math.square on complex, real part

Percentage Accurate: 93.5% → 100.0%

Time: 6.2s

Alternatives: 5

Speedup: 1.0×

• 44.8% 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: 93.5% 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 94.1%

   \[re \cdot re - im \cdot im \]
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 48.3%

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

   4. sqrt-unprod 75.5%

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

   5. sqr-neg 75.5%

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

   6. sqrt-prod 28.2%

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

   7. add-sqr-sqrt 55.2%

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

4. Applied egg-rr 55.2%

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

   1. add-sqr-sqrt 28.2%

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

   2. sqrt-prod 75.5%

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

   3. add-sqr-sqrt 28.2%

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

   4. add-sqr-sqrt 28.2%

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

   5. sqr-neg 28.2%

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

   6. swap-sqr 28.2%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 51.4%

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

   9. distribute-rgt-neg-out 51.4%

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

   10. add-sqr-sqrt 100.0%

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

   11. sub-neg 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{-185}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;im \cdot im \leq 2 \cdot 10^{-133}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{elif}\;im \cdot im \leq 2 \cdot 10^{-61}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 2e-185)
   (* re re)
   (if (<= (* im im) 2e-133)
     (* im (- im))
     (if (<= (* im im) 2e-61) (* re re) (* im (- re im))))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 2e-185) {
		tmp = re * re;
	} else if ((im * im) <= 2e-133) {
		tmp = im * -im;
	} else if ((im * im) <= 2e-61) {
		tmp = re * re;
	} else {
		tmp = im * (re - im);
	}
	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-185) then
        tmp = re * re
    else if ((im * im) <= 2d-133) then
        tmp = im * -im
    else if ((im * im) <= 2d-61) then
        tmp = re * re
    else
        tmp = im * (re - im)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 2e-185) {
		tmp = re * re;
	} else if ((im * im) <= 2e-133) {
		tmp = im * -im;
	} else if ((im * im) <= 2e-61) {
		tmp = re * re;
	} else {
		tmp = im * (re - im);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 2e-185:
		tmp = re * re
	elif (im * im) <= 2e-133:
		tmp = im * -im
	elif (im * im) <= 2e-61:
		tmp = re * re
	else:
		tmp = im * (re - im)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 2e-185)
		tmp = Float64(re * re);
	elseif (Float64(im * im) <= 2e-133)
		tmp = Float64(im * Float64(-im));
	elseif (Float64(im * im) <= 2e-61)
		tmp = Float64(re * re);
	else
		tmp = Float64(im * Float64(re - im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 2e-185)
		tmp = re * re;
	elseif ((im * im) <= 2e-133)
		tmp = im * -im;
	elseif ((im * im) <= 2e-61)
		tmp = re * re;
	else
		tmp = im * (re - im);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 2e-185], N[(re * re), $MachinePrecision], If[LessEqual[N[(im * im), $MachinePrecision], 2e-133], N[(im * (-im)), $MachinePrecision], If[LessEqual[N[(im * im), $MachinePrecision], 2e-61], N[(re * re), $MachinePrecision], N[(im * N[(re - im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 im im) < 2e-185 or 2.0000000000000001e-133 < (*.f64 im im) < 2.0000000000000001e-61

   1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]
   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 45.1%

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

      4. sqrt-unprod 93.7%

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

      5. sqr-neg 93.7%

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

      6. sqrt-prod 48.6%

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

      7. add-sqr-sqrt 91.2%

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in re around inf 91.4%

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

   6. Taylor expanded in re around inf 91.9%

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

   if 2e-185 < (*.f64 im im) < 2.0000000000000001e-133

   1. Initial program 99.9%

      \[re \cdot re - im \cdot im \]
   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 43.4%

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

      4. sqrt-unprod 57.7%

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

      5. sqr-neg 57.7%

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

      6. sqrt-prod 13.9%

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

      7. add-sqr-sqrt 27.4%

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

   4. Applied egg-rr 27.4%

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

      1. add-sqr-sqrt 13.9%

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

      2. sqrt-prod 57.7%

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

      3. add-sqr-sqrt 13.9%

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

      4. add-sqr-sqrt 13.9%

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

      5. sqr-neg 13.9%

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

      6. swap-sqr 13.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 55.8%

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

      9. distribute-rgt-neg-out 55.8%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 72.5%

