math.square on complex, real part

Percentage Accurate: 93.7% → 100.0%

Time: 4.9s

Alternatives: 5

Speedup: 0.8×

• 43.9% 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: 93.7% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. difference-of-squares N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(re - im\right) \cdot re\right), \color{blue}{\left(\left(re - im\right) \cdot im\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(re - im\right), re\right), \left(\color{blue}{\left(re - im\right)} \cdot im\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(re, im\right), re\right), \left(\left(\color{blue}{re} - im\right) \cdot im\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(re, im\right), re\right), \mathsf{*.f64}\left(\left(re - im\right), \color{blue}{im}\right)\right) \]

   8. --lowering--.f64 93.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(re, im\right), re\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(re, im\right), im\right)\right) \]

4. Applied egg-rr 93.7%

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

5. Final simplification 93.7%

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

Alternative 2: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \cdot im\_m \leq \infty:\\ \;\;\;\;re\_m \cdot re\_m - im\_m \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;0 - im\_m \cdot im\_m\\ \end{array} \end{array} \]

FPCore

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

C

re_m = fabs(re);
im_m = fabs(im);
double re_sqr(double re_m, double im_m) {
	double tmp;
	if ((im_m * im_m) <= ((double) INFINITY)) {
		tmp = (re_m * re_m) - (im_m * im_m);
	} else {
		tmp = 0.0 - (im_m * im_m);
	}
	return tmp;
}

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
public static double re_sqr(double re_m, double im_m) {
	double tmp;
	if ((im_m * im_m) <= Double.POSITIVE_INFINITY) {
		tmp = (re_m * re_m) - (im_m * im_m);
	} else {
		tmp = 0.0 - (im_m * im_m);
	}
	return tmp;
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
def re_sqr(re_m, im_m):
	tmp = 0
	if (im_m * im_m) <= math.inf:
		tmp = (re_m * re_m) - (im_m * im_m)
	else:
		tmp = 0.0 - (im_m * im_m)
	return tmp

Julia

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

MATLAB

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

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
re$95$sqr[re$95$m_, im$95$m_] := If[LessEqual[N[(im$95$m * im$95$m), $MachinePrecision], Infinity], N[(N[(re$95$m * re$95$m), $MachinePrecision] - N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.6%

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

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

   1. Initial program 90.6%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({im}^{2}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{im}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({im}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(im \cdot \color{blue}{im}\right)\right) \]

      5. *-lowering-*.f64 56.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

   5. Simplified 56.8%

      \[\leadsto \color{blue}{0 - im \cdot im} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot im\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(im \cdot im\right)\right) \]

      3. *-lowering-*.f64 56.8%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(im, im\right)\right) \]

   7. Applied egg-rr 56.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq \infty:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;0 - im \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re\_m \cdot re\_m \leq 3.7 \cdot 10^{+74}:\\ \;\;\;\;0 - im\_m \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;re\_m \cdot re\_m\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
(FPCore re_sqr (re_m im_m)
 :precision binary64
 (if (<= (* re_m re_m) 3.7e+74) (- 0.0 (* im_m im_m)) (* re_m re_m)))

C

re_m = fabs(re);
im_m = fabs(im);
double re_sqr(double re_m, double im_m) {
	double tmp;
	if ((re_m * re_m) <= 3.7e+74) {
		tmp = 0.0 - (im_m * im_m);
	} else {
		tmp = re_m * re_m;
	}
	return tmp;
}

Fortran

re_m = abs(re)
im_m = abs(im)
real(8) function re_sqr(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((re_m * re_m) <= 3.7d+74) then
        tmp = 0.0d0 - (im_m * im_m)
    else
        tmp = re_m * re_m
    end if
    re_sqr = tmp
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
public static double re_sqr(double re_m, double im_m) {
	double tmp;
	if ((re_m * re_m) <= 3.7e+74) {
		tmp = 0.0 - (im_m * im_m);
	} else {
		tmp = re_m * re_m;
	}
	return tmp;
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
def re_sqr(re_m, im_m):
	tmp = 0
	if (re_m * re_m) <= 3.7e+74:
		tmp = 0.0 - (im_m * im_m)
	else:
		tmp = re_m * re_m
	return tmp

Julia

re_m = abs(re)
im_m = abs(im)
function re_sqr(re_m, im_m)
	tmp = 0.0
	if (Float64(re_m * re_m) <= 3.7e+74)
		tmp = Float64(0.0 - Float64(im_m * im_m));
	else
		tmp = Float64(re_m * re_m);
	end
	return tmp
end

