math.square on complex, real part

Percentage Accurate: 93.5% → 98.3%

Time: 2.8s

Alternatives: 4

Speedup: 0.8×

• 43.2% 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: 93.5% 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: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} \mathbf{if}\;re_m \leq 5 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(re_m, re_m, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re_m - im\right) \cdot \left(re_m - im\right)\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (if (<= re_m 5e+211)
   (fma re_m re_m (* im (- im)))
   (* (- re_m im) (- re_m im))))

C

re_m = fabs(re);
double re_sqr(double re_m, double im) {
	double tmp;
	if (re_m <= 5e+211) {
		tmp = fma(re_m, re_m, (im * -im));
	} else {
		tmp = (re_m - im) * (re_m - im);
	}
	return tmp;
}

Julia

re_m = abs(re)
function re_sqr(re_m, im)
	tmp = 0.0
	if (re_m <= 5e+211)
		tmp = fma(re_m, re_m, Float64(im * Float64(-im)));
	else
		tmp = Float64(Float64(re_m - im) * Float64(re_m - im));
	end
	return tmp
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
re$95$sqr[re$95$m_, im_] := If[LessEqual[re$95$m, 5e+211], N[(re$95$m * re$95$m + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m - im), $MachinePrecision] * N[(re$95$m - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.9999999999999995e211

   1. Initial program 95.9%

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

      1. sqr-neg 95.9%

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

      2. cancel-sign-sub 95.9%

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

      3. fma-def 98.3%

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

   3. Simplified 98.3%

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

   if 4.9999999999999995e211 < re

   1. Initial program 78.6%

      \[re \cdot re - im \cdot im \]
   2. 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 57.1%

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

      3. sqrt-prod 100.0%

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

      4. sqr-neg 100.0%

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

      5. sqrt-unprod 42.9%

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

      6. add-sqr-sqrt 100.0%

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

      7. sub-neg 100.0%

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

      8. pow1 100.0%

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

      9. pow1 100.0%

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

      10. pow-prod-up 100.0%

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

      11. metadata-eval 100.0%

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

      12. add-sqr-sqrt 100.0%

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

      13. add-sqr-sqrt 57.1%

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

      14. difference-of-squares 57.1%

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

      15. unpow-prod-down 57.1%

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

   3. Applied egg-rr 57.1%

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

      1. unpow2 57.1%

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

      2. unpow2 57.1%

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

      3. unswap-sqr 57.1%

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

      4. difference-of-squares 57.1%

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

      5. unpow1/2 57.1%

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

      6. unpow1/2 57.1%

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

      7. pow-sqr 57.1%

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

      8. metadata-eval 57.1%

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

      9. unpow1 57.1%

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

      10. unpow1/2 57.1%

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

      11. unpow1/2 57.1%

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

      12. pow-sqr 57.1%

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

      13. metadata-eval 57.1%

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

      14. unpow1 57.1%

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

      15. difference-of-squares 57.1%

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

      16. unpow1/2 57.1%

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

      17. unpow1/2 57.1%

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

      18. pow-sqr 57.1%

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

      19. metadata-eval 57.1%

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

      20. unpow1 57.1%

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

   5. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re - im\right) \cdot \left(re - im\right)\\ \end{array} \]

Alternative 2: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2000000:\\ \;\;\;\;\left(re_m - im\right) \cdot \left(re_m - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (if (<= (* im im) 2000000.0) (* (- re_m im) (- re_m im)) (* im (- im))))

C

re_m = fabs(re);
double re_sqr(double re_m, double im) {
	double tmp;
	if ((im * im) <= 2000000.0) {
		tmp = (re_m - im) * (re_m - im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Fortran

re_m = abs(re)
real(8) function re_sqr(re_m, im)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im * im) <= 2000000.0d0) then
        tmp = (re_m - im) * (re_m - im)
    else
        tmp = im * -im
    end if
    re_sqr = tmp
end function

Java

re_m = Math.abs(re);
public static double re_sqr(double re_m, double im) {
	double tmp;
	if ((im * im) <= 2000000.0) {
		tmp = (re_m - im) * (re_m - im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

re_m = math.fabs(re)
def re_sqr(re_m, im):
	tmp = 0
	if (im * im) <= 2000000.0:
		tmp = (re_m - im) * (re_m - im)
	else:
		tmp = im * -im
	return tmp