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

   8. Taylor expanded in re around 0 74.2%

      \[\leadsto \color{blue}{\left(-1 \cdot im\right)} \cdot im \]
   9. Step-by-step derivation

      1. neg-mul-1 74.2%

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

   10. Simplified 74.2%

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

   if 2.0000000000000001e-61 < (*.f64 im im)

   1. Initial program 88.2%

      \[re \cdot re - im \cdot im \]
   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 51.8%

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

      4. sqrt-unprod 61.5%

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

      5. sqr-neg 61.5%

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

      6. sqrt-prod 11.9%

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

      7. add-sqr-sqrt 26.8%

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

   4. Applied egg-rr 26.8%

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

      1. add-sqr-sqrt 11.9%

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

      2. sqrt-prod 61.5%

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

      3. add-sqr-sqrt 11.9%

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

      4. add-sqr-sqrt 11.9%

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

      5. sqr-neg 11.9%

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

      6. swap-sqr 11.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 47.9%

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

      9. distribute-rgt-neg-out 47.9%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 78.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{-185}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;im \cdot im \leq 2 \cdot 10^{-133}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{elif}\;im \cdot im \leq 2 \cdot 10^{-61}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 6 \cdot 10^{-72} \lor \neg \left(re \cdot re \leq 1.5 \cdot 10^{+61}\right) \land re \cdot re \leq 10^{+167}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (or (<= (* re re) 6e-72)
         (and (not (<= (* re re) 1.5e+61)) (<= (* re re) 1e+167)))
   (* im (- im))
   (* re re)))

C

double re_sqr(double re, double im) {
	double tmp;
	if (((re * re) <= 6e-72) || (!((re * re) <= 1.5e+61) && ((re * re) <= 1e+167))) {
		tmp = 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) <= 6d-72) .or. (.not. ((re * re) <= 1.5d+61)) .and. ((re * re) <= 1d+167)) then
        tmp = 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) <= 6e-72) || (!((re * re) <= 1.5e+61) && ((re * re) <= 1e+167))) {
		tmp = im * -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if ((re * re) <= 6e-72) or (not ((re * re) <= 1.5e+61) and ((re * re) <= 1e+167)):
		tmp = im * -im
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if ((Float64(re * re) <= 6e-72) || (!(Float64(re * re) <= 1.5e+61) && (Float64(re * re) <= 1e+167)))
		tmp = Float64(im * Float64(-im));
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if (((re * re) <= 6e-72) || (~(((re * re) <= 1.5e+61)) && ((re * re) <= 1e+167)))
		tmp = im * -im;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[Or[LessEqual[N[(re * re), $MachinePrecision], 6e-72], And[N[Not[LessEqual[N[(re * re), $MachinePrecision], 1.5e+61]], $MachinePrecision], LessEqual[N[(re * re), $MachinePrecision], 1e+167]]], N[(im * (-im)), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 6e-72 or 1.5e61 < (*.f64 re re) < 1e167

   1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]
   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 46.1%

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

      4. sqrt-unprod 65.1%

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

      5. sqr-neg 65.1%

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

      6. sqrt-prod 18.8%

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

      7. add-sqr-sqrt 29.9%

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

   4. Applied egg-rr 29.9%

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

      1. add-sqr-sqrt 18.8%

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

      2. sqrt-prod 65.1%

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

      3. add-sqr-sqrt 18.8%

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

      4. add-sqr-sqrt 18.8%

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

      5. sqr-neg 18.8%

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

      6. swap-sqr 18.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 53.5%

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

      9. distribute-rgt-neg-out 53.5%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 83.2%

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

   8. Taylor expanded in re around 0 83.7%

      \[\leadsto \color{blue}{\left(-1 \cdot im\right)} \cdot im \]
   9. Step-by-step derivation

      1. neg-mul-1 83.7%

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

   10. Simplified 83.7%

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

   if 6e-72 < (*.f64 re re) < 1.5e61 or 1e167 < (*.f64 re re)

   1. Initial program 87.7%

      \[re \cdot re - im \cdot im \]
   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 50.8%

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

      4. sqrt-unprod 86.9%

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

      5. sqr-neg 86.9%

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

      6. sqrt-prod 38.5%

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

      7. add-sqr-sqrt 83.0%

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in re around inf 89.2%

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

   6. Taylor expanded in re around inf 83.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 6 \cdot 10^{-72} \lor \neg \left(re \cdot re \leq 1.5 \cdot 10^{+61}\right) \land re \cdot re \leq 10^{+167}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.0% 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 94.1%

   \[re \cdot re - im \cdot im \]
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 48.3%

