math.square on complex, real part

Percentage Accurate: 94.2% → 96.8%

Time: 2.0s

Alternatives: 3

Speedup: 0.6×

• 44.3% 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: 94.2% 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: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 2 \cdot 10^{+302}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* re re) 2e+302) (- (* re re) (* im im)) (* re re)))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 2e+302) {
		tmp = (re * re) - (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) <= 2d+302) then
        tmp = (re * re) - (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) <= 2e+302) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (re * re) <= 2e+302:
		tmp = (re * re) - (im * im)
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(re * re) <= 2e+302)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((re * re) <= 2e+302)
		tmp = (re * re) - (im * im);
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(re * re), $MachinePrecision], 2e+302], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 2.0000000000000002e302

   1. Initial program 100.0%

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

   if 2.0000000000000002e302 < (*.f64 re re)

   1. Initial program 69.7%

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

   2. Taylor expanded in re around inf 89.4%

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

      1. unpow2 89.4%

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

   4. Simplified 89.4%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 2 \cdot 10^{+302}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 2: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+77}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re -4.2e+77) (* re re) (if (<= re 2.15e-26) (* im (- im)) (* re re))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (re <= -4.2e+77) {
		tmp = re * re;
	} else if (re <= 2.15e-26) {
		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 <= (-4.2d+77)) then
        tmp = re * re
    else if (re <= 2.15d-26) 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 <= -4.2e+77) {
		tmp = re * re;
	} else if (re <= 2.15e-26) {
		tmp = im * -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if re <= -4.2e+77:
		tmp = re * re
	elif re <= 2.15e-26:
		tmp = im * -im
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (re <= -4.2e+77)
		tmp = Float64(re * re);
	elseif (re <= 2.15e-26)
		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 <= -4.2e+77)
		tmp = re * re;
	elseif (re <= 2.15e-26)
		tmp = im * -im;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[re, -4.2e+77], N[(re * re), $MachinePrecision], If[LessEqual[re, 2.15e-26], N[(im * (-im)), $MachinePrecision], N[(re * re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -4.1999999999999997e77 or 2.14999999999999994e-26 < re

   1. Initial program 84.6%

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

   2. Taylor expanded in re around inf 79.6%

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

      1. unpow2 79.6%

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

   4. Simplified 79.6%

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

   if -4.1999999999999997e77 < re < 2.14999999999999994e-26

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 86.3%

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

      1. unpow2 86.3%

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

      2. mul-1-neg 86.3%

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

      3. distribute-rgt-neg-in 86.3%

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

   4. Simplified 86.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+77}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;re \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 3: 53.4% 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 92.2%

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

2. Taylor expanded in re around inf 56.6%

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

   1. unpow2 56.6%

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

4. Simplified 56.6%

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

5. Final simplification 56.6%

   \[\leadsto re \cdot re \]

Reproduce

herbie shell --seed 2023187 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))