Julia

re_m = abs(re)
function re_sqr(re_m, im)
	tmp = 0.0
	if (Float64(im * im) <= 2000000.0)
		tmp = Float64(Float64(re_m - im) * Float64(re_m - im));
	else
		tmp = Float64(im * Float64(-im));
	end
	return tmp
end

MATLAB

re_m = abs(re);
function tmp_2 = re_sqr(re_m, im)
	tmp = 0.0;
	if ((im * im) <= 2000000.0)
		tmp = (re_m - im) * (re_m - im);
	else
		tmp = im * -im;
	end
	tmp_2 = tmp;
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
re$95$sqr[re$95$m_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 2000000.0], N[(N[(re$95$m - im), $MachinePrecision] * N[(re$95$m - im), $MachinePrecision]), $MachinePrecision], N[(im * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 2e6

   1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]
   2. 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 50.7%

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

      3. sqrt-prod 92.1%

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

      4. sqr-neg 92.1%

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

      5. sqrt-unprod 41.3%

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

      6. add-sqr-sqrt 83.1%

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

      7. sub-neg 83.1%

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

      8. pow1 83.1%

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

      9. pow1 83.1%

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

      10. pow-prod-up 83.1%

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

      11. metadata-eval 83.1%

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

      12. add-sqr-sqrt 45.4%

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

      13. add-sqr-sqrt 21.7%

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

      14. difference-of-squares 21.7%

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

      15. unpow-prod-down 21.7%

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

   3. Applied egg-rr 21.7%

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

      1. unpow2 21.7%

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

      2. unpow2 21.7%

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

      3. unswap-sqr 21.7%

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

      4. difference-of-squares 21.7%

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

      5. unpow1/2 21.7%

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

      6. unpow1/2 21.7%

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

      7. pow-sqr 21.8%

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

      8. metadata-eval 21.8%

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

      9. unpow1 21.8%

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

      10. unpow1/2 21.8%

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

      11. unpow1/2 21.8%

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

      12. pow-sqr 21.8%

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

      13. metadata-eval 21.8%

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

      14. unpow1 21.8%

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

      15. difference-of-squares 21.8%

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

      16. unpow1/2 21.8%

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

      17. unpow1/2 21.8%

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

      18. pow-sqr 41.9%

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

      19. metadata-eval 41.9%

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

      20. unpow1 41.9%

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

   5. Simplified 83.1%

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

   if 2e6 < (*.f64 im im)

   1. Initial program 90.0%

      \[re \cdot re - im \cdot im \]

   2. Taylor expanded in re around 0 80.8%

      \[\leadsto \color{blue}{-1 \cdot {im}^{2}} \]
   3. Step-by-step derivation

      1. mul-1-neg 80.8%

         \[\leadsto \color{blue}{-{im}^{2}} \]

   4. Simplified 80.8%

      \[\leadsto \color{blue}{-{im}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 80.8%

         \[\leadsto -\color{blue}{im \cdot im} \]

   6. Applied egg-rr 80.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2000000:\\ \;\;\;\;\left(re - im\right) \cdot \left(re - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternative 3: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} \mathbf{if}\;re_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;re_m \cdot re_m - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re_m - im\right) \cdot \left(re_m - im\right)\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (if (<= re_m 1.32e+154)
   (- (* re_m re_m) (* im im))
   (* (- re_m im) (- re_m im))))

C

re_m = fabs(re);
double re_sqr(double re_m, double im) {
	double tmp;
	if (re_m <= 1.32e+154) {
		tmp = (re_m * re_m) - (im * im);
	} else {
		tmp = (re_m - im) * (re_m - im);
	}
	return tmp;
}

Fortran

re_m = abs(re)
real(8) function re_sqr(re_m, im)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re_m <= 1.32d+154) then
        tmp = (re_m * re_m) - (im * im)
    else
        tmp = (re_m - im) * (re_m - im)
    end if
    re_sqr = tmp
end function

