math.square on complex, real part

Percentage Accurate: 93.8% → 99.7%

Time: 3.2s

Alternatives: 4

Speedup: 0.5×

• 43.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ re \cdot re - im \cdot im \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

C

double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Python

def re_sqr(re, im):
	return (re * re) - (im * im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ re \cdot re - im \cdot im \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

C

double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Python

def re_sqr(re, im):
	return (re * re) - (im * im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re\_m \cdot re\_m \leq 5 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(re\_m, re\_m, im\_m \cdot \left(-im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\_m \cdot \left(re\_m + im\_m \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
(FPCore re_sqr (re_m im_m)
 :precision binary64
 (if (<= (* re_m re_m) 5e+252)
   (fma re_m re_m (* im_m (- im_m)))
   (* re_m (+ re_m (* im_m -2.0)))))

C

re_m = fabs(re);
im_m = fabs(im);
double re_sqr(double re_m, double im_m) {
	double tmp;
	if ((re_m * re_m) <= 5e+252) {
		tmp = fma(re_m, re_m, (im_m * -im_m));
	} else {
		tmp = re_m * (re_m + (im_m * -2.0));
	}
	return tmp;
}

Julia

re_m = abs(re)
im_m = abs(im)
function re_sqr(re_m, im_m)
	tmp = 0.0
	if (Float64(re_m * re_m) <= 5e+252)
		tmp = fma(re_m, re_m, Float64(im_m * Float64(-im_m)));
	else
		tmp = Float64(re_m * Float64(re_m + Float64(im_m * -2.0)));
	end
	return tmp
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
re$95$sqr[re$95$m_, im$95$m_] := If[LessEqual[N[(re$95$m * re$95$m), $MachinePrecision], 5e+252], N[(re$95$m * re$95$m + N[(im$95$m * (-im$95$m)), $MachinePrecision]), $MachinePrecision], N[(re$95$m * N[(re$95$m + N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 4.9999999999999997e252

   1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]
   2. Step-by-step derivation

      1. sqr-neg 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)} \]
   4. Add Preprocessing

   if 4.9999999999999997e252 < (*.f64 re re)

   1. Initial program 76.4%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. difference-of-squares 100.0%

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

      2. add-sqr-sqrt 54.2%

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

      3. sqrt-prod 79.2%

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

      4. sqr-neg 79.2%

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

      5. sqrt-unprod 34.7%

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

      6. add-sqr-sqrt 83.3%

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

      7. sub-neg 83.3%

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

      8. pow1 83.3%

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

      9. pow1 83.3%

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

      10. pow-prod-up 83.3%

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

      11. add-sqr-sqrt 40.2%

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

      12. add-sqr-sqrt 25.0%

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

      13. difference-of-squares 25.0%

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

      14. metadata-eval 25.0%

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

      15. unpow-prod-down 25.0%

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

   4. Applied egg-rr 25.0%

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

      1. unpow2 25.0%

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

      2. unpow2 25.0%

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

      3. unswap-sqr 25.0%

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

      4. difference-of-squares 25.0%

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

      5. unpow1/2 25.0%

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

      6. unpow1/2 25.0%

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

      7. pow-sqr 25.0%

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

      8. metadata-eval 25.0%

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

      9. unpow1 25.0%

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

      10. unpow1/2 25.0%

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

      11. unpow1/2 25.0%

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

      12. pow-sqr 25.0%

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

      13. metadata-eval 25.0%

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

      14. unpow1 25.0%

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

      15. difference-of-squares 25.0%

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

      16. unpow1/2 25.0%

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

      17. unpow1/2 25.0%

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

      18. pow-sqr 48.6%

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

      19. metadata-eval 48.6%

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

      20. unpow1 48.6%

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

   6. Simplified 83.3%

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

   7. Taylor expanded in re around inf 72.2%

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

      1. associate-*r* 72.2%

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

      2. unpow2 72.2%

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

      3. distribute-rgt-out 88.9%

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

      4. *-commutative 88.9%

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

   9. Simplified 88.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 5 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re + im \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re\_m \cdot re\_m \leq 5 \cdot 10^{+252}:\\ \;\;\;\;re\_m \cdot re\_m - im\_m \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;re\_m \cdot \left(re\_m + im\_m \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
(FPCore re_sqr (re_m im_m)
 :precision binary64
 (if (<= (* re_m re_m) 5e+252)
   (- (* re_m re_m) (* im_m im_m))
   (* re_m (+ re_m (* im_m -2.0)))))

C

re_m = fabs(re);
im_m = fabs(im);
double re_sqr(double re_m, double im_m) {
	double tmp;
	if ((re_m * re_m) <= 5e+252) {
		tmp = (re_m * re_m) - (im_m * im_m);
	} else {
		tmp = re_m * (re_m + (im_m * -2.0));
	}
	return tmp;
}

