math.sqrt on complex, real part

Percentage Accurate: 41.2% → 84.8%

Time: 2.7s

Alternatives: 9

Speedup: 2.1×

• Could not converge on a ground truth

  1. re = 4.614906735260092e-21
  2. im = +nan.0

Specification

Math

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

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)
use fmin_fmax_functions
    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: 41.2% accurate, 1.0× speedup

Math

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

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)
use fmin_fmax_functions
    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: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{-1 \cdot \frac{{im}^{2}}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<=
     (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))))
     0.0)
  (* 0.5 (sqrt (* -1.0 (/ (pow im 2.0) re))))
  (* 0.5 (sqrt (* 2.0 (+ (hypot re im) re))))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)))) <= 0.0) {
		tmp = 0.5 * sqrt((-1.0 * (pow(im, 2.0) / 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 ((0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)))) <= 0.0) {
		tmp = 0.5 * Math.sqrt((-1.0 * (Math.pow(im, 2.0) / re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) + re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))) <= 0.0:
		tmp = 0.5 * math.sqrt((-1.0 * (math.pow(im, 2.0) / 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(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re)))) <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(-1.0 * Float64((im ^ 2.0) / 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 ((0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)))) <= 0.0)
		tmp = 0.5 * sqrt((-1.0 * ((im ^ 2.0) / re)));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[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], 0.0], N[(0.5 * N[Sqrt[N[(-1.0 * N[(N[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision]), $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 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 0.0

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 25.9%

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in im around inf

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

      1. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

   8. Taylor expanded in re around -inf

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 14.6%

         \[\leadsto 0.5 \cdot \sqrt{-1 \cdot \frac{{im}^{2}}{re}} \]

   10. Applied rewrites 14.6%

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]

   if 0.0 < (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))

   1. Initial program 41.2%

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

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-+.f64 N/A

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

      9. add-flip N/A

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

      10. sub-flip N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. lower-hypot.f64 79.7%

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

   3. Applied rewrites 79.7%

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

Alternative 2: 79.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

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

   3. lift-sqrt.f64 N/A

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

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

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-+.f64 N/A

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

   9. add-flip N/A

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

   10. sub-flip N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. distribute-lft-neg-out N/A

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

   15. sqr-neg-rev N/A

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

   16. lower-hypot.f64 79.7%

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

3. Applied rewrites 79.7%

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

Alternative 3: 65.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \leq 1.25 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \mathbf{elif}\;\left|im\right| \leq 4.6 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(\left|im\right|, \left|im\right|, re \cdot re\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{\left|\left(-\left|im\right|\right) + re\right|}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (fabs im) 1.25e-132)
  (* 0.5 (sqrt (* 4.0 re)))
  (if (<= (fabs im) 4.6e+144)
    (*
     (sqrt (* (+ (sqrt (fma (fabs im) (fabs im) (* re re))) re) 2.0))
     0.5)
    (* (* 0.5 (sqrt 2.0)) (sqrt (fabs (+ (- (fabs im)) re)))))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) <= 1.25e-132) {
		tmp = 0.5 * sqrt((4.0 * re));
	} else if (fabs(im) <= 4.6e+144) {
		tmp = sqrt(((sqrt(fma(fabs(im), fabs(im), (re * re))) + re) * 2.0)) * 0.5;
	} else {
		tmp = (0.5 * sqrt(2.0)) * sqrt(fabs((-fabs(im) + re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) <= 1.25e-132)
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	elseif (abs(im) <= 4.6e+144)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(abs(im), abs(im), Float64(re * re))) + re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * sqrt(abs(Float64(Float64(-abs(im)) + re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Abs[im], $MachinePrecision], 1.25e-132], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[im], $MachinePrecision], 4.6e+144], N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision] + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Abs[N[((-N[Abs[im], $MachinePrecision]) + re), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.25e-132

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 25.9%

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in im around inf

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

      1. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

   8. Taylor expanded in re around inf

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
   9. Step-by-step derivation

      1. lower-*.f64 26.3%

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

   10. Applied rewrites 26.3%

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

   if 1.25e-132 < im < 4.6000000000000003e144

   1. Initial program 41.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.2%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 41.2%

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

   3. Applied rewrites 41.2%

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

   if 4.6000000000000003e144 < im

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 29.5%

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

   4. Applied rewrites 29.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqrt-prod N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-fabs.f64 55.7%

