math.square on complex, real part

Percentage Accurate: 92.9% → 100.0%

Time: 2.4s

Alternatives: 3

Speedup: 1.0×

• 44.3% 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: 92.9% 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 93.0%

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

   1. difference-of-squares 100.0%

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

   2. *-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.8 \cdot 10^{+15}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-49}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re -8.8e+15) (* re re) (if (<= re 2.9e-49) (* im (- im)) (* re re))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (re <= -8.8e+15) {
		tmp = re * re;
	} else if (re <= 2.9e-49) {
		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 <= (-8.8d+15)) then
        tmp = re * re
    else if (re <= 2.9d-49) 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 <= -8.8e+15) {
		tmp = re * re;
	} else if (re <= 2.9e-49) {
		tmp = im * -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if re <= -8.8e+15:
		tmp = re * re
	elif re <= 2.9e-49:
		tmp = im * -im
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (re <= -8.8e+15)
		tmp = Float64(re * re);
	elseif (re <= 2.9e-49)
		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 <= -8.8e+15)
		tmp = re * re;
	elseif (re <= 2.9e-49)
		tmp = im * -im;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[re, -8.8e+15], N[(re * re), $MachinePrecision], If[LessEqual[re, 2.9e-49], N[(im * (-im)), $MachinePrecision], N[(re * re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -8.8e15 or 2.9e-49 < re

   1. Initial program 86.0%

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

   2. Taylor expanded in re around inf 81.1%

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

      1. unpow2 81.1%

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

   4. Simplified 81.1%

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

   if -8.8e15 < re < 2.9e-49

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. unpow2 85.5%

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

      3. distribute-rgt-neg-in 85.5%

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

   4. Simplified 85.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.8 \cdot 10^{+15}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-49}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 3: 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 93.0%

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

2. Taylor expanded in re around inf 52.6%

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

   1. unpow2 52.6%

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

4. Simplified 52.6%

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

5. Final simplification 52.6%

   \[\leadsto re \cdot re \]

Reproduce

herbie shell --seed 2023297 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))