math.square on complex, real part

Percentage Accurate: 93.8% → 100.0%

Time: 4.6s

Alternatives: 3

Speedup: 1.0×

• 43.8% 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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

   \[re \cdot re - im \cdot im \]
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 49.5%

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

   3. sqrt-prod 79.0%

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

   4. sqr-neg 79.0%

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

   5. sqrt-unprod 29.8%

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

   6. add-sqr-sqrt 49.6%

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

   7. sub-neg 49.6%

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

   8. pow1 49.6%

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

   9. pow1 49.6%

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

   10. pow-prod-up 49.6%

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

   11. add-sqr-sqrt 26.4%

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

   12. add-sqr-sqrt 10.2%

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

   13. difference-of-squares 10.2%

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

   14. metadata-eval 10.2%

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

   15. unpow-prod-down 10.2%

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

4. Applied egg-rr 10.2%

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

   1. unpow2 10.2%

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

   2. unpow2 10.2%

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

   3. unswap-sqr 10.2%

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

   4. difference-of-squares 10.2%

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

   5. unpow1/2 10.2%

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

   6. unpow1/2 10.2%

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

   7. pow-sqr 10.2%

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

   8. metadata-eval 10.2%

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

   9. unpow1 10.2%

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

   10. unpow1/2 10.2%

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

   11. unpow1/2 10.2%

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

   12. pow-sqr 10.2%

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

   13. metadata-eval 10.2%

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

   14. unpow1 10.2%

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

   15. difference-of-squares 10.2%

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

   16. unpow1/2 10.2%

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

   17. unpow1/2 10.2%

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

   18. pow-sqr 19.8%

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

   19. metadata-eval 19.8%

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

   20. unpow1 19.8%

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

6. Simplified 49.6%

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

8. Applied egg-rr 22.5%

   \[\leadsto \color{blue}{\frac{{\left(re + im\right)}^{2} \cdot \left({re}^{3} - {im}^{3}\right)}{\left(re + im\right) \cdot \mathsf{fma}\left(im, re + im, {re}^{2}\right)}} \]
9. Step-by-step derivation

   1. times-frac 37.0%

      \[\leadsto \color{blue}{\frac{{\left(re + im\right)}^{2}}{re + im} \cdot \frac{{re}^{3} - {im}^{3}}{\mathsf{fma}\left(im, re + im, {re}^{2}\right)}} \]

   2. associate-*r/ 30.0%

      \[\leadsto \color{blue}{\frac{\frac{{\left(re + im\right)}^{2}}{re + im} \cdot \left({re}^{3} - {im}^{3}\right)}{\mathsf{fma}\left(im, re + im, {re}^{2}\right)}} \]

   3. unpow2 30.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(re + im\right) \cdot \left(re + im\right)}}{re + im} \cdot \left({re}^{3} - {im}^{3}\right)}{\mathsf{fma}\left(im, re + im, {re}^{2}\right)} \]

   4. associate-/l* 30.0%

      \[\leadsto \frac{\color{blue}{\frac{re + im}{\frac{re + im}{re + im}}} \cdot \left({re}^{3} - {im}^{3}\right)}{\mathsf{fma}\left(im, re + im, {re}^{2}\right)} \]

   5. *-inverses 30.0%

      \[\leadsto \frac{\frac{re + im}{\color{blue}{1}} \cdot \left({re}^{3} - {im}^{3}\right)}{\mathsf{fma}\left(im, re + im, {re}^{2}\right)} \]

   6. /-rgt-identity 30.0%

      \[\leadsto \frac{\color{blue}{\left(re + im\right)} \cdot \left({re}^{3} - {im}^{3}\right)}{\mathsf{fma}\left(im, re + im, {re}^{2}\right)} \]

   7. associate-/l* 36.9%

      \[\leadsto \color{blue}{\frac{re + im}{\frac{\mathsf{fma}\left(im, re + im, {re}^{2}\right)}{{re}^{3} - {im}^{3}}}} \]

   8. +-commutative 36.9%

      \[\leadsto \frac{\color{blue}{im + re}}{\frac{\mathsf{fma}\left(im, re + im, {re}^{2}\right)}{{re}^{3} - {im}^{3}}} \]

   9. +-commutative 36.9%

      \[\leadsto \frac{im + re}{\frac{\mathsf{fma}\left(im, \color{blue}{im + re}, {re}^{2}\right)}{{re}^{3} - {im}^{3}}} \]

10. Simplified 36.9%

    \[\leadsto \color{blue}{\frac{im + re}{\frac{\mathsf{fma}\left(im, im + re, {re}^{2}\right)}{{re}^{3} - {im}^{3}}}} \]
11. Step-by-step derivation

    1. clear-num 36.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\mathsf{fma}\left(im, im + re, {re}^{2}\right)}{{re}^{3} - {im}^{3}}}{im + re}}} \]

    2. associate-/r/ 36.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(im, im + re, {re}^{2}\right)}{{re}^{3} - {im}^{3}}} \cdot \left(im + re\right)} \]

    3. clear-num 37.0%

       \[\leadsto \color{blue}{\frac{{re}^{3} - {im}^{3}}{\mathsf{fma}\left(im, im + re, {re}^{2}\right)}} \cdot \left(im + re\right) \]

    4. fma-udef 36.9%

       \[\leadsto \frac{{re}^{3} - {im}^{3}}{\color{blue}{im \cdot \left(im + re\right) + {re}^{2}}} \cdot \left(im + re\right) \]

    5. +-commutative 36.9%

       \[\leadsto \frac{{re}^{3} - {im}^{3}}{\color{blue}{{re}^{2} + im \cdot \left(im + re\right)}} \cdot \left(im + re\right) \]

