math.square on complex, real part

Percentage Accurate: 93.7% → 97.0%

Time: 3.1s

Alternatives: 4

Speedup: 0.5×

• 43.0% 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: 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 94.1%

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

   1. sqr-neg 94.1%

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

   2. cancel-sign-sub 94.1%

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

   3. fma-def 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+44}:\\ \;\;\;\;\left(re - im\right) \cdot \left(re - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 1e+44) (* (- re im) (- re im)) (* im (- im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1e+44) {
		tmp = (re - im) * (re - im);
	} else {
		tmp = im * -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) <= 1d+44) then
        tmp = (re - im) * (re - im)
    else
        tmp = im * -im
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1e+44) {
		tmp = (re - im) * (re - im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 1e+44:
		tmp = (re - im) * (re - im)
	else:
		tmp = im * -im
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 1e+44)
		tmp = Float64(Float64(re - im) * Float64(re - im));
	else
		tmp = Float64(im * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 1e+44)
		tmp = (re - im) * (re - im);
	else
		tmp = im * -im;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 1e+44], N[(N[(re - im), $MachinePrecision] * N[(re - im), $MachinePrecision]), $MachinePrecision], N[(im * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 1.0000000000000001e44

   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. add-sqr-sqrt 44.6%

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

      3. sqrt-prod 91.1%

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

      4. sqr-neg 91.1%

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

      5. sqrt-unprod 46.4%

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

      6. add-sqr-sqrt 83.0%

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

      7. sub-neg 83.0%

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

      8. pow1 83.0%

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

      9. pow1 83.0%

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

      10. pow-prod-up 83.0%

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

      11. add-sqr-sqrt 46.1%

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

      12. add-sqr-sqrt 19.7%

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

      13. difference-of-squares 19.7%

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

      14. metadata-eval 19.7%

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

      15. unpow-prod-down 19.7%

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

   4. Applied egg-rr 19.7%

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

      1. unpow2 19.7%

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

      2. unpow2 19.7%

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

      3. unswap-sqr 19.7%

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

      4. difference-of-squares 19.7%

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

      5. unpow1/2 19.7%

         \[\leadsto \left(\color{blue}{{re}^{0.5}} \cdot \sqrt{re} - \sqrt{im} \cdot \sqrt{im}\right) \cdot \left(\left(\sqrt{re} + \sqrt{im}\right) \cdot \left(\sqrt{re} - \sqrt{im}\right)\right) \]

      6. unpow1/2 19.7%

         \[\leadsto \left({re}^{0.5} \cdot \color{blue}{{re}^{0.5}} - \sqrt{im} \cdot \sqrt{im}\right) \cdot \left(\left(\sqrt{re} + \sqrt{im}\right) \cdot \left(\sqrt{re} - \sqrt{im}\right)\right) \]

      7. pow-sqr 19.8%

         \[\leadsto \left(\color{blue}{{re}^{\left(2 \cdot 0.5\right)}} - \sqrt{im} \cdot \sqrt{im}\right) \cdot \left(\left(\sqrt{re} + \sqrt{im}\right) \cdot \left(\sqrt{re} - \sqrt{im}\right)\right) \]

      8. metadata-eval 19.8%

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

      9. unpow1 19.8%

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

      10. unpow1/2 19.8%

          \[\leadsto \left(re - \color{blue}{{im}^{0.5}} \cdot \sqrt{im}\right) \cdot \left(\left(\sqrt{re} + \sqrt{im}\right) \cdot \left(\sqrt{re} - \sqrt{im}\right)\right) \]

      11. unpow1/2 19.8%

          \[\leadsto \left(re - {im}^{0.5} \cdot \color{blue}{{im}^{0.5}}\right) \cdot \left(\left(\sqrt{re} + \sqrt{im}\right) \cdot \left(\sqrt{re} - \sqrt{im}\right)\right) \]

      12. pow-sqr 19.8%

          \[\leadsto \left(re - \color{blue}{{im}^{\left(2 \cdot 0.5\right)}}\right) \cdot \left(\left(\sqrt{re} + \sqrt{im}\right) \cdot \left(\sqrt{re} - \sqrt{im}\right)\right) \]

      13. metadata-eval 19.8%

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

      14. unpow1 19.8%

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

      15. difference-of-squares 19.8%

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

      16. unpow1/2 19.8%

          \[\leadsto \left(re - im\right) \cdot \left(\color{blue}{{re}^{0.5}} \cdot \sqrt{re} - \sqrt{im} \cdot \sqrt{im}\right) \]

      17. unpow1/2 19.8%

          \[\leadsto \left(re - im\right) \cdot \left({re}^{0.5} \cdot \color{blue}{{re}^{0.5}} - \sqrt{im} \cdot \sqrt{im}\right) \]

      18. pow-sqr 36.6%

          \[\leadsto \left(re - im\right) \cdot \left(\color{blue}{{re}^{\left(2 \cdot 0.5\right)}} - \sqrt{im} \cdot \sqrt{im}\right) \]

      19. metadata-eval 36.6%

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

      20. unpow1 36.6%

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

   6. Simplified 83.0%

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

   if 1.0000000000000001e44 < (*.f64 im im)

   1. Initial program 87.9%

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

   3. Taylor expanded in re around 0 83.5%

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

      1. mul-1-neg 83.5%

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

   5. Simplified 83.5%

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

      1. unpow2 83.5%

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

   7. Applied egg-rr 83.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+44}:\\ \;\;\;\;\left(re - im\right) \cdot \left(re - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 5 \cdot 10^{+291}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 5e+291) (- (* re re) (* im im)) (* im (- im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 5e+291) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * -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) <= 5d+291) then
        tmp = (re * re) - (im * im)
    else
        tmp = im * -im
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 5e+291) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 5e+291:
		tmp = (re * re) - (im * im)
	else:
		tmp = im * -im
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 5e+291)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(im * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 5e+291)
		tmp = (re * re) - (im * im);
	else
		tmp = im * -im;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 5e+291], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(im * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 5.0000000000000001e291

   1. Initial program 100.0%

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

   if 5.0000000000000001e291 < (*.f64 im im)

   1. Initial program 78.3%

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

   3. Taylor expanded in re around 0 92.7%

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

      1. mul-1-neg 92.7%

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

   5. Simplified 92.7%

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

      1. unpow2 92.8%

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

   7. Applied egg-rr 92.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 5 \cdot 10^{+291}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.3% accurate, 1.8× speedup

Math

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

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 * Float64(-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 56.3%

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

   1. mul-1-neg 56.3%

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

5. Simplified 56.3%

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

   1. unpow2 56.3%

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

7. Applied egg-rr 56.3%

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

8. Final simplification 56.3%

   \[\leadsto im \cdot \left(-im\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))