math.square on complex, real part

Percentage Accurate: 94.1% → 96.9%

Time: 3.6s

Alternatives: 4

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: 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: 96.9% 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 94.5%

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

   1. sqr-neg 94.5%

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

   2. cancel-sign-sub 94.5%

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

   3. fma-define 97.3%

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

3. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 2: 96.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 5 \cdot 10^{+300}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;-{im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 5e+300) (- (* re re) (* im im)) (- (pow im 2.0))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 5e+300) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = -pow(im, 2.0);
	}
	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) <= 5d+300) then
        tmp = (re * re) - (im * im)
    else
        tmp = -(im ** 2.0d0)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 5e+300) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = -Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 5e+300:
		tmp = (re * re) - (im * im)
	else:
		tmp = -math.pow(im, 2.0)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 5e+300)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(-(im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 5e+300)
		tmp = (re * re) - (im * im);
	else
		tmp = -(im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 5e+300], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], (-N[Power[im, 2.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 5.00000000000000026e300

   1. Initial program 100.0%

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

   if 5.00000000000000026e300 < (*.f64 im im)

   1. Initial program 79.1%

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

   3. Taylor expanded in re around 0 89.6%

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

      1. mul-1-neg 89.6%

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

   5. Simplified 89.6%

      \[\leadsto \color{blue}{-{im}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 5 \cdot 10^{+300}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;-{im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 2 \cdot 10^{+287}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* re re) 2e+287) (- (* re re) (* im im)) (* (+ re im) (+ re im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 2e+287) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = (re + 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 ((re * re) <= 2d+287) then
        tmp = (re * re) - (im * im)
    else
        tmp = (re + im) * (re + im)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 2e+287) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = (re + im) * (re + im);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (re * re) <= 2e+287:
		tmp = (re * re) - (im * im)
	else:
		tmp = (re + im) * (re + im)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(re * re) <= 2e+287)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(Float64(re + im) * Float64(re + im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((re * re) <= 2e+287)
		tmp = (re * re) - (im * im);
	else
		tmp = (re + im) * (re + im);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(re * re), $MachinePrecision], 2e+287], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(re + im), $MachinePrecision] * N[(re + im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 2.0000000000000002e287

   1. Initial program 100.0%

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

   if 2.0000000000000002e287 < (*.f64 re re)

   1. Initial program 81.6%

      \[re \cdot re - im \cdot im \]
   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 53.9%

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

      4. sqrt-unprod 92.1%

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

      5. sqr-neg 92.1%

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

      6. sqrt-prod 42.1%

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

      7. add-sqr-sqrt 90.8%

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

   4. Applied egg-rr 90.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 2 \cdot 10^{+287}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.9% 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 94.5%

   \[re \cdot re - im \cdot im \]
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 50.7%

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

   4. sqrt-unprod 76.8%

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

   5. sqr-neg 76.8%

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

   6. sqrt-prod 27.2%

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

   7. add-sqr-sqrt 52.8%

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

4. Applied egg-rr 52.8%

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

5. Final simplification 52.8%

   \[\leadsto \left(re + im\right) \cdot \left(re + im\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))