math.square on complex, real part

Percentage Accurate: 93.3% → 99.8%

Time: 4.9s

Alternatives: 4

Speedup: 0.5×

• 45.2% 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: 93.3% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(1 - \frac{im}{re} \cdot \frac{im}{re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 2e+300)
   (fma re re (* im (- im)))
   (* re (* re (- 1.0 (* (/ im re) (/ im re)))))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 2e+300) {
		tmp = fma(re, re, (im * -im));
	} else {
		tmp = re * (re * (1.0 - ((im / re) * (im / re))));
	}
	return tmp;
}

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 2e+300)
		tmp = fma(re, re, Float64(im * Float64(-im)));
	else
		tmp = Float64(re * Float64(re * Float64(1.0 - Float64(Float64(im / re) * Float64(im / re)))));
	end
	return tmp
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 2e+300], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(1.0 - N[(N[(im / re), $MachinePrecision] * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 2.0000000000000001e300

   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-define 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 2.0000000000000001e300 < (*.f64 im im)

   1. Initial program 76.0%

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

   3. Taylor expanded in re around inf 53.5%

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

      1. mul-1-neg 53.5%

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

      2. unsub-neg 53.5%

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

   5. Simplified 53.5%

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

      1. add-sqr-sqrt 1.4%

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

      2. pow2 1.4%

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

      3. sqrt-prod 1.4%

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

      4. sqrt-pow1 1.4%

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

      5. metadata-eval 1.4%

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

      6. pow1 1.4%

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

      7. add-sqr-sqrt 1.4%

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

      8. pow2 1.4%

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

      9. sqrt-div 1.4%

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

      10. sqrt-pow1 11.3%

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

      11. metadata-eval 11.3%

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

      12. pow1 11.3%

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

      13. sqrt-pow1 11.3%

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

      14. metadata-eval 11.3%

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

      15. pow1 11.3%

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

   7. Applied egg-rr 11.3%

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

      1. unpow2 11.3%

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

      2. *-commutative 11.3%

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

      3. *-commutative 11.3%

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

      4. swap-sqr 11.2%

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

      5. pow2 11.2%

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

      6. pow2 11.2%

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

      7. add-sqr-sqrt 77.4%

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

      8. associate-*r* 100.0%

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

      9. pow2 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. pow2 100.0%

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

   11. Applied egg-rr 100.0%

       \[\leadsto \left(\left(1 - \color{blue}{\frac{im}{re} \cdot \frac{im}{re}}\right) \cdot re\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(1 - \frac{im}{re} \cdot \frac{im}{re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+300}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(1 - \frac{im}{re} \cdot \frac{im}{re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 2e+300)
   (- (* re re) (* im im))
   (* re (* re (- 1.0 (* (/ im re) (/ im re)))))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 2e+300) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = re * (re * (1.0 - ((im / re) * (im / re))));
	}
	return tmp;
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im * im) <= 2d+300) then
        tmp = (re * re) - (im * im)
    else
        tmp = re * (re * (1.0d0 - ((im / re) * (im / re))))
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 2e+300) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = re * (re * (1.0 - ((im / re) * (im / re))));
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 2e+300:
		tmp = (re * re) - (im * im)
	else:
		tmp = re * (re * (1.0 - ((im / re) * (im / re))))
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 2e+300)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(re * Float64(re * Float64(1.0 - Float64(Float64(im / re) * Float64(im / re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 2e+300)
		tmp = (re * re) - (im * im);
	else
		tmp = re * (re * (1.0 - ((im / re) * (im / re))));
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 2e+300], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(1.0 - N[(N[(im / re), $MachinePrecision] * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 2.0000000000000001e300

   1. Initial program 100.0%

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

   if 2.0000000000000001e300 < (*.f64 im im)

