math.square on complex, real part

Percentage Accurate: 93.8% → 100.0%

Time: 4.3s

Alternatives: 5

Speedup: 0.6×

• 43.5% 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \mathsf{fma}\left(re, \frac{1}{im} \cdot \frac{re}{im}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 2e+209)
   (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+209) {
		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+209)
		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+209], 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) < 2.0000000000000001e209

   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 2.0000000000000001e209 < (*.f64 im im)

   1. Initial program 81.4%

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

   3. Taylor expanded in im around inf 81.4%

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

      1. unpow2 81.4%

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

      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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \left(-1 + \frac{\frac{re}{im}}{\frac{im}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 1e+210)
   (fma re re (* im (- im)))
   (* (pow im 2.0) (+ -1.0 (/ (/ re im) (/ im re))))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1e+210) {
		tmp = fma(re, re, (im * -im));
	} else {
		tmp = pow(im, 2.0) * (-1.0 + ((re / im) / (im / re)));
	}
	return tmp;
}

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 1e+210)
		tmp = fma(re, re, Float64(im * Float64(-im)));
	else
		tmp = Float64((im ^ 2.0) * Float64(-1.0 + Float64(Float64(re / im) / Float64(im / re))));
	end
	return tmp
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 1e+210], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 2.0], $MachinePrecision] * N[(-1.0 + N[(N[(re / im), $MachinePrecision] / N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 9.99999999999999927e209

   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 9.99999999999999927e209 < (*.f64 im im)

   1. Initial program 81.2%

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

   3. Taylor expanded in im around inf 81.2%

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

      1. unpow2 81.2%

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

      2. associate-/l* 89.4%

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

      3. fma-neg 89.4%

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

      4. metadata-eval 89.4%

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

   5. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. unpow2 89.4%

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

      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%

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

      1. fma-undefine 100.0%

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

      2. associate-*r* 100.0%

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

      3. div-inv 100.0%

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

      4. pow2 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow2 100.0%

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

       2. clear-num 100.0%

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

       3. un-div-inv 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \left(-1 + \frac{\frac{re}{im}}{\frac{im}{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im + re\right) \cdot \left(im + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 5e+263) (fma re re (* im (- im))) (* (+ im re) (+ im re))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (re <= 5e+263) {
		tmp = fma(re, re, (im * -im));
	} else {
		tmp = (im + re) * (im + re);
	}
	return tmp;
}

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (re <= 5e+263)
		tmp = fma(re, re, Float64(im * Float64(-im)));
	else
		tmp = Float64(Float64(im + re) * Float64(im + re));
	end
	return tmp
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[re, 5e+263], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(N[(im + re), $MachinePrecision] * N[(im + re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.00000000000000022e263

   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-define 98.3%

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

   3. Simplified 98.3%

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

   if 5.00000000000000022e263 < re

   1. Initial program 57.1%

      \[re \cdot re - im \cdot im \]
   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 50.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-prod 50.0%

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

      7. add-sqr-sqrt 92.9%

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

   4. Applied egg-rr 92.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im + re\right) \cdot \left(im + re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.3 \cdot 10^{+136}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(im + re\right) \cdot \left(im + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 3.3e+136) (- (* re re) (* im im)) (* (+ im re) (+ im re))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (re <= 3.3e+136) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = (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 (re <= 3.3d+136) then
        tmp = (re * re) - (im * im)
    else
        tmp = (im + re) * (im + re)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if (re <= 3.3e+136) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = (im + re) * (im + re);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if re <= 3.3e+136:
		tmp = (re * re) - (im * im)
	else:
		tmp = (im + re) * (im + re)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (re <= 3.3e+136)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(Float64(im + re) * Float64(im + re));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if (re <= 3.3e+136)
		tmp = (re * re) - (im * im);
	else
		tmp = (im + re) * (im + re);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[re, 3.3e+136], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(im + re), $MachinePrecision] * N[(im + re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.29999999999999992e136

   1. Initial program 96.8%

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

   if 3.29999999999999992e136 < re

   1. Initial program 75.0%

      \[re \cdot re - im \cdot im \]
   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 41.7%

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

      4. sqrt-unprod 94.4%

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

      5. sqr-neg 94.4%

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

      6. sqrt-prod 52.8%

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

      7. add-sqr-sqrt 91.7%

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

   4. Applied egg-rr 91.7%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.3 \cdot 10^{+136}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(im + re\right) \cdot \left(im + re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.1% 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 93.7%

   \[re \cdot re - im \cdot im \]
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 46.8%

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

   4. sqrt-unprod 73.0%

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

   5. sqr-neg 73.0%

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

   6. sqrt-prod 27.3%

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

   7. add-sqr-sqrt 54.2%

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

4. Applied egg-rr 54.2%

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

5. Final simplification 54.2%

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

Reproduce

herbie shell --seed 2024093 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))