math.square on complex, real part

Percentage Accurate: 93.4% → 96.6%

Time: 1.9s

Alternatives: 4

Speedup: 0.6×

• 43.2% 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: 93.4% 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: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+304}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 1e+304) (- (* re re) (* im im)) (* im (- im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1e+304) {
		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) <= 1d+304) 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) <= 1e+304) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 1e+304:
		tmp = (re * re) - (im * im)
	else:
		tmp = im * -im
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 1e+304)
		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) <= 1e+304)
		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], 1e+304], 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) < 9.9999999999999994e303

   1. Initial program 100.0%

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

   if 9.9999999999999994e303 < (*.f64 im im)

   1. Initial program 79.0%

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

   2. Taylor expanded in re around 0 90.3%

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

      1. unpow2 90.3%

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

      2. mul-1-neg 90.3%

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

      3. distribute-rgt-neg-in 90.3%

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

   4. Simplified 90.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+304}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternative 2: 96.7% 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 94.9%

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

   1. fma-neg 97.6%

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

   2. distribute-rgt-neg-in 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 3: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8 \cdot 10^{+109} \lor \neg \left(im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (or (<= im -8e+109) (not (<= im 1.4e-57))) (* im (- im)) (* re re)))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im <= -8e+109) || !(im <= 1.4e-57)) {
		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 ((im <= (-8d+109)) .or. (.not. (im <= 1.4d-57))) 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 ((im <= -8e+109) || !(im <= 1.4e-57)) {
		tmp = im * -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im <= -8e+109) or not (im <= 1.4e-57):
		tmp = im * -im
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if ((im <= -8e+109) || !(im <= 1.4e-57))
		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 ((im <= -8e+109) || ~((im <= 1.4e-57)))
		tmp = im * -im;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[Or[LessEqual[im, -8e+109], N[Not[LessEqual[im, 1.4e-57]], $MachinePrecision]], N[(im * (-im)), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -7.99999999999999985e109 or 1.4e-57 < im

   1. Initial program 88.8%

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

   2. Taylor expanded in re around 0 84.5%

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

      1. unpow2 84.5%

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

      2. mul-1-neg 84.5%

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

      3. distribute-rgt-neg-in 84.5%

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

   4. Simplified 84.5%

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

   if -7.99999999999999985e109 < im < 1.4e-57

   1. Initial program 100.0%

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

   2. Taylor expanded in re around inf 86.2%

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

      1. unpow2 86.2%

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

   4. Simplified 86.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8 \cdot 10^{+109} \lor \neg \left(im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 4: 53.8% 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 94.9%

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

2. Taylor expanded in re around inf 54.6%

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

   1. unpow2 54.6%

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

4. Simplified 54.6%

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

5. Final simplification 54.6%

   \[\leadsto re \cdot re \]

Reproduce

herbie shell --seed 2023178 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))