Fortran

re_m = abs(re)
im_m = abs(im)
real(8) function re_sqr(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((re_m * re_m) <= 5d+252) then
        tmp = (re_m * re_m) - (im_m * im_m)
    else
        tmp = re_m * (re_m + (im_m * (-2.0d0)))
    end if
    re_sqr = tmp
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
public static double re_sqr(double re_m, double im_m) {
	double tmp;
	if ((re_m * re_m) <= 5e+252) {
		tmp = (re_m * re_m) - (im_m * im_m);
	} else {
		tmp = re_m * (re_m + (im_m * -2.0));
	}
	return tmp;
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
def re_sqr(re_m, im_m):
	tmp = 0
	if (re_m * re_m) <= 5e+252:
		tmp = (re_m * re_m) - (im_m * im_m)
	else:
		tmp = re_m * (re_m + (im_m * -2.0))
	return tmp

Julia

re_m = abs(re)
im_m = abs(im)
function re_sqr(re_m, im_m)
	tmp = 0.0
	if (Float64(re_m * re_m) <= 5e+252)
		tmp = Float64(Float64(re_m * re_m) - Float64(im_m * im_m));
	else
		tmp = Float64(re_m * Float64(re_m + Float64(im_m * -2.0)));
	end
	return tmp
end

MATLAB

re_m = abs(re);
im_m = abs(im);
function tmp_2 = re_sqr(re_m, im_m)
	tmp = 0.0;
	if ((re_m * re_m) <= 5e+252)
		tmp = (re_m * re_m) - (im_m * im_m);
	else
		tmp = re_m * (re_m + (im_m * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
re$95$sqr[re$95$m_, im$95$m_] := If[LessEqual[N[(re$95$m * re$95$m), $MachinePrecision], 5e+252], N[(N[(re$95$m * re$95$m), $MachinePrecision] - N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision], N[(re$95$m * N[(re$95$m + N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 4.9999999999999997e252

   1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing

   if 4.9999999999999997e252 < (*.f64 re re)

   1. Initial program 76.4%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. difference-of-squares 100.0%

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

      2. add-sqr-sqrt 54.2%

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

      3. sqrt-prod 79.2%

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

      4. sqr-neg 79.2%

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

      5. sqrt-unprod 34.7%

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

      6. add-sqr-sqrt 83.3%

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

      7. sub-neg 83.3%

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

      8. pow1 83.3%

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

      9. pow1 83.3%

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

      10. pow-prod-up 83.3%

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

      11. add-sqr-sqrt 40.2%

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

      12. add-sqr-sqrt 25.0%

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

      13. difference-of-squares 25.0%

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

      14. metadata-eval 25.0%

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

      15. unpow-prod-down 25.0%

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

   4. Applied egg-rr 25.0%

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

      1. unpow2 25.0%

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

      2. unpow2 25.0%

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

      3. unswap-sqr 25.0%

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

      4. difference-of-squares 25.0%

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

      5. unpow1/2 25.0%

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

      6. unpow1/2 25.0%

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

      7. pow-sqr 25.0%

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

      8. metadata-eval 25.0%

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

      9. unpow1 25.0%

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

      10. unpow1/2 25.0%

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

      11. unpow1/2 25.0%

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

      12. pow-sqr 25.0%

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

      13. metadata-eval 25.0%

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

      14. unpow1 25.0%

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

      15. difference-of-squares 25.0%

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

      16. unpow1/2 25.0%

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

      17. unpow1/2 25.0%

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

      18. pow-sqr 48.6%

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

      19. metadata-eval 48.6%

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

      20. unpow1 48.6%

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

   6. Simplified 83.3%

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

   7. Taylor expanded in re around inf 72.2%

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

      1. associate-*r* 72.2%

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

      2. unpow2 72.2%

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

      3. distribute-rgt-out 88.9%

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

      4. *-commutative 88.9%

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

   9. Simplified 88.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 5 \cdot 10^{+252}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re + im \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ re\_m \cdot \left(re\_m + im\_m \cdot -2\right) \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
(FPCore re_sqr (re_m im_m) :precision binary64 (* re_m (+ re_m (* im_m -2.0))))

C

re_m = fabs(re);
im_m = fabs(im);
double re_sqr(double re_m, double im_m) {
	return re_m * (re_m + (im_m * -2.0));
}

Fortran

re_m = abs(re)
im_m = abs(im)
real(8) function re_sqr(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    re_sqr = re_m * (re_m + (im_m * (-2.0d0)))
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
public static double re_sqr(double re_m, double im_m) {
	return re_m * (re_m + (im_m * -2.0));
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
def re_sqr(re_m, im_m):
	return re_m * (re_m + (im_m * -2.0))

Julia

re_m = abs(re)
im_m = abs(im)
function re_sqr(re_m, im_m)
	return Float64(re_m * Float64(re_m + Float64(im_m * -2.0)))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
function tmp = re_sqr(re_m, im_m)
	tmp = re_m * (re_m + (im_m * -2.0));
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
re$95$sqr[re$95$m_, im$95$m_] := N[(re$95$m * N[(re$95$m + N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.4%

   \[re \cdot re - im \cdot im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. difference-of-squares 100.0%

