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

Percentage Accurate: 42.1% → 87.7%

Time: 8.6s

Alternatives: 7

Speedup: 1.7×

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: 42.1% 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: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.8e+51)
   (* (sqrt (* (- (hypot re im) re) 2.0)) 0.5)
   (/ (* im 0.5) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.8e+51) {
		tmp = sqrt(((hypot(re, im) - re) * 2.0)) * 0.5;
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.8e+51) {
		tmp = Math.sqrt(((Math.hypot(re, im) - re) * 2.0)) * 0.5;
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.8e+51:
		tmp = math.sqrt(((math.hypot(re, im) - re) * 2.0)) * 0.5
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.8e+51)
		tmp = Float64(sqrt(Float64(Float64(hypot(re, im) - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.8e+51)
		tmp = sqrt(((hypot(re, im) - re) * 2.0)) * 0.5;
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.8e+51], N[(N[Sqrt[N[(N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.7999999999999997e51

   1. Initial program 46.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      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-*.f64 N/A

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

      5. lower-hypot.f64 93.7

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

   4. Applied rewrites 93.7%

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

   if 4.7999999999999997e51 < re

   1. Initial program 6.6%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 77.4%

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

   9. Applied rewrites 77.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im \cdot 2\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e-45)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 4.8e+51)
     (* (sqrt (fma (- (/ re im) 2.0) re (* im 2.0))) 0.5)
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e-45) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 4.8e+51) {
		tmp = sqrt(fma(((re / im) - 2.0), re, (im * 2.0))) * 0.5;
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e-45)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 4.8e+51)
		tmp = Float64(sqrt(fma(Float64(Float64(re / im) - 2.0), re, Float64(im * 2.0))) * 0.5);
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e-45], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 4.8e+51], N[(N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + N[(im * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.8e-45

   1. Initial program 43.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -1.8e-45 < re < 4.7999999999999997e51

   1. Initial program 47.5%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if 4.7999999999999997e51 < re

   1. Initial program 6.6%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 77.4%

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

   9. Applied rewrites 77.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im \cdot 2\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im}, -2, 2\right) \cdot im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e-45)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 4.8e+51)
     (* (sqrt (* (fma (/ re im) -2.0 2.0) im)) 0.5)
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e-45) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 4.8e+51) {
		tmp = sqrt((fma((re / im), -2.0, 2.0) * im)) * 0.5;
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e-45)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 4.8e+51)
		tmp = Float64(sqrt(Float64(fma(Float64(re / im), -2.0, 2.0) * im)) * 0.5);
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e-45], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 4.8e+51], N[(N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] * -2.0 + 2.0), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.8e-45

   1. Initial program 43.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -1.8e-45 < re < 4.7999999999999997e51

   1. Initial program 47.5%

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

   3. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if 4.7999999999999997e51 < re

   1. Initial program 6.6%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 77.4%

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

   9. Applied rewrites 77.6%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im}, -2, 2\right) \cdot im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e-45)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 4.8e+51)
     (* (sqrt (* (- im re) 2.0)) 0.5)
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e-45) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 4.8e+51) {
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.8d-45)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else if (re <= 4.8d+51) then
        tmp = sqrt(((im - re) * 2.0d0)) * 0.5d0
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e-45) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 4.8e+51) {
		tmp = Math.sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e-45:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	elif re <= 4.8e+51:
		tmp = math.sqrt(((im - re) * 2.0)) * 0.5
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e-45)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 4.8e+51)
		tmp = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e-45)
		tmp = sqrt((-4.0 * re)) * 0.5;
	elseif (re <= 4.8e+51)
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e-45], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 4.8e+51], N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.8e-45

   1. Initial program 43.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -1.8e-45 < re < 4.7999999999999997e51

   1. Initial program 47.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 81.5

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

   5. Applied rewrites 81.5%

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

   if 4.7999999999999997e51 < re

   1. Initial program 6.6%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 77.4%

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

   9. Applied rewrites 77.6%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e-45) (* (sqrt (* -4.0 re)) 0.5) (* (sqrt (* im 2.0)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e-45) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = sqrt((im * 2.0)) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.8d-45)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else
        tmp = sqrt((im * 2.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e-45) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = Math.sqrt((im * 2.0)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e-45:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	else:
		tmp = math.sqrt((im * 2.0)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e-45)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(im * 2.0)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e-45)
		tmp = sqrt((-4.0 * re)) * 0.5;
	else
		tmp = sqrt((im * 2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e-45], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.8e-45

   1. Initial program 43.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -1.8e-45 < re

   1. Initial program 35.3%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1e-309) (* (sqrt (* -4.0 re)) 0.5) (* (sqrt 0.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e-309) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = sqrt(0.0) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1d-309)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else
        tmp = sqrt(0.0d0) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1e-309) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = Math.sqrt(0.0) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1e-309:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	else:
		tmp = math.sqrt(0.0) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e-309)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	else
		tmp = Float64(sqrt(0.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1e-309)
		tmp = sqrt((-4.0 * re)) * 0.5;
	else
		tmp = sqrt(0.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e-309], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[0.0], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.000000000000002e-309

   1. Initial program 50.7%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   if -1.000000000000002e-309 < re

   1. Initial program 23.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

   4. Applied rewrites 22.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}, \frac{\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}}{\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} + re}, -\frac{re \cdot re}{\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} + re}\right)}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 21.5%

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

   7. Taylor expanded in im around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{0}} \]

      4. metadata-eval 7.7

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

   9. Applied rewrites 7.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 5.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{0} \cdot 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sqrt 0.0) 0.5))

C

double code(double re, double im) {
	return sqrt(0.0) * 0.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(0.0d0) * 0.5d0
end function

Java

public static double code(double re, double im) {
	return Math.sqrt(0.0) * 0.5;
}

Python

def code(re, im):
	return math.sqrt(0.0) * 0.5

Julia

function code(re, im)
	return Float64(sqrt(0.0) * 0.5)
end

MATLAB

function tmp = code(re, im)
	tmp = sqrt(0.0) * 0.5;
end

Wolfram

code[re_, im_] := N[(N[Sqrt[0.0], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 37.6%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. lift-*.f64 N/A

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

   7. div-sub N/A

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

   8. sub-neg N/A

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

4. Applied rewrites 24.4%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}, \frac{\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}}{\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} + re}, -\frac{re \cdot re}{\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} + re}\right)}} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. div-inv N/A

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

   7. lower-fma.f64 N/A

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

6. Applied rewrites 24.1%

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

7. Taylor expanded in im around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{0}} \]

   4. metadata-eval 5.2

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

9. Applied rewrites 5.2%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

10. Final simplification 5.2%

    \[\leadsto \sqrt{0} \cdot 0.5 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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)))))