math.square on complex, real part

Percentage Accurate: 93.7% → 100.0%

Time: 4.4s

Alternatives: 3

Speedup: 1.0×

• 44.1% 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: 93.7% 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, 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-- 20.5%

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

   4. div-sub 20.5%

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

5. Applied egg-rr 20.5%

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

   1. +-commutative 20.5%

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

   2. associate-+l+ 20.5%

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

   3. *-commutative 20.5%

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

   4. +-commutative 20.5%

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

   5. associate-+l+ 20.5%

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

   6. *-commutative 20.5%

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

7. Simplified 20.5%

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

   1. *-un-lft-identity 20.5%

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

   2. *-commutative 20.5%

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

   3. sub-div 20.5%

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

   4. *-commutative 20.5%

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

9. Applied egg-rr 20.5%

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

    1. *-rgt-identity 20.5%

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

11. Simplified 20.5%

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

12. Taylor expanded in re around 0 94.1%

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

    1. unpow2 94.1%

       \[\leadsto \color{blue}{re \cdot re} + -1 \cdot {im}^{2} \]

    2. +-commutative 94.1%

       \[\leadsto \color{blue}{-1 \cdot {im}^{2} + re \cdot re} \]

    3. neg-mul-1 94.1%

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

    4. unpow2 94.1%

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

    5. distribute-rgt-neg-in 94.1%

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

    6. fma-def 97.3%

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

14. Simplified 97.3%

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

15. Taylor expanded in im around 0 94.1%

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

    1. unpow2 94.1%

       \[\leadsto \color{blue}{re \cdot re} + -1 \cdot {im}^{2} \]

    2. unpow2 94.1%

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

    3. mul-1-neg 94.1%

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

    4. unsub-neg 94.1%

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

    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: 76.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 9.5 \cdot 10^{-134} \lor \neg \left(re \cdot re \leq 6.8 \cdot 10^{-93}\right) \land re \cdot re \leq 9.5 \cdot 10^{-52}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (or (<= (* re re) 9.5e-134)
         (and (not (<= (* re re) 6.8e-93)) (<= (* re re) 9.5e-52)))
   (* im (- im))
   (* re re)))

C

double re_sqr(double re, double im) {
	double tmp;
	if (((re * re) <= 9.5e-134) || (!((re * re) <= 6.8e-93) && ((re * re) <= 9.5e-52))) {
		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) <= 9.5d-134) .or. (.not. ((re * re) <= 6.8d-93)) .and. ((re * re) <= 9.5d-52)) 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) <= 9.5e-134) || (!((re * re) <= 6.8e-93) && ((re * re) <= 9.5e-52))) {
		tmp = im * -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if ((re * re) <= 9.5e-134) or (not ((re * re) <= 6.8e-93) and ((re * re) <= 9.5e-52)):
		tmp = im * -im
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if ((Float64(re * re) <= 9.5e-134) || (!(Float64(re * re) <= 6.8e-93) && (Float64(re * re) <= 9.5e-52)))
		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) <= 9.5e-134) || (~(((re * re) <= 6.8e-93)) && ((re * re) <= 9.5e-52)))
		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], 9.5e-134], And[N[Not[LessEqual[N[(re * re), $MachinePrecision], 6.8e-93]], $MachinePrecision], LessEqual[N[(re * re), $MachinePrecision], 9.5e-52]]], N[(im * (-im)), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 9.5000000000000008e-134 or 6.80000000000000002e-93 < (*.f64 re re) < 9.50000000000000007e-52

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 88.7%

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

      1. unpow2 88.7%

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

      2. mul-1-neg 88.7%

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

      3. distribute-rgt-neg-in 88.7%

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

   4. Simplified 88.7%

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

   if 9.5000000000000008e-134 < (*.f64 re re) < 6.80000000000000002e-93 or 9.50000000000000007e-52 < (*.f64 re re)

   1. Initial program 90.2%

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

   2. Taylor expanded in re around inf 75.6%

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

      1. unpow2 75.6%

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

   4. Simplified 75.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 9.5 \cdot 10^{-134} \lor \neg \left(re \cdot re \leq 6.8 \cdot 10^{-93}\right) \land re \cdot re \leq 9.5 \cdot 10^{-52}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 3: 52.9% 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 55.9%

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

   1. unpow2 55.9%

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

4. Simplified 55.9%

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

5. Final simplification 55.9%

   \[\leadsto re \cdot re \]

Reproduce

herbie shell --seed 2023171 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))