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

Percentage Accurate: 41.5% → 90.3%

Time: 11.3s

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: 41.5% 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

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* 0.5 (* im (pow re -0.5)))
   (* 0.5 (sqrt (* 2.0 (- (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 * pow(re, -0.5));
	} else {
		tmp = 0.5 * sqrt((2.0 * (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.pow(re, -0.5));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (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.pow(re, -0.5))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (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 * (re ^ -0.5)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 * (re ^ -0.5));
	else
		tmp = 0.5 * sqrt((2.0 * (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[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $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 8.8%

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

      1. hypot-udef 18.5%

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

      2. add-cbrt-cube 17.8%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right) \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}} \]

      3. pow1/3 17.3%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right) \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333}} \]

      4. add-sqr-sqrt 17.3%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333} \]

      5. pow1 17.3%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333} \]

      6. pow1/2 17.3%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      7. pow-prod-up 17.3%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      8. metadata-eval 17.3%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 17.3%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. unpow1/3 17.8%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}}} \]

   6. Simplified 17.8%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}}} \]

   7. Taylor expanded in im around 0 92.2%

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

      1. unpow1/2 92.2%

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

      2. exp-to-pow 86.8%

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

      3. *-commutative 86.8%

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

      4. log-rec 86.8%

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

      5. neg-mul-1 86.8%

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

      6. associate-*r* 86.8%

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

      7. metadata-eval 86.8%

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

      8. log-pow 86.8%

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

      9. rem-exp-log 92.2%

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

   9. Simplified 92.2%

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

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

   1. Initial program 45.7%

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

      1. sub-neg 45.7%

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

      2. sqr-neg 45.7%

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

      3. sub-neg 45.7%

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

      4. sqr-neg 45.7%

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

      5. hypot-def 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
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

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

Alternative 2: 74.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq -2.7 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re -2.7e+30)
     (* 0.5 (sqrt (* 2.0 (* re -2.0))))
     (if (<= re 7.5e-54)
       t_0
       (if (<= re 2.2e+32)
         (* 0.5 (* im (pow re -0.5)))
         (if (<= re 1e+70) t_0 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((im * 2.0));
	double tmp;
	if (re <= -2.7e+30) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 7.5e-54) {
		tmp = t_0;
	} else if (re <= 2.2e+32) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else if (re <= 1e+70) {
		tmp = t_0;
	} 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) :: tmp
    t_0 = 0.5d0 * sqrt((im * 2.0d0))
    if (re <= (-2.7d+30)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 7.5d-54) then
        tmp = t_0
    else if (re <= 2.2d+32) then
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    else if (re <= 1d+70) then
        tmp = t_0
    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((im * 2.0));
	double tmp;
	if (re <= -2.7e+30) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 7.5e-54) {
		tmp = t_0;
	} else if (re <= 2.2e+32) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else if (re <= 1e+70) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((im * 2.0))
	tmp = 0
	if re <= -2.7e+30:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 7.5e-54:
		tmp = t_0
	elif re <= 2.2e+32:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	elif re <= 1e+70:
		tmp = t_0
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	tmp = 0.0
	if (re <= -2.7e+30)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 7.5e-54)
		tmp = t_0;
	elseif (re <= 2.2e+32)
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	elseif (re <= 1e+70)
		tmp = t_0;
	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((im * 2.0));
	tmp = 0.0;
	if (re <= -2.7e+30)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 7.5e-54)
		tmp = t_0;
	elseif (re <= 2.2e+32)
		tmp = 0.5 * (im * (re ^ -0.5));
	elseif (re <= 1e+70)
		tmp = t_0;
	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[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.7e+30], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.5e-54], t$95$0, If[LessEqual[re, 2.2e+32], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+70], t$95$0, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.6999999999999999e30

   1. Initial program 35.2%

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

   3. Taylor expanded in re around -inf 83.0%

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

      1. *-commutative 83.0%

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

   5. Simplified 83.0%

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

   if -2.6999999999999999e30 < re < 7.5000000000000005e-54 or 2.20000000000000001e32 < re < 1.00000000000000007e70

