math.square on complex, real part

Percentage Accurate: 93.0% → 100.0%

Time: 4.3s

Alternatives: 3

Speedup: 1.0×

• 45.1% 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: 93.0% 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. Step-by-step derivation

   1. fma-neg 96.9%

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

   2. distribute-rgt-neg-in 96.9%

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

3. Simplified 96.9%

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

   1. distribute-rgt-neg-out 96.9%

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

   2. fma-neg 94.1%

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

   3. flip3-- 17.8%

      \[\leadsto \color{blue}{\frac{{\left(re \cdot re\right)}^{3} - {\left(im \cdot im\right)}^{3}}{\left(re \cdot re\right) \cdot \left(re \cdot re\right) + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) + \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)}} \]

   4. div-sub 17.8%

      \[\leadsto \color{blue}{\frac{{\left(re \cdot re\right)}^{3}}{\left(re \cdot re\right) \cdot \left(re \cdot re\right) + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) + \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)} - \frac{{\left(im \cdot im\right)}^{3}}{\left(re \cdot re\right) \cdot \left(re \cdot re\right) + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) + \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)}} \]

5. Applied egg-rr 17.8%

   \[\leadsto \color{blue}{\frac{{re}^{6}}{\left({re}^{4} + {im}^{4}\right) + {\left(re \cdot im\right)}^{2}} - \frac{{im}^{6}}{\left({re}^{4} + {im}^{4}\right) + {\left(re \cdot im\right)}^{2}}} \]
6. Step-by-step derivation

   1. +-commutative 17.8%

      \[\leadsto \frac{{re}^{6}}{\color{blue}{\left({im}^{4} + {re}^{4}\right)} + {\left(re \cdot im\right)}^{2}} - \frac{{im}^{6}}{\left({re}^{4} + {im}^{4}\right) + {\left(re \cdot im\right)}^{2}} \]

   2. associate-+l+ 17.8%

      \[\leadsto \frac{{re}^{6}}{\color{blue}{{im}^{4} + \left({re}^{4} + {\left(re \cdot im\right)}^{2}\right)}} - \frac{{im}^{6}}{\left({re}^{4} + {im}^{4}\right) + {\left(re \cdot im\right)}^{2}} \]

   3. *-commutative 17.8%

      \[\leadsto \frac{{re}^{6}}{{im}^{4} + \left({re}^{4} + {\color{blue}{\left(im \cdot re\right)}}^{2}\right)} - \frac{{im}^{6}}{\left({re}^{4} + {im}^{4}\right) + {\left(re \cdot im\right)}^{2}} \]

   4. +-commutative 17.8%

      \[\leadsto \frac{{re}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)} - \frac{{im}^{6}}{\color{blue}{\left({im}^{4} + {re}^{4}\right)} + {\left(re \cdot im\right)}^{2}} \]

   5. associate-+l+ 17.8%

      \[\leadsto \frac{{re}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)} - \frac{{im}^{6}}{\color{blue}{{im}^{4} + \left({re}^{4} + {\left(re \cdot im\right)}^{2}\right)}} \]

   6. *-commutative 17.8%

      \[\leadsto \frac{{re}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)} - \frac{{im}^{6}}{{im}^{4} + \left({re}^{4} + {\color{blue}{\left(im \cdot re\right)}}^{2}\right)} \]

7. Simplified 17.8%

   \[\leadsto \color{blue}{\frac{{re}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)} - \frac{{im}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)}} \]
8. Step-by-step derivation

   1. *-un-lft-identity 17.8%

      \[\leadsto \color{blue}{1 \cdot \left(\frac{{re}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)} - \frac{{im}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)}\right)} \]

   2. *-commutative 17.8%

      \[\leadsto \color{blue}{\left(\frac{{re}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)} - \frac{{im}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)}\right) \cdot 1} \]

   3. sub-div 17.8%

      \[\leadsto \color{blue}{\frac{{re}^{6} - {im}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)}} \cdot 1 \]

9. Applied egg-rr 17.8%

   \[\leadsto \color{blue}{\frac{{re}^{6} - {im}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)} \cdot 1} \]
10. Step-by-step derivation

    1. *-rgt-identity 17.8%

       \[\leadsto \color{blue}{\frac{{re}^{6} - {im}^{6}}{{im}^{4} + \left({re}^{4} + {\left(im \cdot re\right)}^{2}\right)}} \]

    2. +-commutative 17.8%

       \[\leadsto \frac{{re}^{6} - {im}^{6}}{\color{blue}{\left({re}^{4} + {\left(im \cdot re\right)}^{2}\right) + {im}^{4}}} \]

