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

Percentage Accurate: 41.7% → 86.6%

Time: 8.1s

Alternatives: 7

Speedup: 2.2×

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: 41.7% 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: 86.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.1 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.1e-91)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (/ (* 0.5 im) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.1e-91) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.1e-91) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.1e-91:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.1e-91)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.1e-91)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.1e-91], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.09999999999999981e-91

   1. Initial program 52.7%

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

      1. lower-hypot.f64 97.9

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

   4. Applied egg-rr 97.9%

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

   if 3.09999999999999981e-91 < re

   1. Initial program 13.3%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 51.3

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

   5. Simplified 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 80.3

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

   7. Applied egg-rr 80.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.1 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.85 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - im \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.8e+69)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re -1.85e-120)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 1.6e-97)
       (* 0.5 (sqrt (* 2.0 (- im (* im (/ re im))))))
       (/ (* 0.5 im) (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.8e+69) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= -1.85e-120) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 1.6e-97) {
		tmp = 0.5 * sqrt((2.0 * (im - (im * (re / im)))));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.8e+69)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= -1.85e-120)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 1.6e-97)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - Float64(im * Float64(re / im))))));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.8e+69], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.85e-120], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.6e-97], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - N[(im * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -5.7999999999999997e69

   1. Initial program 25.1%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 79.8

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

   5. Simplified 79.8%

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

   if -5.7999999999999997e69 < re < -1.85e-120

   1. Initial program 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 80.0

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

   4. Applied egg-rr 80.0%

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

   if -1.85e-120 < re < 1.5999999999999999e-97

   1. Initial program 52.3%

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

   3. Taylor expanded in im around inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 90.6

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

   5. Simplified 90.6%

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

   if 1.5999999999999999e-97 < re

   1. Initial program 13.3%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 51.3

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

   5. Simplified 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 80.3

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

   7. Applied egg-rr 80.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{+69}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.85 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - im \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - im \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.5e-31)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.7e-94)
     (* 0.5 (sqrt (* 2.0 (- im (* im (/ re im))))))
     (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.5e-31) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.7e-94) {
		tmp = 0.5 * sqrt((2.0 * (im - (im * (re / im)))));
	} 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 <= (-2.5d-31)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.7d-94) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - (im * (re / im)))))
    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 <= -2.5e-31) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.7e-94) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - (im * (re / im)))));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.5e-31:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.7e-94:
		tmp = 0.5 * math.sqrt((2.0 * (im - (im * (re / im)))))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.5e-31)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.7e-94)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - Float64(im * Float64(re / im))))));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.5e-31)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.7e-94)
		tmp = 0.5 * sqrt((2.0 * (im - (im * (re / im)))));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.5e-31], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.7e-94], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - N[(im * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.5e-31

   1. Initial program 45.3%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 73.5

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

   5. Simplified 73.5%

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

   if -2.5e-31 < re < 4.70000000000000003e-94

   1. Initial program 57.6%

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

   3. Taylor expanded in im around inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 85.3

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

   5. Simplified 85.3%

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

   if 4.70000000000000003e-94 < re

   1. Initial program 13.3%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 51.3

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

   5. Simplified 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 80.3

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

   7. Applied egg-rr 80.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.7 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - im \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.6 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.6e-32)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.5e-101)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.6e-32) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.5e-101) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} 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 <= (-8.6d-32)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.5d-101) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    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 <= -8.6e-32) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.5e-101) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8.6e-32:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.5e-101:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8.6e-32)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.5e-101)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8.6e-32)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.5e-101)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8.6e-32], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.5e-101], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.5999999999999998e-32

   1. Initial program 45.3%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 73.5

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

   5. Simplified 73.5%

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

   if -8.5999999999999998e-32 < re < 2.5e-101

   1. Initial program 57.6%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 85.3

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

   5. Simplified 85.3%

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

   if 2.5e-101 < re

   1. Initial program 13.3%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 51.3

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

   5. Simplified 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 80.3

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

   7. Applied egg-rr 80.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.6 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.3 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.3e-31)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.1e-94)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.3e-31) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.1e-94) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.3d-31)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.1d-94) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.3e-31) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.1e-94) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.3e-31:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.1e-94:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.3e-31)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.1e-94)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.3e-31)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.1e-94)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.3e-31], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.1e-94], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.3e-31

   1. Initial program 45.3%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 73.5

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

   5. Simplified 73.5%

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

   if -4.3e-31 < re < 4.10000000000000001e-94

   1. Initial program 57.6%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 85.3

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

   5. Simplified 85.3%

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

   if 4.10000000000000001e-94 < re

   1. Initial program 13.3%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 51.3

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

   5. Simplified 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 80.3

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

   7. Applied egg-rr 80.3%

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

      1. lift-sqrt.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 80.2

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

   9. Applied egg-rr 80.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.3 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.8 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.8e-32) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.8e-32) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-9.8d-32)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9.8e-32) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.8e-32:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.8e-32)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9.8e-32)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.8e-32], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -9.7999999999999996e-32

   1. Initial program 45.3%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 73.5

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

   5. Simplified 73.5%

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

   if -9.7999999999999996e-32 < re

   1. Initial program 38.8%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{im \cdot 2}} \]

      2. lower-*.f64 58.1

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

   5. Simplified 58.1%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.8 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.6%

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

3. Taylor expanded in re around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   2. lower-*.f64 50.6

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

5. Simplified 50.6%

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

6. Final simplification 50.6%

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

Reproduce

herbie shell --seed 2024207 
(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)))))