   1. Initial program 57.5%

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

      1. hypot-udef 91.4%

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

      2. sub-neg 91.4%

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

      3. +-commutative 91.4%

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

      4. add-cbrt-cube 85.8%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(-\color{blue}{\sqrt[3]{\left(re \cdot re\right) \cdot re}}\right) + \mathsf{hypot}\left(re, im\right)\right)} \]

      5. cbrt-prod 89.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(-\color{blue}{\sqrt[3]{re \cdot re} \cdot \sqrt[3]{re}}\right) + \mathsf{hypot}\left(re, im\right)\right)} \]

      6. distribute-rgt-neg-in 89.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt[3]{re \cdot re} \cdot \left(-\sqrt[3]{re}\right)} + \mathsf{hypot}\left(re, im\right)\right)} \]

      7. fma-def 89.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{re \cdot re}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

      8. cbrt-prod 91.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{\sqrt[3]{re} \cdot \sqrt[3]{re}}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)} \]

      9. pow2 91.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{re}\right)}^{2}}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)} \]

   4. Applied egg-rr 91.3%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{re}\right)}^{2}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

   5. Taylor expanded in re around 0 77.3%

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

   if 7.5000000000000005e-54 < re < 2.20000000000000001e32

   1. Initial program 20.1%

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

      1. hypot-udef 39.3%

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

      2. add-cbrt-cube 24.6%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right) \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}} \]

      3. pow1/3 23.4%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right) \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333}} \]

      4. add-sqr-sqrt 23.4%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333} \]

      5. pow1 23.4%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333} \]

      6. pow1/2 23.4%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      7. pow-prod-up 23.4%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      8. metadata-eval 23.4%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 23.4%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. unpow1/3 24.7%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}}} \]

   6. Simplified 24.7%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}}} \]

   7. Taylor expanded in im around 0 66.7%

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

      1. unpow1/2 66.7%

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

      2. exp-to-pow 64.1%

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

      3. *-commutative 64.1%

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

      4. log-rec 64.1%

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

      5. neg-mul-1 64.1%

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

      6. associate-*r* 64.1%

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

      7. metadata-eval 64.1%

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

      8. log-pow 64.1%

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

      9. rem-exp-log 66.8%

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

   9. Simplified 66.8%

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

   if 1.00000000000000007e70 < re

   1. Initial program 7.8%

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

   3. Taylor expanded in im around 0 81.6%

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

      1. *-commutative 81.6%

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

   5. Simplified 81.6%

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

      1. expm1-log1p-u 81.1%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      2. expm1-udef 32.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} - 1\right)} \]

      3. sqrt-unprod 32.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\left(im \cdot \color{blue}{\sqrt{2 \cdot 0.5}}\right) \cdot \sqrt{\frac{1}{re}}\right)} - 1\right) \]

      4. metadata-eval 32.3%

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

      5. metadata-eval 32.3%

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

      6. *-rgt-identity 32.3%

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

      7. add-sqr-sqrt 32.3%

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

      8. sqrt-prod 27.9%

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

      9. unpow2 27.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{im}^{2}}} \cdot \sqrt{\frac{1}{re}}\right)} - 1\right) \]

      10. sqrt-prod 27.9%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{im}^{2} \cdot \frac{1}{re}}}\right)} - 1\right) \]

      11. div-inv 27.9%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{im}^{2}}{re}}}\right)} - 1\right) \]

      12. sqrt-div 27.9%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{im}^{2}}}{\sqrt{re}}}\right)} - 1\right) \]

