math.square on complex, real part

Percentage Accurate: 94.1% → 100.0%

Time: 4.4s

Alternatives: 4

Speedup: 1.0×

• 42.6% 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: 94.1% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

   1. difference-of-squares N/A

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

   2. *-commutative N/A

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

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

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

   4. --lowering--.f64 N/A

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

   5. +-lowering-+.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+85}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 1e+85) (* re re) (* im (- re im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1e+85) {
		tmp = re * re;
	} else {
		tmp = im * (re - 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+85) then
        tmp = re * re
    else
        tmp = im * (re - im)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1e+85) {
		tmp = re * re;
	} else {
		tmp = im * (re - im);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 1e+85:
		tmp = re * re
	else:
		tmp = im * (re - im)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 1e+85)
		tmp = Float64(re * re);
	else
		tmp = Float64(im * Float64(re - im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 1e+85)
		tmp = re * re;
	else
		tmp = im * (re - im);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 1e+85], N[(re * re), $MachinePrecision], N[(im * N[(re - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 1e85

   1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]
   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 79.6%

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

   5. Simplified 79.6%

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

   if 1e85 < (*.f64 im im)

   1. Initial program 83.2%

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

      1. difference-of-squares N/A

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

      2. *-commutative N/A

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

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

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

      4. --lowering--.f64 N/A

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

      5. +-lowering-+.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(re, im\right), \color{blue}{im}\right) \]
   6. Step-by-step derivation

   7. Simplified 85.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+85}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 3.8 \cdot 10^{-49}:\\ \;\;\;\;0 - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* re re) 3.8e-49) (- 0.0 (* im im)) (* re re)))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 3.8e-49) {
		tmp = 0.0 - (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) <= 3.8d-49) then
        tmp = 0.0d0 - (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) <= 3.8e-49) {
		tmp = 0.0 - (im * im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (re * re) <= 3.8e-49:
		tmp = 0.0 - (im * im)
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(re * re) <= 3.8e-49)
		tmp = Float64(0.0 - 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) <= 3.8e-49)
		tmp = 0.0 - (im * im);
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(re * re), $MachinePrecision], 3.8e-49], N[(0.0 - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 3.7999999999999997e-49

   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 86.4%

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

   5. Simplified 86.4%

      \[\leadsto \color{blue}{0 - im \cdot im} \]
   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 86.4%

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

   7. Applied egg-rr 86.4%

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

   if 3.7999999999999997e-49 < (*.f64 re re)

   1. Initial program 88.1%

      \[re \cdot re - im \cdot im \]
   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 78.3%

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

   5. Simplified 78.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 3.8 \cdot 10^{-49}:\\ \;\;\;\;0 - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.3% 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 93.3%

   \[re \cdot re - im \cdot im \]
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 58.1%

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

5. Simplified 58.1%

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

Reproduce

herbie shell --seed 2024161 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))