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

   6. Applied rewrites 55.7%

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

Alternative 4: 64.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \leq 1.25 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \mathbf{elif}\;\left|im\right| \leq 4.6 \cdot 10^{+144}:\\ \;\;\;\;0.7071067811865476 \cdot \sqrt{\left|\sqrt{\mathsf{fma}\left(\left|im\right|, \left|im\right|, re \cdot re\right)} + re\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{\left|\left(-\left|im\right|\right) + re\right|}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (fabs im) 1.25e-132)
  (* 0.5 (sqrt (* 4.0 re)))
  (if (<= (fabs im) 4.6e+144)
    (*
     0.7071067811865476
     (sqrt (fabs (+ (sqrt (fma (fabs im) (fabs im) (* re re))) re))))
    (* (* 0.5 (sqrt 2.0)) (sqrt (fabs (+ (- (fabs im)) re)))))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) <= 1.25e-132) {
		tmp = 0.5 * sqrt((4.0 * re));
	} else if (fabs(im) <= 4.6e+144) {
		tmp = 0.7071067811865476 * sqrt(fabs((sqrt(fma(fabs(im), fabs(im), (re * re))) + re)));
	} else {
		tmp = (0.5 * sqrt(2.0)) * sqrt(fabs((-fabs(im) + re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) <= 1.25e-132)
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	elseif (abs(im) <= 4.6e+144)
		tmp = Float64(0.7071067811865476 * sqrt(abs(Float64(sqrt(fma(abs(im), abs(im), Float64(re * re))) + re))));
	else
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * sqrt(abs(Float64(Float64(-abs(im)) + re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Abs[im], $MachinePrecision], 1.25e-132], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[im], $MachinePrecision], 4.6e+144], N[(0.7071067811865476 * N[Sqrt[N[Abs[N[(N[Sqrt[N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision] + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Abs[N[((-N[Abs[im], $MachinePrecision]) + re), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.25e-132

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 25.9%

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in im around inf

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

      1. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

   8. Taylor expanded in re around inf

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
   9. Step-by-step derivation

      1. lower-*.f64 26.3%

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

   10. Applied rewrites 26.3%

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

   if 1.25e-132 < im < 4.6000000000000003e144

   1. Initial program 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqrt-prod N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-fabs.f64 41.3%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 41.3%

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

   3. Applied rewrites 41.3%

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

   4. Evaluated real constant 41.3%

      \[\leadsto \color{blue}{0.7071067811865476} \cdot \sqrt{\left|\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} + re\right|} \]

   if 4.6000000000000003e144 < im

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 29.5%

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

   4. Applied rewrites 29.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqrt-prod N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-fabs.f64 55.7%

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

   6. Applied rewrites 55.7%

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

Alternative 5: 64.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+222}:\\ \;\;\;\;\sqrt{\left|im\right| + \left|im\right|} \cdot 0\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(\left|im\right| + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= re -4.2e+222)
  (* (sqrt (+ (fabs im) (fabs im))) 0.0)
  (if (<= re 1.2e+79)
    (* (sqrt (* (+ (fabs im) re) 2.0)) 0.5)
    (* 0.5 (sqrt (* 4.0 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.2e+222) {
		tmp = sqrt((fabs(im) + fabs(im))) * 0.0;
	} else if (re <= 1.2e+79) {
		tmp = sqrt(((fabs(im) + re) * 2.0)) * 0.5;
	} else {
		tmp = 0.5 * sqrt((4.0 * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.2d+222)) then
        tmp = sqrt((abs(im) + abs(im))) * 0.0d0
    else if (re <= 1.2d+79) then
        tmp = sqrt(((abs(im) + re) * 2.0d0)) * 0.5d0
    else
        tmp = 0.5d0 * sqrt((4.0d0 * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.2e+222) {
		tmp = Math.sqrt((Math.abs(im) + Math.abs(im))) * 0.0;
	} else if (re <= 1.2e+79) {
		tmp = Math.sqrt(((Math.abs(im) + re) * 2.0)) * 0.5;
	} else {
		tmp = 0.5 * Math.sqrt((4.0 * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.2e+222:
		tmp = math.sqrt((math.fabs(im) + math.fabs(im))) * 0.0
	elif re <= 1.2e+79:
		tmp = math.sqrt(((math.fabs(im) + re) * 2.0)) * 0.5
	else:
		tmp = 0.5 * math.sqrt((4.0 * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.2e+222)
		tmp = Float64(sqrt(Float64(abs(im) + abs(im))) * 0.0);
	elseif (re <= 1.2e+79)
		tmp = Float64(sqrt(Float64(Float64(abs(im) + re) * 2.0)) * 0.5);
	else
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.2e+222)
		tmp = sqrt((abs(im) + abs(im))) * 0.0;
	elseif (re <= 1.2e+79)
		tmp = sqrt(((abs(im) + re) * 2.0)) * 0.5;
	else
		tmp = 0.5 * sqrt((4.0 * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.2e+222], N[(N[Sqrt[N[(N[Abs[im], $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.0), $MachinePrecision], If[LessEqual[re, 1.2e+79], N[(N[Sqrt[N[(N[(N[Abs[im], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.2000000000000002e222

