math.square on complex, real part

Percentage Accurate: 93.6% → 96.8%

Time: 4.1s

Alternatives: 5

Speedup: 0.5×

• 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: 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: 96.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. sqr-neg 94.9%

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

   2. cancel-sign-sub 94.9%

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

   3. fma-define 97.7%

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

3. Simplified 97.7%

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

Alternative 2: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+295}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 2e+295) (- (* re re) (* im im)) (* im (- im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 2e+295) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * -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) <= 2d+295) then
        tmp = (re * re) - (im * im)
    else
        tmp = im * -im
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 2e+295) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 2e+295:
		tmp = (re * re) - (im * im)
	else:
		tmp = im * -im
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 2e+295)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(im * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 2e+295)
		tmp = (re * re) - (im * im);
	else
		tmp = im * -im;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 2e+295], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(im * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 2e295

   1. Initial program 100.0%

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

   if 2e295 < (*.f64 im im)

   1. Initial program 77.2%

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

   3. Taylor expanded in re around 0 89.5%

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

      1. neg-mul-1 89.5%

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

   5. Simplified 89.5%

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

      1. unpow2 89.5%

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

      2. distribute-lft-neg-in 89.5%

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

   7. Applied egg-rr 89.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+295}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* re re) 2e-7) (* im (- im)) (* re re)))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 2e-7) {
		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) <= 2d-7) 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) <= 2e-7) {
		tmp = im * -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (re * re) <= 2e-7:
		tmp = im * -im
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(re * re) <= 2e-7)
		tmp = Float64(im * Float64(-im));
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((re * re) <= 2e-7)
		tmp = im * -im;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(re * re), $MachinePrecision], 2e-7], N[(im * (-im)), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 1.9999999999999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 85.2%

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

      1. neg-mul-1 85.2%

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

   5. Simplified 85.2%

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

      1. unpow2 85.2%

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

      2. distribute-lft-neg-in 85.2%

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

   7. Applied egg-rr 85.2%

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

   if 1.9999999999999999e-7 < (*.f64 re re)

   1. Initial program 89.3%

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

      1. difference-of-squares 100.0%

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

      2. sub-neg 100.0%

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

      3. add-sqr-sqrt 50.4%

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

      4. sqrt-unprod 86.6%

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

      5. sqr-neg 86.6%

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

      6. sqrt-prod 39.5%

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

      7. add-sqr-sqrt 76.7%

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

   4. Applied egg-rr 76.7%

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

   5. Taylor expanded in re around inf 86.1%

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

   6. Taylor expanded in re around inf 76.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.6% 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. Step-by-step derivation

   1. difference-of-squares 100.0%

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

   2. sub-neg 100.0%

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

   3. add-sqr-sqrt 48.7%

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

   4. sqrt-unprod 72.1%

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

   5. sqr-neg 72.1%

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

   6. sqrt-prod 24.8%

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

   7. add-sqr-sqrt 50.3%

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

4. Applied egg-rr 50.3%

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

5. Taylor expanded in re around inf 55.6%

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

6. Taylor expanded in re around inf 51.1%

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

Alternative 5: 11.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ im \cdot im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

3. Taylor expanded in re around 0 56.3%

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

   1. neg-mul-1 56.3%

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

5. Simplified 56.3%

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

   1. add-sqr-sqrt 6.0%

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

   2. sqrt-unprod 13.3%

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

   3. sqr-neg 13.3%

      \[\leadsto \sqrt{\color{blue}{{im}^{2} \cdot {im}^{2}}} \]

   4. sqrt-unprod 10.0%

      \[\leadsto \color{blue}{\sqrt{{im}^{2}} \cdot \sqrt{{im}^{2}}} \]

   5. add-sqr-sqrt 10.0%

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

   6. unpow2 10.0%

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

7. Applied egg-rr 10.0%

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

Reproduce

herbie shell --seed 2024181 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))