re)))
	tmp = 0.0
	if (re <= -1.8e+152)
		tmp = t_0;
	elseif (re <= -1.45e+103)
		tmp = t_1;
	elseif (re <= -2.9e+41)
		tmp = t_0;
	elseif (re <= 8e-5)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = sqrt((0.5 * (im - re)));
	tmp = 0.0;
	if (re <= -1.8e+152)
		tmp = t_0;
	elseif (re <= -1.45e+103)
		tmp = t_1;
	elseif (re <= -2.9e+41)
		tmp = t_0;
	elseif (re <= 8e-5)
		tmp = t_1;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[re, -1.8e+152], t$95$0, If[LessEqual[re, -1.45e+103], t$95$1, If[LessEqual[re, -2.9e+41], t$95$0, If[LessEqual[re, 8e-5], t$95$1, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.7999999999999999e152 or -1.4499999999999999e103 < re < -2.89999999999999988e41

   1. Initial program 23.4%

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

      1. sqr-neg 23.4%

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

      2. sqr-neg 23.4%

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

      3. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around -inf 91.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   5. Step-by-step derivation

      1. *-commutative 91.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   6. Simplified 91.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.7999999999999999e152 < re < -1.4499999999999999e103 or -2.89999999999999988e41 < re < 8.00000000000000065e-5

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

      2. hypot-udef 89.7%

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

      3. *-commutative 89.7%

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

      4. add-sqr-sqrt 88.8%

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

      5. sqrt-unprod 89.7%

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

      6. *-commutative 89.7%

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

      7. *-commutative 89.7%

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

      8. swap-sqr 89.7%

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

      9. add-sqr-sqrt 89.7%

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

      10. metadata-eval 89.7%

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

   3. Applied egg-rr 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*r* 89.7%

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

      3. metadata-eval 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in re around 0 75.9%

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

      1. neg-mul-1 75.9%

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

      2. sub-neg 75.9%

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

   8. Simplified 75.9%

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

   if 8.00000000000000065e-5 < re

   1. Initial program 13.9%

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

      1. sqr-neg 13.9%

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

      2. sqr-neg 13.9%

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

      3. hypot-def 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in re around inf 48.9%

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

      1. unpow2 48.9%

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

      2. associate-/l* 59.3%

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

   6. Simplified 59.3%

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

      1. div-inv 59.4%

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

      2. clear-num 59.4%

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

      3. *-un-lft-identity 59.4%

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

      4. add-sqr-sqrt 59.2%

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

      5. times-frac 59.3%

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

      6. metadata-eval 59.3%

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

      7. sqrt-div 59.4%

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

      8. un-div-inv 59.4%

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

      9. metadata-eval 59.4%

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

      10. sqrt-div 59.3%

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

      11. associate-*l* 59.3%

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

      12. sqrt-unprod 75.9%

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

      13. add-sqr-sqrt 76.2%

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

      14. expm1-log1p-u 75.1%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      15. expm1-udef 36.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \sqrt{\frac{1}{re}}\right)} - 1\right)} \]

      16. sqrt-div 36.7%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}\right)} - 1\right) \]

      17. metadata-eval 36.7%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \frac{\color{blue}{1}}{\sqrt{re}}\right)} - 1\right) \]

      18. un-div-inv 36.7%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{im}{\sqrt{re}}}\right)} - 1\right) \]

   8. Applied egg-rr 36.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 75.1%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 76.3%

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

   10. Simplified 76.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.45 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -2.9 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := \sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -1.8 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.36 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -2.9 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 0.00038:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (sqrt (* 0.5 (- im re)))))
   (if (<= re -1.8e+152)
     t_0
     (if (<= re -1.36e+104)
       t_1
       (if (<= re -2.9e+41)
         t_0
         (if (<= re 0.00038) t_1 (/ (* im 0.5) (sqrt re))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -1.8e+152) {
		tmp = t_0;
	} else if (re <= -1.36e+104) {
		tmp = t_1;
	} else if (re <= -2.9e+41) {
		tmp = t_0;
	} else if (re <= 0.00038) {
		tmp = t_1;
	} 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((re * (-4.0d0)))
    t_1 = sqrt((0.5d0 * (im - re)))
    if (re <= (-1.8d+152)) then
        tmp = t_0
    else if (re <= (-1.36d+104)) then
        tmp = t_1
    else if (re <= (-2.9d+41)) then
        tmp = t_0
    else if (re <= 0.00038d0) then
        tmp = t_1
    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((re * -4.0));
	double t_1 = Math.sqrt((0.5 * (im - re)));
	double tmp;
	if (re <= -1.8e+152) {
		tmp = t_0;
	} else if (re <= -1.36e+104) {
		tmp = t_1;
	} else if (re <= -2.9e+41) {
		tmp = t_0;
	} else if (re <= 0.00038) {
		tmp = t_1;
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = math.sqrt((0.5 * (im - re)))
	tmp = 0
	if re <= -1.8e+152:
		tmp = t_0
	elif re <= -1.36e+104:
		tmp = t_1
	elif re <= -2.9e+41:
		tmp = t_0
	elif re <= 0.00038:
		tmp = t_1
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = sqrt(Float64(0.5 * Float64(im - re)))
	tmp = 0.0
	if (re <= -1.8e+152)
		tmp = t_0;
	elseif (re <= -1.36e+104)
		tmp = t_1;
	elseif (re <= -2.9e+41)
		tmp = t_0;
	elseif (re <= 0.00038)
		tmp = t_1;
	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((re * -4.0));
	t_1 = sqrt((0.5 * (im - re)));
	tmp = 0.0;
	if (re <= -1.8e+152)
		tmp = t_0;
	elseif (re <= -1.36e+104)
		tmp = t_1;
	elseif (re <= -2.9e+41)
		tmp = t_0;
	elseif (re <= 0.00038)
		tmp = t_1;
	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[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[re, -1.8e+152], t$95$0, If[LessEqual[re, -1.36e+104], t$95$1, If[LessEqual[re, -2.9e+41], t$95$0, If[LessEqual[re, 0.00038], t$95$1, N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.7999999999999999e152 or -1.3599999999999999e104 < re < -2.89999999999999988e41