MATLAB

re_m = abs(re);
im_m = abs(im);
function tmp_2 = re_sqr(re_m, im_m)
	tmp = 0.0;
	if ((re_m * re_m) <= 3.7e+74)
		tmp = 0.0 - (im_m * im_m);
	else
		tmp = re_m * re_m;
	end
	tmp_2 = tmp;
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
re$95$sqr[re$95$m_, im$95$m_] := If[LessEqual[N[(re$95$m * re$95$m), $MachinePrecision], 3.7e+74], N[(0.0 - N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision], N[(re$95$m * re$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 3.7000000000000001e74

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({im}^{2}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{im}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({im}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(im \cdot \color{blue}{im}\right)\right) \]

      5. *-lowering-*.f64 84.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

   5. Simplified 84.0%

      \[\leadsto \color{blue}{0 - im \cdot im} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot im\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(im \cdot im\right)\right) \]

      3. *-lowering-*.f64 84.0%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(im, im\right)\right) \]

   7. Applied egg-rr 84.0%

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

   if 3.7000000000000001e74 < (*.f64 re re)

   1. Initial program 79.1%

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

   3. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 77.2%

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{re}\right) \]

   5. Simplified 77.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 3.7 \cdot 10^{+74}:\\ \;\;\;\;0 - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ \frac{re\_m + im\_m}{\frac{1}{re\_m - im\_m}} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
(FPCore re_sqr (re_m im_m)
 :precision binary64
 (/ (+ re_m im_m) (/ 1.0 (- re_m im_m))))

C

re_m = fabs(re);
im_m = fabs(im);
double re_sqr(double re_m, double im_m) {
	return (re_m + im_m) / (1.0 / (re_m - im_m));
}

Fortran

re_m = abs(re)
im_m = abs(im)
real(8) function re_sqr(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    re_sqr = (re_m + im_m) / (1.0d0 / (re_m - im_m))
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
public static double re_sqr(double re_m, double im_m) {
	return (re_m + im_m) / (1.0 / (re_m - im_m));
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
def re_sqr(re_m, im_m):
	return (re_m + im_m) / (1.0 / (re_m - im_m))

Julia

re_m = abs(re)
im_m = abs(im)
function re_sqr(re_m, im_m)
	return Float64(Float64(re_m + im_m) / Float64(1.0 / Float64(re_m - im_m)))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
function tmp = re_sqr(re_m, im_m)
	tmp = (re_m + im_m) / (1.0 / (re_m - im_m));
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
re$95$sqr[re$95$m_, im$95$m_] := N[(N[(re$95$m + im$95$m), $MachinePrecision] / N[(1.0 / N[(re$95$m - im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.6%

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

   1. difference-of-squares N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(re - im\right) \cdot re\right), \color{blue}{\left(\left(re - im\right) \cdot im\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(re - im\right), re\right), \left(\color{blue}{\left(re - im\right)} \cdot im\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(re, im\right), re\right), \left(\left(\color{blue}{re} - im\right) \cdot im\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(re, im\right), re\right), \mathsf{*.f64}\left(\left(re - im\right), \color{blue}{im}\right)\right) \]

   8. --lowering--.f64 93.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(re, im\right), re\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(re, im\right), im\right)\right) \]

4. Applied egg-rr 93.7%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

      \[\leadsto \frac{re + im}{\color{blue}{\frac{re + im}{re \cdot re - im \cdot im}}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(re + im\right), \color{blue}{\left(\frac{re + im}{re \cdot re - im \cdot im}\right)}\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(re, im\right), \left(\frac{\color{blue}{re + im}}{re \cdot re - im \cdot im}\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(re, im\right), \left(\frac{1}{\color{blue}{\frac{re \cdot re - im \cdot im}{re + im}}}\right)\right) \]

   9. flip-- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(re, im\right), \left(\frac{1}{re - \color{blue}{im}}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(re, im\right), \mathsf{/.f64}\left(1, \color{blue}{\left(re - im\right)}\right)\right) \]

   11. --lowering--.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(re, im\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(re, \color{blue}{im}\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{re + im}{\frac{1}{re - im}}} \]
7. Add Preprocessing

Alternative 5: 54.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ re\_m \cdot re\_m \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

3. Taylor expanded in re around inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 49.2%

      \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{re}\right) \]

5. Simplified 49.2%

   \[\leadsto \color{blue}{re \cdot re} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))