math.square on complex, real part

Percentage Accurate: 93.9% → 97.0%

Time: 2.6s

Alternatives: 4

Speedup: 0.6×

• 42.9% 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: 93.9% 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: 97.0% 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 91.4%

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

   1. sqr-neg 91.4%

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

   2. cancel-sign-sub 91.4%

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

   3. fma-def 96.1%

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

3. Simplified 96.1%

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

4. Final simplification 96.1%

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

Alternative 2: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 5 \cdot 10^{+305}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 5e+305) (- (* re re) (* im im)) (* im (- im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 5e+305) {
		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) <= 5d+305) 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) <= 5e+305) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 5e+305:
		tmp = (re * re) - (im * im)
	else:
		tmp = im * -im
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 5e+305)
		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) <= 5e+305)
		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], 5e+305], 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) < 5.00000000000000009e305

   1. Initial program 100.0%

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

   if 5.00000000000000009e305 < (*.f64 im im)

   1. Initial program 68.1%

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

   2. Taylor expanded in re around 0 85.5%

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

      1. unpow2 85.5%

         \[\leadsto -1 \cdot \color{blue}{\left(im \cdot im\right)} \]

      2. mul-1-neg 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 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 5 \cdot 10^{+305}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternative 3: 65.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.46 \cdot 10^{-12}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= im 1.46e-12) (* re re) (* im (- im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (im <= 1.46e-12) {
		tmp = re * re;
	} 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 <= 1.46d-12) then
        tmp = re * re
    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 <= 1.46e-12) {
		tmp = re * re;
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if im <= 1.46e-12:
		tmp = re * re
	else:
		tmp = im * -im
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (im <= 1.46e-12)
		tmp = Float64(re * re);
	else
		tmp = Float64(im * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if (im <= 1.46e-12)
		tmp = re * re;
	else
		tmp = im * -im;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[im, 1.46e-12], N[(re * re), $MachinePrecision], N[(im * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.46000000000000004e-12

   1. Initial program 94.9%

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

   2. Taylor expanded in re around inf 61.9%

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

      1. unpow2 61.9%

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

   4. Simplified 61.9%

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

   if 1.46000000000000004e-12 < im

   1. Initial program 80.0%

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

   2. Taylor expanded in re around 0 75.1%

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

      1. unpow2 75.1%

         \[\leadsto -1 \cdot \color{blue}{\left(im \cdot im\right)} \]

      2. mul-1-neg 75.1%

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

      3. distribute-rgt-neg-in 75.1%

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

   4. Simplified 75.1%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.46 \cdot 10^{-12}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

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 91.4%

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

2. Taylor expanded in re around inf 53.5%

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

   1. unpow2 53.5%

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

4. Simplified 53.5%

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

5. Final simplification 53.5%

   \[\leadsto re \cdot re \]

Reproduce

herbie shell --seed 2023279 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))