      13. unpow2 27.9%

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

      14. sqrt-prod 32.3%

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

      15. add-sqr-sqrt 32.3%

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

   7. Applied egg-rr 32.3%

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

      1. expm1-def 81.9%

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

      2. expm1-log1p 82.5%

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

   9. Simplified 82.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.7e+35)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 3.1e-54)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 6.8e+31)
       (* 0.5 (* im (pow re -0.5)))
       (if (<= re 1.1e+69)
         (* 0.5 (sqrt (* im 2.0)))
         (* 0.5 (/ im (sqrt re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+35) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.1e-54) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 6.8e+31) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else if (re <= 1.1e+69) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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) :: tmp
    if (re <= (-2.7d+35)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 3.1d-54) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 6.8d+31) then
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    else if (re <= 1.1d+69) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+35) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.1e-54) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 6.8e+31) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else if (re <= 1.1e+69) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.7e+35:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 3.1e-54:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 6.8e+31:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	elif re <= 1.1e+69:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.7e+35)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 3.1e-54)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 6.8e+31)
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	elseif (re <= 1.1e+69)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.7e+35)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 3.1e-54)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 6.8e+31)
		tmp = 0.5 * (im * (re ^ -0.5));
	elseif (re <= 1.1e+69)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.7e+35], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.1e-54], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.8e+31], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.1e+69], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -2.70000000000000003e35

   1. Initial program 34.6%

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

   3. Taylor expanded in re around -inf 84.0%

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

      1. *-commutative 84.0%

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

   5. Simplified 84.0%

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

   if -2.70000000000000003e35 < re < 3.10000000000000004e-54

   1. Initial program 57.1%

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

   3. Taylor expanded in re around 0 77.9%

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

   if 3.10000000000000004e-54 < re < 6.7999999999999996e31

   1. Initial program 20.1%

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

      1. hypot-udef 39.3%

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

      2. add-cbrt-cube 24.6%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right) \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}} \]

      3. pow1/3 23.4%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right) \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333}} \]

      4. add-sqr-sqrt 23.4%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333} \]

      5. pow1 23.4%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333} \]

      6. pow1/2 23.4%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      7. pow-prod-up 23.4%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      8. metadata-eval 23.4%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 23.4%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. unpow1/3 24.7%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}}} \]

   6. Simplified 24.7%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}}} \]

   7. Taylor expanded in im around 0 66.7%

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

      1. unpow1/2 66.7%

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

      2. exp-to-pow 64.1%

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

      3. *-commutative 64.1%

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

      4. log-rec 64.1%

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

      5. neg-mul-1 64.1%

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

      6. associate-*r* 64.1%

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

      7. metadata-eval 64.1%

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

      8. log-pow 64.1%

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

      9. rem-exp-log 66.8%

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

   9. Simplified 66.8%

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

   if 6.7999999999999996e31 < re < 1.1000000000000001e69

   1. Initial program 64.4%

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

      1. hypot-udef 96.9%

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

      2. sub-neg 96.9%

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

      3. +-commutative 96.9%

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

      4. add-cbrt-cube 96.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(-\color{blue}{\sqrt[3]{\left(re \cdot re\right) \cdot re}}\right) + \mathsf{hypot}\left(re, im\right)\right)} \]

      5. cbrt-prod 96.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(-\color{blue}{\sqrt[3]{re \cdot re} \cdot \sqrt[3]{re}}\right) + \mathsf{hypot}\left(re, im\right)\right)} \]

      6. distribute-rgt-neg-in 96.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt[3]{re \cdot re} \cdot \left(-\sqrt[3]{re}\right)} + \mathsf{hypot}\left(re, im\right)\right)} \]

      7. fma-def 97.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{re \cdot re}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

      8. cbrt-prod 97.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{\sqrt[3]{re} \cdot \sqrt[3]{re}}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)} \]

      9. pow2 97.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{re}\right)}^{2}}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)} \]

   4. Applied egg-rr 97.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{re}\right)}^{2}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

   5. Taylor expanded in re around 0 85.9%

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

   if 1.1000000000000001e69 < re

   1. Initial program 7.8%

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

   3. Taylor expanded in im around 0 81.6%

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

      1. *-commutative 81.6%

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

   5. Simplified 81.6%

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

      1. expm1-log1p-u 81.1%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      2. expm1-udef 32.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} - 1\right)} \]

      3. sqrt-unprod 32.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\left(im \cdot \color{blue}{\sqrt{2 \cdot 0.5}}\right) \cdot \sqrt{\frac{1}{re}}\right)} - 1\right) \]

