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

Percentage Accurate: 41.4% → 90.4%

Time: 7.2s

Alternatives: 9

Speedup: 1.9×

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: 41.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 91.4

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

   5. Applied rewrites 91.4%

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

   7. Applied rewrites 92.1%

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

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

   1. Initial program 41.8%

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

      1. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-hypot.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-hypot.f64 N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

   6. Applied rewrites 92.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 92.1

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

      7. lift-hypot.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-hypot.f64 92.1

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

      10. lift-hypot.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-hypot.f64 92.1

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

   8. Applied rewrites 92.1%

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

      1. lift-+.f64 N/A

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

      2. count-2 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 92.1

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

      5. lift-hypot.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-hypot.f64 92.1

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

   10. Applied rewrites 92.1%

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

4. Final simplification 92.1%

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

Alternative 2: 90.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re)) <= 0.0)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	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 ((2.0 * (sqrt(((re * re) + (im * im))) - re)) <= 0.0)
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 91.4

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

   5. Applied rewrites 91.4%

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

   7. Applied rewrites 92.1%

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

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

   1. Initial program 41.8%

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

      1. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 92.0%

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

4. Final simplification 92.1%

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

Alternative 3: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{{re}^{-1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e-39)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 5e-66)
     (* 0.5 (sqrt (+ im im)))
     (* (* im 0.5) (sqrt (pow re -1.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-39) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 5e-66) {
		tmp = 0.5 * sqrt((im + im));
	} else {
		tmp = (im * 0.5) * sqrt(pow(re, -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.9d-39)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 5d-66) then
        tmp = 0.5d0 * sqrt((im + im))
    else
        tmp = (im * 0.5d0) * sqrt((re ** (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-39) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 5e-66) {
		tmp = 0.5 * Math.sqrt((im + im));
	} else {
		tmp = (im * 0.5) * Math.sqrt(Math.pow(re, -1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e-39:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 5e-66:
		tmp = 0.5 * math.sqrt((im + im))
	else:
		tmp = (im * 0.5) * math.sqrt(math.pow(re, -1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e-39)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 5e-66)
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	else
		tmp = Float64(Float64(im * 0.5) * sqrt((re ^ -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.9e-39)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 5e-66)
		tmp = 0.5 * sqrt((im + im));
	else
		tmp = (im * 0.5) * sqrt((re ^ -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e-39], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e-66], N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] * N[Sqrt[N[Power[re, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9000000000000001e-39

   1. Initial program 41.1%

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

      1. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 80.0

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

   7. Applied rewrites 80.0%

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

   if -1.9000000000000001e-39 < re < 4.99999999999999962e-66

   1. Initial program 44.8%

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

      1. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-hypot.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-hypot.f64 N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in re around 0

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

      1. lower-*.f64 78.2

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

   9. Applied rewrites 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   10. Step-by-step derivation

   11. Applied rewrites 78.2%

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

   if 4.99999999999999962e-66 < re

   1. Initial program 16.0%

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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 39.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-hypot.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-hypot.f64 N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

   6. Applied rewrites 39.6%

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

   7. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 78.0

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

   9. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\left(im \cdot 0.5\right) \cdot \sqrt{\frac{1}{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{{re}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{im}{re} \cdot im, -0.5, -2 \cdot re\right)}\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im\right) + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e-39)
   (* 0.5 (sqrt (* 2.0 (fma (* (/ im re) im) -0.5 (* -2.0 re)))))
   (if (<= re 1.06e-66)
     (* 0.5 (sqrt (+ (fma (- (/ re im) 2.0) re im) im)))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-39) {
		tmp = 0.5 * sqrt((2.0 * fma(((im / re) * im), -0.5, (-2.0 * re))));
	} else if (re <= 1.06e-66) {
		tmp = 0.5 * sqrt((fma(((re / im) - 2.0), re, im) + im));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e-39)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * fma(Float64(Float64(im / re) * im), -0.5, Float64(-2.0 * re)))));
	elseif (re <= 1.06e-66)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(Float64(re / im) - 2.0), re, im) + im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e-39], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(N[(im / re), $MachinePrecision] * im), $MachinePrecision] * -0.5 + N[(-2.0 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.06e-66], N[(0.5 * N[Sqrt[N[(N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9000000000000001e-39

   1. Initial program 41.1%

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

   3. Taylor expanded in re around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 76.1%

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

   10. Applied rewrites 80.1%

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

   if -1.9000000000000001e-39 < re < 1.05999999999999994e-66

   1. Initial program 44.8%

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

      1. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-hypot.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-hypot.f64 N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im + re \cdot \left(\frac{re}{im} - 2\right)}} \]
   8. 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 78.4