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

   4. sqrt-unprod 75.5%

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

   5. sqr-neg 75.5%

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

   6. sqrt-prod 28.2%

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

   7. add-sqr-sqrt 55.2%

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

4. Applied egg-rr 55.2%

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

5. Taylor expanded in re around inf 59.1%

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

6. Taylor expanded in re around inf 56.1%

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

Alternative 5: 11.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in re around 0 52.1%

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

   1. neg-mul-1 52.1%

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

5. Simplified 52.1%

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

   1. unpow2 52.1%

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

   2. add-sqr-sqrt 28.5%

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

   3. associate-*r* 28.5%

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

   4. distribute-rgt-neg-out 28.5%

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

   5. associate-*r* 28.5%

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

   6. add-sqr-sqrt 4.7%

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

   7. sqrt-unprod 7.6%

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

   8. swap-sqr 7.6%

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

   9. unpow2 7.6%

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

   10. metadata-eval 7.6%

       \[\leadsto \sqrt{{im}^{\color{blue}{\left(0.3333333333333333 \cdot 6\right)}} \cdot \left(\left(\sqrt{im} \cdot \left(-\sqrt{im}\right)\right) \cdot \left(\sqrt{im} \cdot \left(-\sqrt{im}\right)\right)\right)} \]

   11. pow-pow 7.6%

       \[\leadsto \sqrt{\color{blue}{{\left({im}^{0.3333333333333333}\right)}^{6}} \cdot \left(\left(\sqrt{im} \cdot \left(-\sqrt{im}\right)\right) \cdot \left(\sqrt{im} \cdot \left(-\sqrt{im}\right)\right)\right)} \]

   12. pow1/3 7.6%

       \[\leadsto \sqrt{{\color{blue}{\left(\sqrt[3]{im}\right)}}^{6} \cdot \left(\left(\sqrt{im} \cdot \left(-\sqrt{im}\right)\right) \cdot \left(\sqrt{im} \cdot \left(-\sqrt{im}\right)\right)\right)} \]

   13. swap-sqr 7.6%

       \[\leadsto \sqrt{{\left(\sqrt[3]{im}\right)}^{6} \cdot \color{blue}{\left(\left(\sqrt{im} \cdot \sqrt{im}\right) \cdot \left(\left(-\sqrt{im}\right) \cdot \left(-\sqrt{im}\right)\right)\right)}} \]

   14. sqr-neg 7.6%

       \[\leadsto \sqrt{{\left(\sqrt[3]{im}\right)}^{6} \cdot \left(\left(\sqrt{im} \cdot \sqrt{im}\right) \cdot \color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)}\right)} \]

   15. add-sqr-sqrt 7.6%

       \[\leadsto \sqrt{{\left(\sqrt[3]{im}\right)}^{6} \cdot \left(\color{blue}{im} \cdot \left(\sqrt{im} \cdot \sqrt{im}\right)\right)} \]

   16. add-sqr-sqrt 14.8%

       \[\leadsto \sqrt{{\left(\sqrt[3]{im}\right)}^{6} \cdot \left(im \cdot \color{blue}{im}\right)} \]

   17. unpow2 14.8%

       \[\leadsto \sqrt{{\left(\sqrt[3]{im}\right)}^{6} \cdot \color{blue}{{im}^{2}}} \]

   18. metadata-eval 14.8%

       \[\leadsto \sqrt{{\left(\sqrt[3]{im}\right)}^{6} \cdot {im}^{\color{blue}{\left(0.3333333333333333 \cdot 6\right)}}} \]

   19. pow-pow 7.6%

       \[\leadsto \sqrt{{\left(\sqrt[3]{im}\right)}^{6} \cdot \color{blue}{{\left({im}^{0.3333333333333333}\right)}^{6}}} \]

   20. pow1/3 14.8%

       \[\leadsto \sqrt{{\left(\sqrt[3]{im}\right)}^{6} \cdot {\color{blue}{\left(\sqrt[3]{im}\right)}}^{6}} \]

7. Applied egg-rr 12.5%

   \[\leadsto \color{blue}{im \cdot im} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))