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

Percentage Accurate: 41.0% → 89.9%

Time: 8.9s

Alternatives: 10

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: 41.0% 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: 89.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 8.3%

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

   3. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.4

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

   7. Applied rewrites 99.7%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 42.6%

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

      1. lift-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 93.1

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

   4. Applied rewrites 93.1%

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

Alternative 2: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+116}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(re + im\right) \cdot \frac{\mathsf{fma}\left(re, re, im \cdot im\right)}{re + im}} - re\right)}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(\frac{re \cdot \left(-2 + \frac{re}{im}\right)}{im} - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.5e+116)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re -1.9e-84)
     (*
      0.5
      (sqrt
       (*
        2.0
        (- (sqrt (* (+ re im) (/ (fma re re (* im im)) (+ re im)))) re))))
     (if (<= re 5.5e+24)
       (* 0.5 (sqrt (* im (- (/ (* re (+ -2.0 (/ re im))) im) -2.0))))
       (* (/ im (sqrt re)) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.5e+116) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= -1.9e-84) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re + im) * (fma(re, re, (im * im)) / (re + im)))) - re)));
	} else if (re <= 5.5e+24) {
		tmp = 0.5 * sqrt((im * (((re * (-2.0 + (re / im))) / im) - -2.0)));
	} else {
		tmp = (im / sqrt(re)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.5e+116)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= -1.9e-84)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re + im) * Float64(fma(re, re, Float64(im * im)) / Float64(re + im)))) - re))));
	elseif (re <= 5.5e+24)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(Float64(Float64(re * Float64(-2.0 + Float64(re / im))) / im) - -2.0))));
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.5e+116], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.9e-84], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re + im), $MachinePrecision] * N[(N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(re + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.5e+24], N[(0.5 * N[Sqrt[N[(im * N[(N[(N[(re * N[(-2.0 + N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.4999999999999999e116

   1. Initial program 16.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. *-commutative N/A

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

      2. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if -1.4999999999999999e116 < re < -1.89999999999999993e-84

   1. Initial program 76.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. flip-+ N/A

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

      3. difference-of-squares N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. flip-+ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 76.2

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

   4. Applied rewrites 76.2%

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

   if -1.89999999999999993e-84 < re < 5.5000000000000002e24

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.3

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 52.3

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

   8. Taylor expanded in im around -inf

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

   10. Applied rewrites 77.4%

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

   if 5.5000000000000002e24 < re

   1. Initial program 10.7%

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

   3. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 77.6

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

   5. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.6

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

   7. Applied rewrites 78.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+116}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(re + im\right) \cdot \frac{\mathsf{fma}\left(re, re, im \cdot im\right)}{re + im}} - re\right)}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(\frac{re \cdot \left(-2 + \frac{re}{im}\right)}{im} - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(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)))))