      4. metadata-eval 32.3%

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

      5. metadata-eval 32.3%

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

      6. *-rgt-identity 32.3%

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

      7. add-sqr-sqrt 32.3%

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

      8. sqrt-prod 27.9%

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

      9. unpow2 27.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{im}^{2}}} \cdot \sqrt{\frac{1}{re}}\right)} - 1\right) \]

      10. sqrt-prod 27.9%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{im}^{2} \cdot \frac{1}{re}}}\right)} - 1\right) \]

      11. div-inv 27.9%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{im}^{2}}{re}}}\right)} - 1\right) \]

      12. sqrt-div 27.9%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{im}^{2}}}{\sqrt{re}}}\right)} - 1\right) \]

      13. unpow2 27.9%

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

      14. sqrt-prod 32.3%

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

      15. add-sqr-sqrt 32.3%

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

   7. Applied egg-rr 32.3%

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

      1. expm1-def 81.9%

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

      2. expm1-log1p 82.5%

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

   9. Simplified 82.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.4 \cdot 10^{-54} \lor \neg \left(re \leq 5.6 \cdot 10^{+32}\right) \land re \leq 1.85 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 2.4e-54) (and (not (<= re 5.6e+32)) (<= re 1.85e+68)))
   (* 0.5 (sqrt (* im 2.0)))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 2.4e-54) || (!(re <= 5.6e+32) && (re <= 1.85e+68))) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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) :: tmp
    if ((re <= 2.4d-54) .or. (.not. (re <= 5.6d+32)) .and. (re <= 1.85d+68)) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 2.4e-54) || (!(re <= 5.6e+32) && (re <= 1.85e+68))) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 2.4e-54) or (not (re <= 5.6e+32) and (re <= 1.85e+68)):
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 2.4e-54) || (!(re <= 5.6e+32) && (re <= 1.85e+68)))
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 2.4e-54) || (~((re <= 5.6e+32)) && (re <= 1.85e+68)))
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 2.4e-54], And[N[Not[LessEqual[re, 5.6e+32]], $MachinePrecision], LessEqual[re, 1.85e+68]]], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.40000000000000013e-54 or 5.6e32 < re < 1.84999999999999999e68

   1. Initial program 49.9%

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

      1. hypot-udef 94.3%

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

      2. sub-neg 94.3%

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

      3. +-commutative 94.3%

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

      4. add-cbrt-cube 63.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(-\color{blue}{\sqrt[3]{\left(re \cdot re\right) \cdot re}}\right) + \mathsf{hypot}\left(re, im\right)\right)} \]

      5. cbrt-prod 72.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(-\color{blue}{\sqrt[3]{re \cdot re} \cdot \sqrt[3]{re}}\right) + \mathsf{hypot}\left(re, im\right)\right)} \]

      6. distribute-rgt-neg-in 72.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt[3]{re \cdot re} \cdot \left(-\sqrt[3]{re}\right)} + \mathsf{hypot}\left(re, im\right)\right)} \]

      7. fma-def 72.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{re \cdot re}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

      8. cbrt-prod 94.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{\sqrt[3]{re} \cdot \sqrt[3]{re}}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)} \]

      9. pow2 94.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{re}\right)}^{2}}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)} \]

   4. Applied egg-rr 94.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{re}\right)}^{2}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

   5. Taylor expanded in re around 0 59.0%

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

   if 2.40000000000000013e-54 < re < 5.6e32 or 1.84999999999999999e68 < re

   1. Initial program 11.6%

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

   3. Taylor expanded in im around 0 76.7%

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

      1. *-commutative 76.7%

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

   5. Simplified 76.7%

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

      1. expm1-log1p-u 76.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      2. expm1-udef 25.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} - 1\right)} \]

      3. sqrt-unprod 25.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\left(im \cdot \color{blue}{\sqrt{2 \cdot 0.5}}\right) \cdot \sqrt{\frac{1}{re}}\right)} - 1\right) \]

