math.square on complex, real part

Percentage Accurate: 94.0% → 100.0%

Time: 6.5s

Alternatives: 3

Speedup: 1.2×

• 43.4% 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

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

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 94.5%

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

   1. difference-of-squares N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. +-lowering-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re - im \cdot im \leq -1 \cdot 10^{-285}:\\ \;\;\;\;im \cdot \left(0 - im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (- (* re re) (* im im)) -1e-285) (* im (- 0.0 im)) (* re re)))

C

double re_sqr(double re, double im) {
	double tmp;
	if (((re * re) - (im * im)) <= -1e-285) {
		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) - (im * im)) <= (-1d-285)) 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) - (im * im)) <= -1e-285) {
		tmp = im * (0.0 - im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if ((re * re) - (im * im)) <= -1e-285:
		tmp = im * (0.0 - im)
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(Float64(re * re) - Float64(im * im)) <= -1e-285)
		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) - (im * im)) <= -1e-285)
		tmp = im * (0.0 - 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], -1e-285], N[(im * N[(0.0 - im), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 re re) (*.f64 im im)) < -1.00000000000000007e-285

   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. neg-sub0 N/A

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

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

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

      4. +-rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left({im}^{2} + 0\right)} \]

      5. unpow2 N/A

         \[\leadsto 0 - \left(\color{blue}{im \cdot im} + 0\right) \]

      6. accelerator-lowering-fma.f64 98.0

         \[\leadsto 0 - \color{blue}{\mathsf{fma}\left(im, im, 0\right)} \]

   5. Simplified 98.0%

      \[\leadsto \color{blue}{0 - \mathsf{fma}\left(im, im, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 98.0

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

   7. Applied egg-rr 98.0%

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

   if -1.00000000000000007e-285 < (-.f64 (*.f64 re re) (*.f64 im im))

   1. Initial program 90.2%

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

   3. Taylor expanded in re around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{re}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{re \cdot re} + 0 \]

      3. accelerator-lowering-fma.f64 95.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, 0\right)} \]

   5. Simplified 95.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 95.6

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

   7. Applied egg-rr 95.6%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re - im \cdot im \leq -1 \cdot 10^{-285}:\\ \;\;\;\;im \cdot \left(0 - im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.5% 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 94.5%

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

3. Taylor expanded in re around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{{re}^{2} + 0} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{re \cdot re} + 0 \]

   3. accelerator-lowering-fma.f64 54.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, 0\right)} \]

5. Simplified 54.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 54.4

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

7. Applied egg-rr 54.4%

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

Reproduce

herbie shell --seed 2024196 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))