math.sqrt on complex, real part

Percentage Accurate: 40.7% → 87.7%

Time: 9.4s

Alternatives: 4

Speedup: 1.9×

Specification

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: 40.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{-65}:\\ \;\;\;\;{\left(0 - re\right)}^{-0.5} \cdot \left(0.5 \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.5e-65)
   (* (pow (- 0.0 re) -0.5) (* 0.5 im_m))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2.5e-65) {
		tmp = pow((0.0 - re), -0.5) * (0.5 * im_m);
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -2.5e-65) {
		tmp = Math.pow((0.0 - re), -0.5) * (0.5 * im_m);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -2.5e-65:
		tmp = math.pow((0.0 - re), -0.5) * (0.5 * im_m)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2.5e-65)
		tmp = Float64((Float64(0.0 - re) ^ -0.5) * Float64(0.5 * im_m));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im_m)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -2.5e-65)
		tmp = ((0.0 - re) ^ -0.5) * (0.5 * im_m);
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2.5e-65], N[(N[Power[N[(0.0 - re), $MachinePrecision], -0.5], $MachinePrecision] * N[(0.5 * im$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.49999999999999991e-65

   1. Initial program 13.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(re + \sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 34.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right)\right)\right)\right) \]

   3. Simplified 34.9%

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

   5. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \frac{{im}^{2}}{re}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(0 - \frac{{im}^{2}}{re}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right)\right) \]

      6. *-lowering-*.f64 41.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right)\right) \]

   7. Simplified 41.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - \frac{im \cdot im}{re}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{im \cdot im}{re}\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\frac{re}{im \cdot im}}\right)\right)\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\frac{re}{im \cdot im}}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{-1}{\frac{re}{im \cdot im}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, \left(\frac{re}{im \cdot im}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(re, \left(im \cdot im\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 41.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(re, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 41.4%

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

       1. *-commutative N/A

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

       2. associate-/r/ N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. sqrt-prod N/A

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

       6. rem-square-sqrt N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. rem-square-sqrt N/A

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

       10. sqrt-prod N/A

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

       11. pow1/2 N/A

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

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{-1}{re}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)}\right) \]

       13. frac-2neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(re\right)}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{\mathsf{neg}\left(re\right)}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       15. inv-pow N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(\mathsf{neg}\left(re\right)\right)}^{-1}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       16. pow-pow N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left(\mathsf{neg}\left(re\right)\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\color{blue}{\frac{1}{2}} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       17. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(re\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       18. neg-sub0 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - re\right), \left(-1 \cdot \frac{1}{2}\right)\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       19. --lowering--.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \left(-1 \cdot \frac{1}{2}\right)\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \frac{-1}{2}\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       21. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \frac{-1}{2}\right), \left(\frac{1}{2} \cdot \sqrt{im \cdot im}\right)\right) \]

       22. sqrt-prod N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \frac{-1}{2}\right), \left(\frac{1}{2} \cdot \left(\sqrt{im} \cdot \color{blue}{\sqrt{im}}\right)\right)\right) \]

       23. rem-square-sqrt N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \frac{-1}{2}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

       24. *-lowering-*.f64 45.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{im}\right)\right) \]

   11. Applied egg-rr 45.6%

       \[\leadsto \color{blue}{{\left(0 - re\right)}^{-0.5} \cdot \left(0.5 \cdot im\right)} \]
   12. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(re\right)\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, im\right)\right) \]

       2. neg-lowering-neg.f64 45.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(re\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, im\right)\right) \]

   13. Applied egg-rr 45.6%

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

   if -2.49999999999999991e-65 < re

   1. Initial program 53.4%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(re + \sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 95.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right)\right)\right)\right) \]

   3. Simplified 95.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{-65}:\\ \;\;\;\;{\left(0 - re\right)}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-63}:\\ \;\;\;\;{\left(0 - re\right)}^{-0.5} \cdot \left(0.5 \cdot im\_m\right)\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -4.5e-63)
   (* (pow (- 0.0 re) -0.5) (* 0.5 im_m))
   (if (<= re 2.2e-25) (* 0.5 (sqrt (* 2.0 (+ re im_m)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -4.5e-63) {
		tmp = pow((0.0 - re), -0.5) * (0.5 * im_m);
	} else if (re <= 2.2e-25) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-4.5d-63)) then
        tmp = ((0.0d0 - re) ** (-0.5d0)) * (0.5d0 * im_m)
    else if (re <= 2.2d-25) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -4.5e-63) {
		tmp = Math.pow((0.0 - re), -0.5) * (0.5 * im_m);
	} else if (re <= 2.2e-25) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -4.5e-63:
		tmp = math.pow((0.0 - re), -0.5) * (0.5 * im_m)
	elif re <= 2.2e-25:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -4.5e-63)
		tmp = Float64((Float64(0.0 - re) ^ -0.5) * Float64(0.5 * im_m));
	elseif (re <= 2.2e-25)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -4.5e-63)
		tmp = ((0.0 - re) ^ -0.5) * (0.5 * im_m);
	elseif (re <= 2.2e-25)
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -4.5e-63], N[(N[Power[N[(0.0 - re), $MachinePrecision], -0.5], $MachinePrecision] * N[(0.5 * im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.2e-25], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.5e-63

   1. Initial program 13.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(re + \sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 34.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right)\right)\right)\right) \]

   3. Simplified 34.9%

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

   5. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \frac{{im}^{2}}{re}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(0 - \frac{{im}^{2}}{re}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right)\right) \]

      6. *-lowering-*.f64 41.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right)\right) \]