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 25.9%

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in im around inf

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

      1. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 26.9%

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 26.9%

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

   9. Applied rewrites 26.9%

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

   10. Taylor expanded in undef-var around zero

       \[\leadsto \sqrt{im + im} \cdot \color{blue}{0} \]
   11. Step-by-step derivation

   12. Applied rewrites 2.8%

       \[\leadsto \sqrt{im + im} \cdot \color{blue}{0} \]

   if -4.2000000000000002e222 < re < 1.1999999999999999e79

   1. Initial program 41.2%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 29.5%

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

   4. Applied rewrites 29.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 29.5%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 29.5%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sum-to-mult-rev N/A

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

      12. lower-+.f64 30.6%

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

   6. Applied rewrites 30.6%

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

   if 1.1999999999999999e79 < re

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 25.9%

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in im around inf

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

      1. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

   8. Taylor expanded in re around inf

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
   9. Step-by-step derivation

      1. lower-*.f64 26.3%

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

   10. Applied rewrites 26.3%

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

Alternative 6: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \leq 6.8 \cdot 10^{-68}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{\left|\left(-\left|im\right|\right) + re\right|}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (fabs im) 6.8e-68)
  (* 0.5 (sqrt (* 4.0 re)))
  (* (* 0.5 (sqrt 2.0)) (sqrt (fabs (+ (- (fabs im)) re))))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) <= 6.8e-68) {
		tmp = 0.5 * sqrt((4.0 * re));
	} else {
		tmp = (0.5 * sqrt(2.0)) * sqrt(fabs((-fabs(im) + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) <= 6.8d-68) then
        tmp = 0.5d0 * sqrt((4.0d0 * re))
    else
        tmp = (0.5d0 * sqrt(2.0d0)) * sqrt(abs((-abs(im) + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) <= 6.8e-68) {
		tmp = 0.5 * Math.sqrt((4.0 * re));
	} else {
		tmp = (0.5 * Math.sqrt(2.0)) * Math.sqrt(Math.abs((-Math.abs(im) + re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) <= 6.8e-68:
		tmp = 0.5 * math.sqrt((4.0 * re))
	else:
		tmp = (0.5 * math.sqrt(2.0)) * math.sqrt(math.fabs((-math.fabs(im) + re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) <= 6.8e-68)
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	else
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * sqrt(abs(Float64(Float64(-abs(im)) + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) <= 6.8e-68)
		tmp = 0.5 * sqrt((4.0 * re));
	else
		tmp = (0.5 * sqrt(2.0)) * sqrt(abs((-abs(im) + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Abs[im], $MachinePrecision], 6.8e-68], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Abs[N[((-N[Abs[im], $MachinePrecision]) + re), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.8000000000000004e-68

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 25.9%

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in im around inf

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

      1. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

   8. Taylor expanded in re around inf

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
   9. Step-by-step derivation

      1. lower-*.f64 26.3%

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

   10. Applied rewrites 26.3%

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

   if 6.8000000000000004e-68 < im

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 29.5%

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

   4. Applied rewrites 29.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqrt-prod N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-fabs.f64 55.7%