    3. +-commutative 17.8%

       \[\leadsto \frac{{re}^{6} - {im}^{6}}{\color{blue}{\left({\left(im \cdot re\right)}^{2} + {re}^{4}\right)} + {im}^{4}} \]

    4. associate-+l+ 17.8%

       \[\leadsto \frac{{re}^{6} - {im}^{6}}{\color{blue}{{\left(im \cdot re\right)}^{2} + \left({re}^{4} + {im}^{4}\right)}} \]

11. Simplified 17.8%

    \[\leadsto \color{blue}{\frac{{re}^{6} - {im}^{6}}{{\left(im \cdot re\right)}^{2} + \left({re}^{4} + {im}^{4}\right)}} \]

12. Taylor expanded in re around 0 10.6%

    \[\leadsto \frac{\color{blue}{-1 \cdot {im}^{6}}}{{\left(im \cdot re\right)}^{2} + \left({re}^{4} + {im}^{4}\right)} \]
13. Step-by-step derivation

    1. neg-mul-1 10.6%

       \[\leadsto \frac{\color{blue}{-{im}^{6}}}{{\left(im \cdot re\right)}^{2} + \left({re}^{4} + {im}^{4}\right)} \]

14. Simplified 10.6%

    \[\leadsto \frac{\color{blue}{-{im}^{6}}}{{\left(im \cdot re\right)}^{2} + \left({re}^{4} + {im}^{4}\right)} \]

15. Taylor expanded in im around inf 94.1%

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

    1. mul-1-neg 94.1%

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

    2. unsub-neg 94.1%

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

    3. unpow2 94.1%

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

    4. unpow2 94.1%

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

    5. difference-of-squares 100.0%

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

17. Simplified 100.0%

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

18. Final simplification 100.0%

    \[\leadsto \left(re + im\right) \cdot \left(re - im\right) \]

Alternative 2: 64.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{-62} \lor \neg \left(im \leq 2.05 \cdot 10^{+80}\right) \land im \leq 4.5 \cdot 10^{+100}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (or (<= im 7.5e-62) (and (not (<= im 2.05e+80)) (<= im 4.5e+100)))
   (* re re)
   (* im (- im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im <= 7.5e-62) || (!(im <= 2.05e+80) && (im <= 4.5e+100))) {
		tmp = re * re;
	} 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 <= 7.5d-62) .or. (.not. (im <= 2.05d+80)) .and. (im <= 4.5d+100)) then
        tmp = re * re
    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 <= 7.5e-62) || (!(im <= 2.05e+80) && (im <= 4.5e+100))) {
		tmp = re * re;
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im <= 7.5e-62) or (not (im <= 2.05e+80) and (im <= 4.5e+100)):
		tmp = re * re
	else:
		tmp = im * -im
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if ((im <= 7.5e-62) || (!(im <= 2.05e+80) && (im <= 4.5e+100)))
		tmp = Float64(re * re);
	else
		tmp = Float64(im * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im <= 7.5e-62) || (~((im <= 2.05e+80)) && (im <= 4.5e+100)))
		tmp = re * re;
	else
		tmp = im * -im;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[Or[LessEqual[im, 7.5e-62], And[N[Not[LessEqual[im, 2.05e+80]], $MachinePrecision], LessEqual[im, 4.5e+100]]], N[(re * re), $MachinePrecision], N[(im * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.5000000000000003e-62 or 2.05000000000000001e80 < im < 4.50000000000000036e100

   1. Initial program 95.8%

      \[re \cdot re - im \cdot im \]

   2. Taylor expanded in re around inf 58.1%

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

      1. unpow2 58.1%

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

   4. Simplified 58.1%

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

   if 7.5000000000000003e-62 < im < 2.05000000000000001e80 or 4.50000000000000036e100 < im

   1. Initial program 88.9%

      \[re \cdot re - im \cdot im \]

   2. Taylor expanded in re around 0 77.9%

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

      1. unpow2 77.9%

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

      2. mul-1-neg 77.9%

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

      3. distribute-rgt-neg-in 77.9%

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

   4. Simplified 77.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{-62} \lor \neg \left(im \leq 2.05 \cdot 10^{+80}\right) \land im \leq 4.5 \cdot 10^{+100}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternative 3: 53.3% 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. Taylor expanded in re around inf 49.6%

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

   1. unpow2 49.6%

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

4. Simplified 49.6%

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

5. Final simplification 49.6%

   \[\leadsto re \cdot re \]

Reproduce

herbie shell --seed 2023214 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))