Java

re_m = Math.abs(re);
public static double re_sqr(double re_m, double im) {
	double tmp;
	if (re_m <= 1.32e+154) {
		tmp = (re_m * re_m) - (im * im);
	} else {
		tmp = (re_m - im) * (re_m - im);
	}
	return tmp;
}

Python

re_m = math.fabs(re)
def re_sqr(re_m, im):
	tmp = 0
	if re_m <= 1.32e+154:
		tmp = (re_m * re_m) - (im * im)
	else:
		tmp = (re_m - im) * (re_m - im)
	return tmp

Julia

re_m = abs(re)
function re_sqr(re_m, im)
	tmp = 0.0
	if (re_m <= 1.32e+154)
		tmp = Float64(Float64(re_m * re_m) - Float64(im * im));
	else
		tmp = Float64(Float64(re_m - im) * Float64(re_m - im));
	end
	return tmp
end

MATLAB

re_m = abs(re);
function tmp_2 = re_sqr(re_m, im)
	tmp = 0.0;
	if (re_m <= 1.32e+154)
		tmp = (re_m * re_m) - (im * im);
	else
		tmp = (re_m - im) * (re_m - im);
	end
	tmp_2 = tmp;
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
re$95$sqr[re$95$m_, im_] := If[LessEqual[re$95$m, 1.32e+154], N[(N[(re$95$m * re$95$m), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m - im), $MachinePrecision] * N[(re$95$m - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.31999999999999998e154

   1. Initial program 97.0%

      \[re \cdot re - im \cdot im \]

   if 1.31999999999999998e154 < re

   1. Initial program 76.0%

      \[re \cdot re - im \cdot im \]
   2. 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 56.0%

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

      3. sqrt-prod 92.0%

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

      4. sqr-neg 92.0%

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

      5. sqrt-unprod 36.0%

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

      6. add-sqr-sqrt 88.0%

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

      7. sub-neg 88.0%

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

      8. pow1 88.0%

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

      9. pow1 88.0%

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

      10. pow-prod-up 88.0%

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

      11. metadata-eval 88.0%

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

      12. add-sqr-sqrt 88.0%

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

      13. add-sqr-sqrt 52.0%

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

      14. difference-of-squares 52.0%

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

      15. unpow-prod-down 52.0%

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

   3. Applied egg-rr 52.0%

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

      1. unpow2 52.0%

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

      2. unpow2 52.0%

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

      3. unswap-sqr 52.0%

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

      4. difference-of-squares 52.0%

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

      5. unpow1/2 52.0%

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

      6. unpow1/2 52.0%

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

      7. pow-sqr 52.0%

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

      8. metadata-eval 52.0%

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

      9. unpow1 52.0%

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

      10. unpow1/2 52.0%

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

      11. unpow1/2 52.0%

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

      12. pow-sqr 52.0%

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

      13. metadata-eval 52.0%

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

      14. unpow1 52.0%

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

      15. difference-of-squares 52.0%

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

      16. unpow1/2 52.0%

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

      17. unpow1/2 52.0%

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

      18. pow-sqr 52.0%

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

      19. metadata-eval 52.0%

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

      20. unpow1 52.0%

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

   5. Simplified 88.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re - im\right) \cdot \left(re - im\right)\\ \end{array} \]

Alternative 4: 52.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

   \[re \cdot re - im \cdot im \]

2. Taylor expanded in re around 0 53.3%

   \[\leadsto \color{blue}{-1 \cdot {im}^{2}} \]
3. Step-by-step derivation

   1. mul-1-neg 53.3%

      \[\leadsto \color{blue}{-{im}^{2}} \]

4. Simplified 53.3%

   \[\leadsto \color{blue}{-{im}^{2}} \]
5. Step-by-step derivation

   1. unpow2 53.3%

      \[\leadsto -\color{blue}{im \cdot im} \]

6. Applied egg-rr 53.3%

   \[\leadsto -\color{blue}{im \cdot im} \]

7. Final simplification 53.3%

   \[\leadsto im \cdot \left(-im\right) \]

Reproduce

herbie shell --seed 2023321 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))