math.square on complex, real part

Percentage Accurate: 94.2% → 100.0%

Time: 5.1s

Alternatives: 6

Speedup: 0.4×

• 43.4% 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: 94.2% 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \mathsf{fma}\left(re, \frac{1}{im} \cdot \frac{re}{im}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 2e+307)
   (fma re re (* im (- im)))
   (* (pow im 2.0) (fma re (* (/ 1.0 im) (/ re im)) -1.0))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 2e+307) {
		tmp = fma(re, re, (im * -im));
	} else {
		tmp = pow(im, 2.0) * fma(re, ((1.0 / im) * (re / im)), -1.0);
	}
	return tmp;
}

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 2e+307)
		tmp = fma(re, re, Float64(im * Float64(-im)));
	else
		tmp = Float64((im ^ 2.0) * fma(re, Float64(Float64(1.0 / im) * Float64(re / im)), -1.0));
	end
	return tmp
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 2e+307], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 2.0], $MachinePrecision] * N[(re * N[(N[(1.0 / im), $MachinePrecision] * N[(re / im), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 1.99999999999999997e307

   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 1.99999999999999997e307 < (*.f64 im im)

   1. Initial program 77.2%

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

   3. Taylor expanded in im around inf 77.2%

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

      1. unpow2 77.2%

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

      2. associate-/l* 89.5%

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

      3. fma-neg 89.5%

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

      4. metadata-eval 89.5%

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

   5. Simplified 89.5%

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

      1. *-un-lft-identity 89.5%

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

      2. unpow2 89.5%

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

      3. times-frac 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto {im}^{2} \cdot \mathsf{fma}\left(re, \color{blue}{\frac{1}{im} \cdot \frac{re}{im}}, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{re}^{2} \cdot \left(1 - \frac{\frac{im}{re}}{\frac{re}{im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* re re) 1e+303)
   (fma re re (* im (- im)))
   (* (pow re 2.0) (- 1.0 (/ (/ im re) (/ re im))))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 1e+303) {
		tmp = fma(re, re, (im * -im));
	} else {
		tmp = pow(re, 2.0) * (1.0 - ((im / re) / (re / im)));
	}
	return tmp;
}

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(re * re) <= 1e+303)
		tmp = fma(re, re, Float64(im * Float64(-im)));
	else
		tmp = Float64((re ^ 2.0) * Float64(1.0 - Float64(Float64(im / re) / Float64(re / im))));
	end
	return tmp
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(re * re), $MachinePrecision], 1e+303], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(N[Power[re, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(im / re), $MachinePrecision] / N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 1e303

   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 1e303 < (*.f64 re re)

   1. Initial program 77.2%

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

   3. Taylor expanded in re around inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. unsub-neg 77.2%

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

   5. Simplified 77.2%

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

      1. unpow2 77.2%

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

      2. associate-/l* 87.7%

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

   7. Applied egg-rr 87.7%

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

      1. *-un-lft-identity 87.7%

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

      2. unpow2 87.7%

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

      3. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-*r* 100.0%

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

       2. div-inv 100.0%

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

       3. clear-num 100.0%

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

       4. un-div-inv 100.0%

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

   11. Applied egg-rr 100.0%

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

Alternative 3: 96.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+307}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;-{im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 2e+307) (- (* re re) (* im im)) (- (pow im 2.0))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 2e+307) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = -pow(im, 2.0);
	}
	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+307) then
        tmp = (re * re) - (im * im)
    else
        tmp = -(im ** 2.0d0)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 2e+307) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = -Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 2e+307:
		tmp = (re * re) - (im * im)
	else:
		tmp = -math.pow(im, 2.0)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 2e+307)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(-(im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 2e+307)
		tmp = (re * re) - (im * im);
	else
		tmp = -(im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 2e+307], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], (-N[Power[im, 2.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 1.99999999999999997e307

   1. Initial program 100.0%

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

   if 1.99999999999999997e307 < (*.f64 im im)

   1. Initial program 77.2%

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

   3. Taylor expanded in re around 0 89.5%

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

      1. mul-1-neg 89.5%

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

   5. Simplified 89.5%

      \[\leadsto \color{blue}{-{im}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. sqr-neg 94.9%

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

   2. cancel-sign-sub 94.9%

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

   3. fma-define 97.6%

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

3. Simplified 97.6%

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

Alternative 5: 97.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot re - im \cdot im\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(im + re\right) \cdot \left(im + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (let* ((t_0 (- (* re re) (* im im))))
   (if (<= t_0 INFINITY) t_0 (* (+ im re) (+ im re)))))

C

double re_sqr(double re, double im) {
	double t_0 = (re * re) - (im * im);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (im + re) * (im + re);
	}
	return tmp;
}

Java

public static double re_sqr(double re, double im) {
	double t_0 = (re * re) - (im * im);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = (im + re) * (im + re);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	t_0 = (re * re) - (im * im)
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = (im + re) * (im + re)
	return tmp

Julia

function re_sqr(re, im)
	t_0 = Float64(Float64(re * re) - Float64(im * im))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(im + re) * Float64(im + re));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	t_0 = (re * re) - (im * im);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = (im + re) * (im + re);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(im + re), $MachinePrecision] * N[(im + re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 re re) (*.f64 im im)) < +inf.0

   1. Initial program 100.0%

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

   if +inf.0 < (-.f64 (*.f64 re re) (*.f64 im im))

   1. Initial program 0.0%

      \[re \cdot re - im \cdot im \]
   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 53.8%

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

      4. sqrt-unprod 69.2%

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

      5. sqr-neg 69.2%

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

      6. sqrt-prod 23.1%

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

      7. add-sqr-sqrt 46.2%

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

   4. Applied egg-rr 46.2%

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

4. Final simplification 97.2%

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

Alternative 6: 52.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

   \[re \cdot re - im \cdot im \]
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. sub-neg 100.0%

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

   3. add-sqr-sqrt 51.1%

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

   4. sqrt-unprod 75.4%

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

   5. sqr-neg 75.4%

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

   6. sqrt-prod 24.7%

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

   7. add-sqr-sqrt 51.7%

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

4. Applied egg-rr 51.7%

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

5. Final simplification 51.7%

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

Reproduce

herbie shell --seed 2024099 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))