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

   9. Applied rewrites 78.4%

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

   11. Applied rewrites 78.4%

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

   if 1.05999999999999994e-66 < re

   1. Initial program 16.0%

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

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 78.1%

      \[\leadsto 0.5 \cdot \frac{1 \cdot \left(im \cdot 1\right)}{\color{blue}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{im}{re} \cdot im, -0.5, -2 \cdot re\right)}\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im\right) + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im\right) + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e-39)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 1.06e-66)
     (* 0.5 (sqrt (+ (fma (- (/ re im) 2.0) re im) im)))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-39) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 1.06e-66) {
		tmp = 0.5 * sqrt((fma(((re / im) - 2.0), re, im) + im));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e-39)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 1.06e-66)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(Float64(re / im) - 2.0), re, im) + im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e-39], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.06e-66], N[(0.5 * N[Sqrt[N[(N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9000000000000001e-39

   1. Initial program 41.1%

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

      1. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 80.0

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

   7. Applied rewrites 80.0%

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

   if -1.9000000000000001e-39 < re < 1.05999999999999994e-66

   1. Initial program 44.8%

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

      1. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-hypot.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-hypot.f64 N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im + re \cdot \left(\frac{re}{im} - 2\right)}} \]
   8. 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 78.4

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

   9. Applied rewrites 78.4%

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

   11. Applied rewrites 78.4%

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

   if 1.05999999999999994e-66 < re

   1. Initial program 16.0%

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

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 78.1%

      \[\leadsto 0.5 \cdot \frac{1 \cdot \left(im \cdot 1\right)}{\color{blue}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im\right) + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e-39)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 5e-66) (* 0.5 (sqrt (+ im im))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-39) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 5e-66) {
		tmp = 0.5 * sqrt((im + 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 <= (-1.9d-39)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 5d-66) then
        tmp = 0.5d0 * sqrt((im + 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 <= -1.9e-39) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 5e-66) {
		tmp = 0.5 * Math.sqrt((im + im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e-39:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 5e-66:
		tmp = 0.5 * math.sqrt((im + im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e-39)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 5e-66)
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.9e-39)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 5e-66)
		tmp = 0.5 * sqrt((im + im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e-39], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e-66], N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9000000000000001e-39

   1. Initial program 41.1%

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

      1. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 80.0

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

   7. Applied rewrites 80.0%

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

   if -1.9000000000000001e-39 < re < 4.99999999999999962e-66

   1. Initial program 44.8%

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

      1. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-hypot.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-hypot.f64 N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in re around 0

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

      1. lower-*.f64 78.2

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

   9. Applied rewrites 78.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   10. Step-by-step derivation

   11. Applied rewrites 78.2%

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

   if 4.99999999999999962e-66 < re

   1. Initial program 16.0%

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

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 78.1%

      \[\leadsto 0.5 \cdot \frac{1 \cdot \left(im \cdot 1\right)}{\color{blue}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+130}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.9e-39)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 2.6e+130) (* 0.5 (sqrt (+ im im))) 0.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-39) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 2.6e+130) {
		tmp = 0.5 * sqrt((im + im));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.9d-39)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 2.6d+130) then
        tmp = 0.5d0 * sqrt((im + im))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.9e-39) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 2.6e+130) {
		tmp = 0.5 * Math.sqrt((im + im));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e-39:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 2.6e+130:
		tmp = 0.5 * math.sqrt((im + im))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e-39)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 2.6e+130)
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.9e-39)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 2.6e+130)
		tmp = 0.5 * sqrt((im + im));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.9e-39], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.6e+130], N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9000000000000001e-39

   1. Initial program 41.1%

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

      1. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 80.0

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

   7. Applied rewrites 80.0%

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

   if -1.9000000000000001e-39 < re < 2.5999999999999998e130

   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. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-hypot.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-hypot.f64 N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

   6. Applied rewrites 72.3%

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

   7. Taylor expanded in re around 0

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

      1. lower-*.f64 66.0

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

   9. Applied rewrites 66.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   10. Step-by-step derivation

   11. Applied rewrites 66.0%

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

   if 2.5999999999999998e130 < re

   1. Initial program 5.6%

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

   4. Applied rewrites 28.3%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+130}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.2 \cdot 10^{-243}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.2e-243) 0.0 (* 0.5 (sqrt (+ im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.2e-243) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * sqrt((im + im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.2d-243) then
        tmp = 0.0d0
    else
        tmp = 0.5d0 * sqrt((im + im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.2e-243) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * Math.sqrt((im + im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.2e-243:
		tmp = 0.0
	else:
		tmp = 0.5 * math.sqrt((im + im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.2e-243)
		tmp = 0.0;
	else
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.2e-243)
		tmp = 0.0;
	else
		tmp = 0.5 * sqrt((im + im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.2e-243], 0.0, N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.2e-243

   1. Initial program 36.1%

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

   4. Applied rewrites 28.4%

      \[\leadsto \color{blue}{0} \]

   if 1.2e-243 < im

   1. Initial program 35.5%

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

      1. 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

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

      4. sqr-abs-rev N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. lift-*.f64 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 78.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-hypot.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-hypot.f64 N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

   6. Applied rewrites 78.9%

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

   7. Taylor expanded in re around 0

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

      1. lower-*.f64 49.3

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

   9. Applied rewrites 49.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.3%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.2 \cdot 10^{-243}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 5.7% accurate, 47.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

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

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 35.6%

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

4. Applied rewrites 6.6%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024343 
(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)))))