   1. Initial program 23.4%

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

      1. sqr-neg 23.4%

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

      2. sqr-neg 23.4%

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

      3. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around -inf 91.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   5. Step-by-step derivation

      1. *-commutative 91.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   6. Simplified 91.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.7999999999999999e152 < re < -1.3599999999999999e104 or -2.89999999999999988e41 < re < 3.8000000000000002e-4

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

      2. hypot-udef 89.7%

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

      3. *-commutative 89.7%

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

      4. add-sqr-sqrt 88.8%

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

      5. sqrt-unprod 89.7%

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

      6. *-commutative 89.7%

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

      7. *-commutative 89.7%

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

      8. swap-sqr 89.7%

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

      9. add-sqr-sqrt 89.7%

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

      10. metadata-eval 89.7%

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

   3. Applied egg-rr 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*r* 89.7%

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

      3. metadata-eval 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in re around 0 75.9%

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

      1. neg-mul-1 75.9%

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

      2. sub-neg 75.9%

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

   8. Simplified 75.9%

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

   if 3.8000000000000002e-4 < re

   1. Initial program 13.9%

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

      1. sqr-neg 13.9%

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

      2. sqr-neg 13.9%

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

      3. hypot-def 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in re around inf 48.9%

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

      1. unpow2 48.9%

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

      2. associate-/l* 59.3%

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

   6. Simplified 59.3%

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

      1. div-inv 59.4%

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

      2. clear-num 59.4%

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

      3. *-un-lft-identity 59.4%

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

      4. add-sqr-sqrt 59.2%

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

      5. times-frac 59.3%

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

      6. metadata-eval 59.3%

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

      7. sqrt-div 59.4%

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

      8. un-div-inv 59.4%

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

      9. metadata-eval 59.4%

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

      10. sqrt-div 59.3%

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

      11. associate-*l* 59.3%

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

      12. sqrt-unprod 75.9%

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

      13. add-sqr-sqrt 76.2%

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

      14. expm1-log1p-u 75.1%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      15. expm1-udef 36.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \sqrt{\frac{1}{re}}\right)} - 1\right)} \]

      16. sqrt-div 36.7%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}\right)} - 1\right) \]

      17. metadata-eval 36.7%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(im \cdot \frac{\color{blue}{1}}{\sqrt{re}}\right)} - 1\right) \]

      18. un-div-inv 36.7%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{im}{\sqrt{re}}}\right)} - 1\right) \]

   8. Applied egg-rr 36.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 75.1%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 76.3%

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

   10. Simplified 76.3%

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

       1. associate-*r/ 76.3%

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

       2. *-commutative 76.3%

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

   12. Applied egg-rr 76.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.36 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -2.9 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 0.00038:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 63.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{if}\;re \leq -1.8 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.8 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -4.75 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))))
   (if (<= re -1.8e+152)
     t_0
     (if (<= re -1.8e+104)
       (sqrt (* 0.5 (- im re)))
       (if (<= re -4.75e+33) t_0 (sqrt (* im 0.5)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double tmp;
	if (re <= -1.8e+152) {
		tmp = t_0;
	} else if (re <= -1.8e+104) {
		tmp = sqrt((0.5 * (im - re)));
	} else if (re <= -4.75e+33) {
		tmp = t_0;
	} else {
		tmp = sqrt((im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    if (re <= (-1.8d+152)) then
        tmp = t_0
    else if (re <= (-1.8d+104)) then
        tmp = sqrt((0.5d0 * (im - re)))
    else if (re <= (-4.75d+33)) then
        tmp = t_0
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double tmp;
	if (re <= -1.8e+152) {
		tmp = t_0;
	} else if (re <= -1.8e+104) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else if (re <= -4.75e+33) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	tmp = 0
	if re <= -1.8e+152:
		tmp = t_0
	elif re <= -1.8e+104:
		tmp = math.sqrt((0.5 * (im - re)))
	elif re <= -4.75e+33:
		tmp = t_0
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	tmp = 0.0
	if (re <= -1.8e+152)
		tmp = t_0;
	elseif (re <= -1.8e+104)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	elseif (re <= -4.75e+33)
		tmp = t_0;
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	tmp = 0.0;
	if (re <= -1.8e+152)
		tmp = t_0;
	elseif (re <= -1.8e+104)
		tmp = sqrt((0.5 * (im - re)));
	elseif (re <= -4.75e+33)
		tmp = t_0;
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.8e+152], t$95$0, If[LessEqual[re, -1.8e+104], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[re, -4.75e+33], t$95$0, N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.7999999999999999e152 or -1.8e104 < re < -4.7500000000000002e33