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

   6. Applied rewrites 55.7%

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

Alternative 7: 59.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \leq 2 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|im\right| + \left|im\right|} \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (fabs im) 2e-64)
  (* 0.5 (sqrt (* 4.0 re)))
  (* (sqrt (+ (fabs im) (fabs im))) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) <= 2e-64) {
		tmp = 0.5 * sqrt((4.0 * re));
	} else {
		tmp = sqrt((fabs(im) + fabs(im))) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) <= 2d-64) then
        tmp = 0.5d0 * sqrt((4.0d0 * re))
    else
        tmp = sqrt((abs(im) + abs(im))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) <= 2e-64) {
		tmp = 0.5 * Math.sqrt((4.0 * re));
	} else {
		tmp = Math.sqrt((Math.abs(im) + Math.abs(im))) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) <= 2e-64:
		tmp = 0.5 * math.sqrt((4.0 * re))
	else:
		tmp = math.sqrt((math.fabs(im) + math.fabs(im))) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) <= 2e-64)
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	else
		tmp = Float64(sqrt(Float64(abs(im) + abs(im))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) <= 2e-64)
		tmp = 0.5 * sqrt((4.0 * re));
	else
		tmp = sqrt((abs(im) + abs(im))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Abs[im], $MachinePrecision], 2e-64], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[Abs[im], $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.9999999999999999e-64

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 25.9%

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in im around inf

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

      1. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

   8. Taylor expanded in re around inf

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
   9. Step-by-step derivation

      1. lower-*.f64 26.3%

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

   10. Applied rewrites 26.3%

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

   if 1.9999999999999999e-64 < im

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 25.9%

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in im around inf

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

      1. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 26.9%

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 26.9%

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

   9. Applied rewrites 26.9%

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

Alternative 8: 53.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\left|im\right| + \left|im\right|}\\ \mathbf{if}\;\left|im\right| \leq 1.02 \cdot 10^{-222}:\\ \;\;\;\;t\_0 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (sqrt (+ (fabs im) (fabs im)))))
  (if (<= (fabs im) 1.02e-222) (* t_0 0.0) (* t_0 0.5))))

C

double code(double re, double im) {
	double t_0 = sqrt((fabs(im) + fabs(im)));
	double tmp;
	if (fabs(im) <= 1.02e-222) {
		tmp = t_0 * 0.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((abs(im) + abs(im)))
    if (abs(im) <= 1.02d-222) then
        tmp = t_0 * 0.0d0
    else
        tmp = t_0 * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt((Math.abs(im) + Math.abs(im)));
	double tmp;
	if (Math.abs(im) <= 1.02e-222) {
		tmp = t_0 * 0.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt((math.fabs(im) + math.fabs(im)))
	tmp = 0
	if math.fabs(im) <= 1.02e-222:
		tmp = t_0 * 0.0
	else:
		tmp = t_0 * 0.5
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(abs(im) + abs(im)))
	tmp = 0.0
	if (abs(im) <= 1.02e-222)
		tmp = Float64(t_0 * 0.0);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt((abs(im) + abs(im)));
	tmp = 0.0;
	if (abs(im) <= 1.02e-222)
		tmp = t_0 * 0.0;
	else
		tmp = t_0 * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[Abs[im], $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[im], $MachinePrecision], 1.02e-222], N[(t$95$0 * 0.0), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.0199999999999999e-222

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 25.9%

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in im around inf

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

      1. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 26.9%

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 26.9%

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

   9. Applied rewrites 26.9%

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

   10. Taylor expanded in undef-var around zero

       \[\leadsto \sqrt{im + im} \cdot \color{blue}{0} \]
   11. Step-by-step derivation

   12. Applied rewrites 2.8%

       \[\leadsto \sqrt{im + im} \cdot \color{blue}{0} \]

   if 1.0199999999999999e-222 < im

   1. Initial program 41.2%

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

   2. Taylor expanded in im around -inf

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

      1. lower-*.f64 25.9%

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in im around inf

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

      1. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 26.9%

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 26.9%

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

   9. Applied rewrites 26.9%

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

Alternative 9: 52.7% accurate, 2.1× speedup

Math

\[\sqrt{\left|im\right| + \left|im\right|} \cdot 0.5 \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt((abs(im) + abs(im))) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

2. Taylor expanded in im around -inf

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

   1. lower-*.f64 25.9%

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

4. Applied rewrites 25.9%

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

5. Taylor expanded in im around inf

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

   1. lower-*.f64 26.9%

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

7. Applied rewrites 26.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 26.9%

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f64 26.9%

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

9. Applied rewrites 26.9%

   \[\leadsto \color{blue}{\sqrt{im + im} \cdot 0.5} \]
10. Add Preprocessing

Developer Target 1: 48.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t\_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t\_0 + re\right)}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
  (if (< re 0.0)
    (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
    (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform c (if (< re 0) (* 1/2 (* (sqrt 2) (sqrt (/ (* im im) (- (modulus re im) re))))) (* 1/2 (sqrt (* 2 (+ (modulus re im) re))))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))