   1. Initial program 76.0%

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

   3. Taylor expanded in re around inf 53.5%

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

      1. mul-1-neg 53.5%

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

      2. unsub-neg 53.5%

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

   5. Simplified 53.5%

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

      1. add-sqr-sqrt 1.4%

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

      2. pow2 1.4%

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

      3. sqrt-prod 1.4%

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

      4. sqrt-pow1 1.4%

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

      5. metadata-eval 1.4%

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

      6. pow1 1.4%

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

      7. add-sqr-sqrt 1.4%

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

      8. pow2 1.4%

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

      9. sqrt-div 1.4%

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

      10. sqrt-pow1 11.3%

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

      11. metadata-eval 11.3%

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

      12. pow1 11.3%

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

      13. sqrt-pow1 11.3%

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

      14. metadata-eval 11.3%

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

      15. pow1 11.3%

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

   7. Applied egg-rr 11.3%

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

      1. unpow2 11.3%

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

      2. *-commutative 11.3%

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

      3. *-commutative 11.3%

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

      4. swap-sqr 11.2%

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

      5. pow2 11.2%

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

      6. pow2 11.2%

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

      7. add-sqr-sqrt 77.4%

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

      8. associate-*r* 100.0%

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

      9. pow2 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. pow2 100.0%

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

   11. Applied egg-rr 100.0%

       \[\leadsto \left(\left(1 - \color{blue}{\frac{im}{re} \cdot \frac{im}{re}}\right) \cdot re\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+300}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(1 - \frac{im}{re} \cdot \frac{im}{re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 5 \cdot 10^{+302}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* re re) 5e+302) (- (* re re) (* im im)) (* re re)))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 5e+302) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re * re) <= 5d+302) then
        tmp = (re * re) - (im * im)
    else
        tmp = re * re
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 5e+302) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (re * re) <= 5e+302:
		tmp = (re * re) - (im * im)
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(re * re) <= 5e+302)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((re * re) <= 5e+302)
		tmp = (re * re) - (im * im);
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(re * re), $MachinePrecision], 5e+302], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 5e302

   1. Initial program 100.0%

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

   if 5e302 < (*.f64 re re)

   1. Initial program 73.8%

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

   3. Taylor expanded in re around inf 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   5. Simplified 73.8%

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

      1. add-sqr-sqrt 73.8%

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

      2. pow2 73.8%

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

      3. sqrt-prod 73.8%

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

      4. sqrt-pow1 73.8%

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

      5. metadata-eval 73.8%

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

      6. pow1 73.8%

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

      7. add-sqr-sqrt 73.8%

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

      8. pow2 73.8%

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

      9. sqrt-div 73.8%

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

      10. sqrt-pow1 84.6%

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

      11. metadata-eval 84.6%

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

      12. pow1 84.6%

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

      13. sqrt-pow1 84.6%

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

      14. metadata-eval 84.6%

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

      15. pow1 84.6%

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

   7. Applied egg-rr 84.6%

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

      1. unpow2 84.6%

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

      2. *-commutative 84.6%

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

      3. *-commutative 84.6%

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

      4. swap-sqr 84.6%

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

      5. pow2 84.6%

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

      6. pow2 84.6%

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

      7. add-sqr-sqrt 100.0%

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

      8. associate-*r* 100.0%

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

      9. pow2 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in im around 0 84.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 5 \cdot 10^{+302}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in re around inf 66.9%

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

   1. mul-1-neg 66.9%

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

   2. unsub-neg 66.9%

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

5. Simplified 66.9%

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

   1. add-sqr-sqrt 40.6%

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

   2. pow2 40.6%

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

   3. sqrt-prod 40.6%

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

   4. sqrt-pow1 40.6%

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

   5. metadata-eval 40.6%

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

   6. pow1 40.6%

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

   7. add-sqr-sqrt 40.6%

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

   8. pow2 40.6%

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

   9. sqrt-div 40.6%

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

   10. sqrt-pow1 43.4%

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

   11. metadata-eval 43.4%

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

   12. pow1 43.4%

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

   13. sqrt-pow1 45.3%

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

   14. metadata-eval 45.3%

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

   15. pow1 45.3%

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

7. Applied egg-rr 45.3%

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

   1. unpow2 45.3%

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

   2. *-commutative 45.3%

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

   3. *-commutative 45.3%

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

   4. swap-sqr 45.3%

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

   5. pow2 45.3%

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

   6. pow2 45.3%

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

   7. add-sqr-sqrt 78.0%

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

   8. associate-*r* 87.9%

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

   9. pow2 87.9%

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

9. Applied egg-rr 87.9%

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

10. Taylor expanded in im around 0 48.0%

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

11. Final simplification 48.0%

    \[\leadsto re \cdot re \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))