   1. Initial program 28.6%

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

      1. sqr-neg 28.6%

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

      2. sqr-neg 28.6%

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

      3. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around -inf 87.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   5. Step-by-step derivation

      1. *-commutative 87.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   6. Simplified 87.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.7999999999999999e152 < re < -1.8e104

   1. Initial program 17.9%

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

      1. *-commutative 17.9%

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

      2. hypot-udef 100.0%

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

      3. *-commutative 100.0%

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

      4. add-sqr-sqrt 99.3%

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

      5. sqrt-unprod 100.0%

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

      6. *-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. swap-sqr 100.0%

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

      9. add-sqr-sqrt 100.0%

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

      10. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 88.7%

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

      1. neg-mul-1 88.7%

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

      2. sub-neg 88.7%

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

   8. Simplified 88.7%

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

   if -4.7500000000000002e33 < re

   1. Initial program 38.1%

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

      1. *-commutative 38.1%

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

      2. hypot-udef 69.9%

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

      3. *-commutative 69.9%

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

      4. add-sqr-sqrt 69.4%

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

      5. sqrt-unprod 69.9%

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

      6. *-commutative 69.9%

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

      7. *-commutative 69.9%

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

      8. swap-sqr 69.9%

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

      9. add-sqr-sqrt 69.9%

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

      10. metadata-eval 69.9%

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

   3. Applied egg-rr 69.9%

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

      1. *-commutative 69.9%

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

      2. associate-*r* 69.9%

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

      3. metadata-eval 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in re around 0 58.0%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.8 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -4.75 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]

Alternative 6: 55.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.6 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.6e-303) (sqrt (* 0.5 (- im re))) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.6e-303) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = sqrt((im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.6d-303)) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.6e-303) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.6e-303:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.6e-303)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.6e-303)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.6e-303], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.60000000000000005e-303

   1. Initial program 46.0%

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

      1. *-commutative 46.0%

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

      2. hypot-udef 100.0%

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

      3. *-commutative 100.0%

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

      4. add-sqr-sqrt 99.1%

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

      5. sqrt-unprod 100.0%

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

      6. *-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. swap-sqr 100.0%

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

      9. add-sqr-sqrt 100.0%

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

      10. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 59.5%

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

      1. neg-mul-1 59.5%

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

      2. sub-neg 59.5%

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

   8. Simplified 59.5%

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

   if -2.60000000000000005e-303 < re

   1. Initial program 25.7%

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

      1. *-commutative 25.7%

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

      2. hypot-udef 54.2%

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

      3. *-commutative 54.2%

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

      4. add-sqr-sqrt 53.8%

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

      5. sqrt-unprod 54.2%

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

      6. *-commutative 54.2%

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

      7. *-commutative 54.2%

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

      8. swap-sqr 54.2%

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

      9. add-sqr-sqrt 54.2%

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

      10. metadata-eval 54.2%

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

   3. Applied egg-rr 54.2%

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

      1. *-commutative 54.2%

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

      2. associate-*r* 54.2%

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

      3. metadata-eval 54.2%

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

   5. Simplified 54.2%

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

   6. Taylor expanded in re around 0 49.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.6 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]

Alternative 7: 52.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

   1. *-commutative 35.7%

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

   2. hypot-udef 76.7%

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

   3. *-commutative 76.7%

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

   4. add-sqr-sqrt 76.1%

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

   5. sqrt-unprod 76.7%

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

   6. *-commutative 76.7%

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

   7. *-commutative 76.7%

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

   8. swap-sqr 76.7%

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

   9. add-sqr-sqrt 76.7%

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

   10. metadata-eval 76.7%

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

3. Applied egg-rr 76.7%

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

   1. *-commutative 76.7%

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

   2. associate-*r* 76.7%

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

   3. metadata-eval 76.7%

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

5. Simplified 76.7%

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

6. Taylor expanded in re around 0 50.8%

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

7. Final simplification 50.8%

   \[\leadsto \sqrt{im \cdot 0.5} \]

Reproduce

herbie shell --seed 2023271 
(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)))))