    6. distribute-rgt-in 36.9%

       \[\leadsto \frac{{re}^{3} - {im}^{3}}{{re}^{2} + \color{blue}{\left(im \cdot im + re \cdot im\right)}} \cdot \left(im + re\right) \]

    7. unpow2 36.9%

       \[\leadsto \frac{{re}^{3} - {im}^{3}}{\color{blue}{re \cdot re} + \left(im \cdot im + re \cdot im\right)} \cdot \left(im + re\right) \]

    8. flip3-- 100.0%

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

    9. +-commutative 100.0%

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

12. Applied egg-rr 100.0%

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

13. Final simplification 100.0%

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

Alternative 2: 57.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double re_sqr(double re, double im) {
	return re * (re + (im * -2.0));
}

Fortran

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

Java

public static double re_sqr(double re, double im) {
	return re * (re + (im * -2.0));
}

Python

def re_sqr(re, im):
	return re * (re + (im * -2.0))

Julia

function re_sqr(re, im)
	return Float64(re * Float64(re + Float64(im * -2.0)))
end

MATLAB

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

Wolfram

re$95$sqr[re_, im_] := N[(re * N[(re + N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[re \cdot re - im \cdot im \]
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 49.5%

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

   3. sqrt-prod 79.0%

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

   4. sqr-neg 79.0%

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

   5. sqrt-unprod 29.8%

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

   6. add-sqr-sqrt 49.6%

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

   7. sub-neg 49.6%

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

   8. pow1 49.6%

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

   9. pow1 49.6%

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

   10. pow-prod-up 49.6%

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

   11. add-sqr-sqrt 26.4%

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

   12. add-sqr-sqrt 10.2%

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

   13. difference-of-squares 10.2%

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

   14. metadata-eval 10.2%

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

   15. unpow-prod-down 10.2%

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

4. Applied egg-rr 10.2%

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

   1. unpow2 10.2%

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

   2. unpow2 10.2%

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

   3. unswap-sqr 10.2%

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

   4. difference-of-squares 10.2%

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

   5. unpow1/2 10.2%

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

   6. unpow1/2 10.2%

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

   7. pow-sqr 10.2%

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

   8. metadata-eval 10.2%

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

   9. unpow1 10.2%

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

   10. unpow1/2 10.2%

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

   11. unpow1/2 10.2%

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

   12. pow-sqr 10.2%

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

   13. metadata-eval 10.2%

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

   14. unpow1 10.2%

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

   15. difference-of-squares 10.2%

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

   16. unpow1/2 10.2%

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

   17. unpow1/2 10.2%

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

   18. pow-sqr 19.8%

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

   19. metadata-eval 19.8%

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

   20. unpow1 19.8%

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

6. Simplified 49.6%

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

7. Taylor expanded in re around inf 49.4%

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

   1. associate-*r* 49.4%

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

   2. unpow2 49.4%

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

   3. distribute-rgt-out 53.7%

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

   4. *-commutative 53.7%

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

9. Simplified 53.7%

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

10. Final simplification 53.7%

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

Alternative 3: 15.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore re_sqr (re im) :precision binary64 (* -2.0 (* re im)))

C

double re_sqr(double re, double im) {
	return -2.0 * (re * im);
}

Fortran

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

Java

public static double re_sqr(double re, double im) {
	return -2.0 * (re * im);
}

Python

def re_sqr(re, im):
	return -2.0 * (re * im)

Julia

function re_sqr(re, im)
	return Float64(-2.0 * Float64(re * im))
end

MATLAB

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

Wolfram

re$95$sqr[re_, im_] := N[(-2.0 * N[(re * im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[re \cdot re - im \cdot im \]
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 49.5%

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

   3. sqrt-prod 79.0%

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

   4. sqr-neg 79.0%

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

   5. sqrt-unprod 29.8%

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

   6. add-sqr-sqrt 49.6%

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

   7. sub-neg 49.6%

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

   8. pow1 49.6%

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

   9. pow1 49.6%

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

   10. pow-prod-up 49.6%

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

   11. add-sqr-sqrt 26.4%

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

   12. add-sqr-sqrt 10.2%

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

   13. difference-of-squares 10.2%

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

   14. metadata-eval 10.2%

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

   15. unpow-prod-down 10.2%

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

4. Applied egg-rr 10.2%

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

   1. unpow2 10.2%

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

   2. unpow2 10.2%

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

   3. unswap-sqr 10.2%

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

   4. difference-of-squares 10.2%

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

   5. unpow1/2 10.2%

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

   6. unpow1/2 10.2%

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

   7. pow-sqr 10.2%

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

   8. metadata-eval 10.2%

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

   9. unpow1 10.2%

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

   10. unpow1/2 10.2%

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

   11. unpow1/2 10.2%

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

   12. pow-sqr 10.2%

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

   13. metadata-eval 10.2%

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

   14. unpow1 10.2%

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

   15. difference-of-squares 10.2%

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

   16. unpow1/2 10.2%

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

   17. unpow1/2 10.2%

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

   18. pow-sqr 19.8%

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

   19. metadata-eval 19.8%

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

   20. unpow1 19.8%

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

6. Simplified 49.6%

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

7. Taylor expanded in re around inf 49.4%

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

   1. associate-*r* 49.4%

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

   2. unpow2 49.4%

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

   3. distribute-rgt-out 53.7%

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

   4. *-commutative 53.7%

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

9. Simplified 53.7%

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

10. Taylor expanded in re around 0 13.6%

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

11. Final simplification 13.6%

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

Reproduce

herbie shell --seed 2024029 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))