math.square on complex, real part

Percentage Accurate: 93.6% → 96.6%

Time: 2.7s

Alternatives: 3

Speedup: 0.5×

• 43.1% 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: 93.6% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 1.8 \cdot 10^{+307}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 1.8e+307) (- (* re re) (* im im)) (* im (- 0.0 im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1.8e+307) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * (0.0 - 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) <= 1.8d+307) then
        tmp = (re * re) - (im * im)
    else
        tmp = im * (0.0d0 - im)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1.8e+307) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * (0.0 - im);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 1.8e+307:
		tmp = (re * re) - (im * im)
	else:
		tmp = im * (0.0 - im)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 1.8e+307)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(im * Float64(0.0 - im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 1.8e+307)
		tmp = (re * re) - (im * im);
	else
		tmp = im * (0.0 - im);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 1.8e+307], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.0 - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 1.8e307

   1. Initial program 100.0%

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

   if 1.8e307 < (*.f64 im im)

   1. Initial program 78.7%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({im}^{2}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{im}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({im}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(im \cdot \color{blue}{im}\right)\right) \]

      5. *-lowering-*.f64 90.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

   5. Simplified 90.2%

      \[\leadsto \color{blue}{0 - im \cdot im} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot im\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(im \cdot im\right)\right) \]

      3. *-lowering-*.f64 90.2%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(im, im\right)\right) \]

   7. Applied egg-rr 90.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 1.8 \cdot 10^{+307}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0 - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 1.7 \cdot 10^{+29}:\\ \;\;\;\;im \cdot \left(0 - im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* re re) 1.7e+29) (* im (- 0.0 im)) (* re re)))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 1.7e+29) {
		tmp = im * (0.0 - 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) <= 1.7d+29) then
        tmp = im * (0.0d0 - 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) <= 1.7e+29) {
		tmp = im * (0.0 - im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (re * re) <= 1.7e+29:
		tmp = im * (0.0 - im)
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(re * re) <= 1.7e+29)
		tmp = Float64(im * Float64(0.0 - im));
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((re * re) <= 1.7e+29)
		tmp = im * (0.0 - im);
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(re * re), $MachinePrecision], 1.7e+29], N[(im * N[(0.0 - im), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 1.69999999999999991e29

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({im}^{2}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{im}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({im}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(im \cdot \color{blue}{im}\right)\right) \]

      5. *-lowering-*.f64 86.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

   5. Simplified 86.3%

      \[\leadsto \color{blue}{0 - im \cdot im} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot im\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(im \cdot im\right)\right) \]

      3. *-lowering-*.f64 86.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(im, im\right)\right) \]

   7. Applied egg-rr 86.3%

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

   if 1.69999999999999991e29 < (*.f64 re re)

   1. Initial program 89.3%

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

   3. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 80.4%

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{re}\right) \]

   5. Simplified 80.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 1.7 \cdot 10^{+29}:\\ \;\;\;\;im \cdot \left(0 - im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.2% 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. Add Preprocessing

3. Taylor expanded in re around inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 53.6%

      \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{re}\right) \]

5. Simplified 53.6%

   \[\leadsto \color{blue}{re \cdot re} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))