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

   2. add-sqr-sqrt 51.1%

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

   3. sqrt-prod 73.1%

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

   4. sqr-neg 73.1%

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

   5. sqrt-unprod 24.7%

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

   6. add-sqr-sqrt 52.2%

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

   7. sub-neg 52.2%

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

   8. pow1 52.2%

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

   9. pow1 52.2%

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

   10. pow-prod-up 52.2%

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

   11. add-sqr-sqrt 27.7%

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

   12. add-sqr-sqrt 15.2%

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

   13. difference-of-squares 15.2%

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

   14. metadata-eval 15.2%

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

   15. unpow-prod-down 15.2%

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

4. Applied egg-rr 15.2%

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

   1. unpow2 15.2%

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

   2. unpow2 15.2%

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

   3. unswap-sqr 15.2%

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

   4. difference-of-squares 15.2%

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

   5. unpow1/2 15.2%

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

   6. unpow1/2 15.2%

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

   7. pow-sqr 15.2%

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

   8. metadata-eval 15.2%

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

   9. unpow1 15.2%

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

   10. unpow1/2 15.2%

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

   11. unpow1/2 15.2%

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

   12. pow-sqr 15.2%

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

   13. metadata-eval 15.2%

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

   14. unpow1 15.2%

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

   15. difference-of-squares 15.2%

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

   16. unpow1/2 15.2%

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

   17. unpow1/2 15.2%

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

   18. pow-sqr 27.5%

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

   19. metadata-eval 27.5%

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

   20. unpow1 27.5%

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

6. Simplified 52.2%

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

7. Taylor expanded in re around inf 50.7%

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

   1. associate-*r* 50.7%

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

   2. unpow2 50.7%

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

   3. distribute-rgt-out 55.4%

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

   4. *-commutative 55.4%

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

9. Simplified 55.4%

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

10. Final simplification 55.4%

    \[\leadsto re \cdot \left(re + im \cdot -2\right) \]
11. Add Preprocessing

Alternative 4: 15.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ re\_m \cdot \left(im\_m \cdot -2\right) \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
(FPCore re_sqr (re_m im_m) :precision binary64 (* re_m (* im_m -2.0)))

C

re_m = fabs(re);
im_m = fabs(im);
double re_sqr(double re_m, double im_m) {
	return re_m * (im_m * -2.0);
}

Fortran

re_m = abs(re)
im_m = abs(im)
real(8) function re_sqr(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    re_sqr = re_m * (im_m * (-2.0d0))
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
public static double re_sqr(double re_m, double im_m) {
	return re_m * (im_m * -2.0);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
def re_sqr(re_m, im_m):
	return re_m * (im_m * -2.0)

Julia

re_m = abs(re)
im_m = abs(im)
function re_sqr(re_m, im_m)
	return Float64(re_m * Float64(im_m * -2.0))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
function tmp = re_sqr(re_m, im_m)
	tmp = re_m * (im_m * -2.0);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
re$95$sqr[re$95$m_, im$95$m_] := N[(re$95$m * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.4%

   \[re \cdot re - im \cdot im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. difference-of-squares 100.0%

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

   2. add-sqr-sqrt 51.1%

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

   3. sqrt-prod 73.1%

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

   4. sqr-neg 73.1%

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

   5. sqrt-unprod 24.7%

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

   6. add-sqr-sqrt 52.2%

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

   7. sub-neg 52.2%

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

   8. pow1 52.2%

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

   9. pow1 52.2%

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

   10. pow-prod-up 52.2%

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

   11. add-sqr-sqrt 27.7%

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

   12. add-sqr-sqrt 15.2%

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

   13. difference-of-squares 15.2%

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

   14. metadata-eval 15.2%

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

   15. unpow-prod-down 15.2%

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

4. Applied egg-rr 15.2%

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

   1. unpow2 15.2%

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

   2. unpow2 15.2%

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

   3. unswap-sqr 15.2%

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

   4. difference-of-squares 15.2%

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

   5. unpow1/2 15.2%

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

   6. unpow1/2 15.2%

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

   7. pow-sqr 15.2%

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

   8. metadata-eval 15.2%

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

   9. unpow1 15.2%

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

   10. unpow1/2 15.2%

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

   11. unpow1/2 15.2%

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

   12. pow-sqr 15.2%

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

   13. metadata-eval 15.2%

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

   14. unpow1 15.2%

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

   15. difference-of-squares 15.2%

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

   16. unpow1/2 15.2%

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

   17. unpow1/2 15.2%

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

   18. pow-sqr 27.5%

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

   19. metadata-eval 27.5%

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

   20. unpow1 27.5%

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

6. Simplified 52.2%

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

7. Taylor expanded in re around inf 50.7%

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

   1. associate-*r* 50.7%

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

   2. unpow2 50.7%

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

   3. distribute-rgt-out 55.4%

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

   4. *-commutative 55.4%

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

9. Simplified 55.4%

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

10. Taylor expanded in re around 0 16.6%

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

    1. *-commutative 16.6%

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

    2. *-commutative 16.6%

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

    3. associate-*r* 16.6%

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

    4. *-commutative 16.6%

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

12. Simplified 16.6%

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

13. Final simplification 16.6%

    \[\leadsto re \cdot \left(im \cdot -2\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))