   7. Simplified 41.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - \frac{im \cdot im}{re}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{im \cdot im}{re}\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\frac{re}{im \cdot im}}\right)\right)\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\frac{re}{im \cdot im}}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{-1}{\frac{re}{im \cdot im}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, \left(\frac{re}{im \cdot im}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(re, \left(im \cdot im\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 41.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(re, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 41.4%

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

       1. *-commutative N/A

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

       2. associate-/r/ N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. sqrt-prod N/A

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

       6. rem-square-sqrt N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. rem-square-sqrt N/A

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

       10. sqrt-prod N/A

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

       11. pow1/2 N/A

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

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{-1}{re}\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)}\right) \]

       13. frac-2neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(re\right)}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{\mathsf{neg}\left(re\right)}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       15. inv-pow N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(\mathsf{neg}\left(re\right)\right)}^{-1}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       16. pow-pow N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left(\mathsf{neg}\left(re\right)\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\color{blue}{\frac{1}{2}} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       17. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(re\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       18. neg-sub0 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - re\right), \left(-1 \cdot \frac{1}{2}\right)\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       19. --lowering--.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \left(-1 \cdot \frac{1}{2}\right)\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \frac{-1}{2}\right), \left(\frac{1}{2} \cdot {\left(im \cdot im\right)}^{\frac{1}{2}}\right)\right) \]

       21. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \frac{-1}{2}\right), \left(\frac{1}{2} \cdot \sqrt{im \cdot im}\right)\right) \]

       22. sqrt-prod N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \frac{-1}{2}\right), \left(\frac{1}{2} \cdot \left(\sqrt{im} \cdot \color{blue}{\sqrt{im}}\right)\right)\right) \]

       23. rem-square-sqrt N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \frac{-1}{2}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

       24. *-lowering-*.f64 45.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, re\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{im}\right)\right) \]

   11. Applied egg-rr 45.6%

       \[\leadsto \color{blue}{{\left(0 - re\right)}^{-0.5} \cdot \left(0.5 \cdot im\right)} \]
   12. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(re\right)\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, im\right)\right) \]

       2. neg-lowering-neg.f64 45.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(re\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, im\right)\right) \]

   13. Applied egg-rr 45.6%

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

   if -4.5e-63 < re < 2.2000000000000002e-25

   1. Initial program 59.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(re + \sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 92.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right)\right)\right)\right) \]

   3. Simplified 92.4%

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

   5. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im + 2 \cdot re\right)}\right)\right) \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(im + re\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(im + re\right)\right)\right)\right) \]

      3. +-lowering-+.f64 46.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(im, re\right)\right)\right)\right) \]

   7. Simplified 46.6%

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

   if 2.2000000000000002e-25 < re

   1. Initial program 43.4%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(re + \sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(1, \color{blue}{\left(\sqrt{re}\right)}\right) \]

      7. sqrt-lowering-sqrt.f64 77.1%

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{sqrt.f64}\left(re\right)\right) \]

   7. Simplified 77.1%

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

      1. *-lft-identity N/A

         \[\leadsto \sqrt{re} \]

      2. sqrt-lowering-sqrt.f64 77.1%

         \[\leadsto \mathsf{sqrt.f64}\left(re\right) \]

   9. Applied egg-rr 77.1%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{-63}:\\ \;\;\;\;{\left(0 - re\right)}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 5.5 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 5.5e-20) (* 0.5 (sqrt (* im_m 2.0))) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 5.5e-20) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 5.5d-20) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 5.5e-20) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 5.5e-20:
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 5.5e-20)
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 5.5e-20)
		tmp = 0.5 * sqrt((im_m * 2.0));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 5.5e-20], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.4999999999999996e-20

   1. Initial program 41.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(re + \sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 70.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right)\right)\right)\right) \]

   3. Simplified 70.0%

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

   5. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 31.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   7. Simplified 31.3%

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

   if 5.4999999999999996e-20 < re

   1. Initial program 43.4%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(re + \sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(1, \color{blue}{\left(\sqrt{re}\right)}\right) \]

      7. sqrt-lowering-sqrt.f64 77.1%

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{sqrt.f64}\left(re\right)\right) \]

   7. Simplified 77.1%

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

      1. *-lft-identity N/A

         \[\leadsto \sqrt{re} \]

      2. sqrt-lowering-sqrt.f64 77.1%

         \[\leadsto \mathsf{sqrt.f64}\left(re\right) \]

   9. Applied egg-rr 77.1%

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

Alternative 4: 26.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sqrt{re} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (sqrt re))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return sqrt(re);
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = sqrt(re)
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return Math.sqrt(re);
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return math.sqrt(re)

Julia

im_m = abs(im)
function code(re, im_m)
	return sqrt(re)
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = sqrt(re);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 42.1%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(re + \sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right)\right)\right) \]

   6. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)\right)\right) \]

   7. hypot-lowering-hypot.f64 78.3%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right)\right)\right)\right) \]

3. Simplified 78.3%

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

5. Taylor expanded in re around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(1, \color{blue}{\left(\sqrt{re}\right)}\right) \]

   7. sqrt-lowering-sqrt.f64 28.0%

      \[\leadsto \mathsf{*.f64}\left(1, \mathsf{sqrt.f64}\left(re\right)\right) \]

7. Simplified 28.0%

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

   1. *-lft-identity N/A

      \[\leadsto \sqrt{re} \]

   2. sqrt-lowering-sqrt.f64 28.0%

      \[\leadsto \mathsf{sqrt.f64}\left(re\right) \]

9. Applied egg-rr 28.0%

   \[\leadsto \color{blue}{\sqrt{re}} \]
10. Add Preprocessing

Developer Target 1: 48.1% accurate, 0.7× speedup

Math

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

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)
    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 2024164 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (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)))))