math.square on complex, real part

Percentage Accurate: 94.0% → 100.0%

Time: 2.4s

Alternatives: 3

Speedup: 1.2×

• 43.3% 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: 94.0% 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, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(re - im\right) \cdot \left(im + re\right) \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (* (- re im) (+ im re)))

C

double re_sqr(double re, double im) {
	return (re - im) * (im + re);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re - im) * (im + re)
end function

Java

public static double re_sqr(double re, double im) {
	return (re - im) * (im + re);
}

Python

def re_sqr(re, im):
	return (re - im) * (im + re)

Julia

function re_sqr(re, im)
	return Float64(Float64(re - im) * Float64(im + re))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re - im) * (im + re);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re - im), $MachinePrecision] * N[(im + re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re - im \cdot im \leq -2 \cdot 10^{-322}:\\ \;\;\;\;\left(-im\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (- (* re re) (* im im)) -2e-322) (* (- im) im) (* re re)))

C

double re_sqr(double re, double im) {
	double tmp;
	if (((re * re) - (im * im)) <= -2e-322) {
		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) - (im * im)) <= (-2d-322)) 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) - (im * im)) <= -2e-322) {
		tmp = -im * im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if ((re * re) - (im * im)) <= -2e-322:
		tmp = -im * im
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(Float64(re * re) - Float64(im * im)) <= -2e-322)
		tmp = Float64(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) - (im * im)) <= -2e-322)
		tmp = -im * im;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], -2e-322], N[((-im) * im), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 re re) (*.f64 im im)) < -1.97626e-322

   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

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

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -1.97626e-322 < (-.f64 (*.f64 re re) (*.f64 im im))

   1. Initial program 89.9%

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

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

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 19.4

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

   5. Applied rewrites 19.4%

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

   6. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 95.3

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

   8. Applied rewrites 95.3%

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

Alternative 3: 52.8% 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. 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

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

   2. unpow2 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 59.2

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

5. Applied rewrites 59.2%

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

6. Taylor expanded in re around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 49.8

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

8. Applied rewrites 49.8%

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

Reproduce

herbie shell --seed 2024339 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))