      4. metadata-eval 25.3%

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

      5. metadata-eval 25.3%

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

      6. *-rgt-identity 25.3%

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

      7. add-sqr-sqrt 25.3%

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

      8. sqrt-prod 22.1%

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

      9. unpow2 22.1%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{im}^{2}}} \cdot \sqrt{\frac{1}{re}}\right)} - 1\right) \]

      10. sqrt-prod 22.1%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{im}^{2} \cdot \frac{1}{re}}}\right)} - 1\right) \]

      11. div-inv 22.1%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{im}^{2}}{re}}}\right)} - 1\right) \]

      12. sqrt-div 22.1%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{im}^{2}}}{\sqrt{re}}}\right)} - 1\right) \]

      13. unpow2 22.1%

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

      14. sqrt-prod 25.3%

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

      15. add-sqr-sqrt 25.3%

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

   7. Applied egg-rr 25.3%

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

      1. expm1-def 77.1%

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

      2. expm1-log1p 77.5%

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

   9. Simplified 77.5%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.4 \cdot 10^{-54} \lor \neg \left(re \leq 5.6 \cdot 10^{+32}\right) \land re \leq 1.85 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{if}\;re \leq 5.6 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3.25 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* im 2.0)))))
   (if (<= re 5.6e-54)
     t_0
     (if (<= re 3.25e+31)
       (* 0.5 (* im (pow re -0.5)))
       (if (<= re 4.5e+68) t_0 (* 0.5 (/ im (sqrt re))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((im * 2.0));
	double tmp;
	if (re <= 5.6e-54) {
		tmp = t_0;
	} else if (re <= 3.25e+31) {
		tmp = 0.5 * (im * pow(re, -0.5));
	} else if (re <= 4.5e+68) {
		tmp = t_0;
	} 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) :: tmp
    t_0 = 0.5d0 * sqrt((im * 2.0d0))
    if (re <= 5.6d-54) then
        tmp = t_0
    else if (re <= 3.25d+31) then
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    else if (re <= 4.5d+68) then
        tmp = t_0
    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((im * 2.0));
	double tmp;
	if (re <= 5.6e-54) {
		tmp = t_0;
	} else if (re <= 3.25e+31) {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	} else if (re <= 4.5e+68) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((im * 2.0))
	tmp = 0
	if re <= 5.6e-54:
		tmp = t_0
	elif re <= 3.25e+31:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	elif re <= 4.5e+68:
		tmp = t_0
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(im * 2.0)))
	tmp = 0.0
	if (re <= 5.6e-54)
		tmp = t_0;
	elseif (re <= 3.25e+31)
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	elseif (re <= 4.5e+68)
		tmp = t_0;
	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((im * 2.0));
	tmp = 0.0;
	if (re <= 5.6e-54)
		tmp = t_0;
	elseif (re <= 3.25e+31)
		tmp = 0.5 * (im * (re ^ -0.5));
	elseif (re <= 4.5e+68)
		tmp = t_0;
	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[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 5.6e-54], t$95$0, If[LessEqual[re, 3.25e+31], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.5e+68], t$95$0, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 5.6000000000000004e-54 or 3.2500000000000002e31 < re < 4.5000000000000003e68

   1. Initial program 49.9%

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

      1. hypot-udef 94.3%

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

      2. sub-neg 94.3%

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

      3. +-commutative 94.3%

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

      4. add-cbrt-cube 63.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(-\color{blue}{\sqrt[3]{\left(re \cdot re\right) \cdot re}}\right) + \mathsf{hypot}\left(re, im\right)\right)} \]

      5. cbrt-prod 72.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(-\color{blue}{\sqrt[3]{re \cdot re} \cdot \sqrt[3]{re}}\right) + \mathsf{hypot}\left(re, im\right)\right)} \]

      6. distribute-rgt-neg-in 72.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt[3]{re \cdot re} \cdot \left(-\sqrt[3]{re}\right)} + \mathsf{hypot}\left(re, im\right)\right)} \]

      7. fma-def 72.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{re \cdot re}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

      8. cbrt-prod 94.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{\sqrt[3]{re} \cdot \sqrt[3]{re}}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)} \]

      9. pow2 94.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{re}\right)}^{2}}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)} \]

