math.square on complex, real part

Percentage Accurate: 93.8% → 100.0%

Time: 2.2s

Alternatives: 4

Speedup: 1.2×

• 56.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function re_sqr(re, im)
use fmin_fmax_functions
    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: 93.8% accurate, 1.0× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function re_sqr(re, im)
use fmin_fmax_functions
    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

\[\left(re - im\right) \cdot \left(im + re\right) \]

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function re_sqr(re, im)
use fmin_fmax_functions
    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 93.8%

   \[re \cdot re - im \cdot im \]
2. 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%

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

3. Applied rewrites 100.0%

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

Alternative 2: 61.2% accurate, 0.9× speedup

Math

\[\left(\left|re\right| - \left|im\right|\right) \cdot \left|im\right| \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

public static double re_sqr(double re, double im) {
	return (Math.abs(re) - Math.abs(im)) * Math.abs(im);
}

Python

def re_sqr(re, im):
	return (math.fabs(re) - math.fabs(im)) * math.fabs(im)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

   \[re \cdot re - im \cdot im \]
2. 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%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in re around 0

   \[\leadsto \left(re - im\right) \cdot \color{blue}{im} \]
5. Step-by-step derivation

6. Applied rewrites 57.1%

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

Alternative 3: 58.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|re\right| \leq 11500000000000000436042548346920998604259158578820579341567866944409666398873300799862902658916296072032798489292663918304046222498495385820247813778374246820388601298157568:\\ \;\;\;\;\left(-\left|im\right|\right) \cdot \left|im\right|\\ \mathbf{else}:\\ \;\;\;\;\left|re\right| \cdot \left|im\right|\\ \end{array} \]

FPCore

(FPCore re_sqr (re im)
  :precision binary64
  (if (<=
     (fabs re)
     11500000000000000436042548346920998604259158578820579341567866944409666398873300799862902658916296072032798489292663918304046222498495385820247813778374246820388601298157568)
  (* (- (fabs im)) (fabs im))
  (* (fabs re) (fabs im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (fabs(re) <= 1.15e+172) {
		tmp = -fabs(im) * fabs(im);
	} else {
		tmp = fabs(re) * fabs(im);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function re_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(re) <= 1.15d+172) then
        tmp = -abs(im) * abs(im)
    else
        tmp = abs(re) * abs(im)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if (Math.abs(re) <= 1.15e+172) {
		tmp = -Math.abs(im) * Math.abs(im);
	} else {
		tmp = Math.abs(re) * Math.abs(im);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if math.fabs(re) <= 1.15e+172:
		tmp = -math.fabs(im) * math.fabs(im)
	else:
		tmp = math.fabs(re) * math.fabs(im)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (abs(re) <= 1.15e+172)
		tmp = Float64(Float64(-abs(im)) * abs(im));
	else
		tmp = Float64(abs(re) * abs(im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if (abs(re) <= 1.15e+172)
		tmp = -abs(im) * abs(im);
	else
		tmp = abs(re) * abs(im);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[Abs[re], $MachinePrecision], 11500000000000000436042548346920998604259158578820579341567866944409666398873300799862902658916296072032798489292663918304046222498495385820247813778374246820388601298157568], N[((-N[Abs[im], $MachinePrecision]) * N[Abs[im], $MachinePrecision]), $MachinePrecision], N[(N[Abs[re], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.15e172

   1. Initial program 93.8%

      \[re \cdot re - im \cdot im \]
   2. 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%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

      \[\leadsto \left(re - im\right) \cdot \color{blue}{im} \]
   5. Step-by-step derivation

   6. Applied rewrites 57.1%

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

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(-1 \cdot im\right)} \cdot im \]
   8. Step-by-step derivation

      1. lower-*.f64 54.0%

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

   9. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\left(-1 \cdot im\right)} \cdot im \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 54.0%

          \[\leadsto \left(-im\right) \cdot im \]

   11. Applied rewrites 54.0%

       \[\leadsto \left(-im\right) \cdot im \]

   if 1.15e172 < re

   1. Initial program 93.8%

      \[re \cdot re - im \cdot im \]
   2. 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%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

      \[\leadsto \left(re - im\right) \cdot \color{blue}{im} \]
   5. Step-by-step derivation

   6. Applied rewrites 57.1%

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

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(-1 \cdot im\right)} \cdot im \]
   8. Step-by-step derivation

      1. lower-*.f64 54.0%

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

   9. Applied rewrites 54.0%

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

   10. Taylor expanded in re around inf

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

   12. Applied rewrites 14.8%

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

Alternative 4: 15.1% accurate, 1.4× speedup

Math

\[\left|re\right| \cdot \left|im\right| \]

FPCore

(FPCore re_sqr (re im)
  :precision binary64
  (* (fabs re) (fabs im)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function re_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = abs(re) * abs(im)
end function

Java

public static double re_sqr(double re, double im) {
	return Math.abs(re) * Math.abs(im);
}

Python

def re_sqr(re, im):
	return math.fabs(re) * math.fabs(im)

Julia

function re_sqr(re, im)
	return Float64(abs(re) * abs(im))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

   \[re \cdot re - im \cdot im \]
2. 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%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in re around 0

   \[\leadsto \left(re - im\right) \cdot \color{blue}{im} \]
5. Step-by-step derivation

6. Applied rewrites 57.1%

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

7. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\left(-1 \cdot im\right)} \cdot im \]
8. Step-by-step derivation

   1. lower-*.f64 54.0%

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

9. Applied rewrites 54.0%

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

10. Taylor expanded in re around inf

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

12. Applied rewrites 14.8%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))