   4. Applied egg-rr 94.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{re}\right)}^{2}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

   5. Taylor expanded in re around 0 59.0%

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

   if 5.6000000000000004e-54 < re < 3.2500000000000002e31

   1. Initial program 20.1%

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

      1. hypot-udef 39.3%

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

      2. add-cbrt-cube 24.6%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right) \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}} \]

      3. pow1/3 23.4%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right) \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333}} \]

      4. add-sqr-sqrt 23.4%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333} \]

      5. pow1 23.4%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.3333333333333333} \]

      6. pow1/2 23.4%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      7. pow-prod-up 23.4%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      8. metadata-eval 23.4%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 23.4%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. unpow1/3 24.7%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}}} \]

   6. Simplified 24.7%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{1.5}}} \]

   7. Taylor expanded in im around 0 66.7%

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

      1. unpow1/2 66.7%

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

      2. exp-to-pow 64.1%

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

      3. *-commutative 64.1%

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

      4. log-rec 64.1%

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

      5. neg-mul-1 64.1%

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

      6. associate-*r* 64.1%

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

      7. metadata-eval 64.1%

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

      8. log-pow 64.1%

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

      9. rem-exp-log 66.8%

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

   9. Simplified 66.8%

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

   if 4.5000000000000003e68 < re

   1. Initial program 7.8%

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

   3. Taylor expanded in im around 0 81.6%

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

      1. *-commutative 81.6%

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

   5. Simplified 81.6%

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

      1. expm1-log1p-u 81.1%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      2. expm1-udef 32.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(im \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} - 1\right)} \]

      3. sqrt-unprod 32.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\left(im \cdot \color{blue}{\sqrt{2 \cdot 0.5}}\right) \cdot \sqrt{\frac{1}{re}}\right)} - 1\right) \]

      4. metadata-eval 32.3%

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

      5. metadata-eval 32.3%

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

      6. *-rgt-identity 32.3%

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

      7. add-sqr-sqrt 32.3%

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

      8. sqrt-prod 27.9%

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

      9. unpow2 27.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{im}^{2}}} \cdot \sqrt{\frac{1}{re}}\right)} - 1\right) \]

      10. sqrt-prod 27.9%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{im}^{2} \cdot \frac{1}{re}}}\right)} - 1\right) \]

      11. div-inv 27.9%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{im}^{2}}{re}}}\right)} - 1\right) \]

      12. sqrt-div 27.9%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{im}^{2}}}{\sqrt{re}}}\right)} - 1\right) \]

      13. unpow2 27.9%

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

      14. sqrt-prod 32.3%

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

      15. add-sqr-sqrt 32.3%

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

   7. Applied egg-rr 32.3%

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

      1. expm1-def 81.9%

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

      2. expm1-log1p 82.5%

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

   9. Simplified 82.5%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.6 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 3.25 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. hypot-udef 80.3%

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

   2. sub-neg 80.3%

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

   3. +-commutative 80.3%

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

   4. add-cbrt-cube 50.5%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(-\color{blue}{\sqrt[3]{\left(re \cdot re\right) \cdot re}}\right) + \mathsf{hypot}\left(re, im\right)\right)} \]

   5. cbrt-prod 60.5%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(-\color{blue}{\sqrt[3]{re \cdot re} \cdot \sqrt[3]{re}}\right) + \mathsf{hypot}\left(re, im\right)\right)} \]

   6. distribute-rgt-neg-in 60.5%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt[3]{re \cdot re} \cdot \left(-\sqrt[3]{re}\right)} + \mathsf{hypot}\left(re, im\right)\right)} \]

   7. fma-def 60.5%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{re \cdot re}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

   8. cbrt-prod 77.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{\sqrt[3]{re} \cdot \sqrt[3]{re}}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)} \]

   9. pow2 77.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{re}\right)}^{2}}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)} \]

4. Applied egg-rr 77.1%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{re}\right)}^{2}, -\sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}} \]

5. Taylor expanded in re around 0 51.3%

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

6. Final simplification 51.3%

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

Reproduce

herbie shell --seed 2024026 
(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)))))