math.square on complex, real part

Percentage Accurate: 93.3% → 98.1%

Time: 1.9s

Alternatives: 4

Speedup: 0.8×

• 44.1% 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: 93.3% 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: 98.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 3.8 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 3.8e+253) (fma re re (* im (- im))) (* re re)))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if (re <= 3.8e+253) {
		tmp = fma(re, re, (im * -im));
	} else {
		tmp = re * re;
	}
	return tmp;
}

Julia

re = abs(re)
function re_sqr(re, im)
	tmp = 0.0
	if (re <= 3.8e+253)
		tmp = fma(re, re, Float64(im * Float64(-im)));
	else
		tmp = Float64(re * re);
	end
	return tmp
end

Wolfram

NOTE: re should be positive before calling this function
re$95$sqr[re_, im_] := If[LessEqual[re, 3.8e+253], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.79999999999999989e253

   1. Initial program 89.7%

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

      1. fma-neg 95.5%

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

      2. distribute-rgt-neg-in 95.5%

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

   3. Simplified 95.5%

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

   if 3.79999999999999989e253 < re

   1. Initial program 66.7%

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

   2. Taylor expanded in re around inf 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.8 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 2: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 7.8 \cdot 10^{+150}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 7.8e+150) (- (* re re) (* im im)) (* re re)))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if (re <= 7.8e+150) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Fortran

NOTE: re should be positive before calling this function
real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 7.8d+150) then
        tmp = (re * re) - (im * im)
    else
        tmp = re * re
    end if
    re_sqr = tmp
end function

Java

re = Math.abs(re);
public static double re_sqr(double re, double im) {
	double tmp;
	if (re <= 7.8e+150) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

re = abs(re)
def re_sqr(re, im):
	tmp = 0
	if re <= 7.8e+150:
		tmp = (re * re) - (im * im)
	else:
		tmp = re * re
	return tmp

Julia

re = abs(re)
function re_sqr(re, im)
	tmp = 0.0
	if (re <= 7.8e+150)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

re = abs(re)
function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if (re <= 7.8e+150)
		tmp = (re * re) - (im * im);
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: re should be positive before calling this function
re$95$sqr[re_, im_] := If[LessEqual[re, 7.8e+150], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 7.79999999999999981e150

   1. Initial program 90.9%

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

   if 7.79999999999999981e150 < re

   1. Initial program 66.7%

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

   2. Taylor expanded in re around inf 87.5%

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

      1. unpow2 87.5%

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

   4. Simplified 87.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.8 \cdot 10^{+150}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 3: 65.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 0.000155:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= im 0.000155) (* re re) (* im (- im))))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if (im <= 0.000155) {
		tmp = re * re;
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Fortran

NOTE: re should be positive before calling this function
real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.000155d0) then
        tmp = re * re
    else
        tmp = im * -im
    end if
    re_sqr = tmp
end function

Java

re = Math.abs(re);
public static double re_sqr(double re, double im) {
	double tmp;
	if (im <= 0.000155) {
		tmp = re * re;
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

re = abs(re)
def re_sqr(re, im):
	tmp = 0
	if im <= 0.000155:
		tmp = re * re
	else:
		tmp = im * -im
	return tmp

Julia

re = abs(re)
function re_sqr(re, im)
	tmp = 0.0
	if (im <= 0.000155)
		tmp = Float64(re * re);
	else
		tmp = Float64(im * Float64(-im));
	end
	return tmp
end

MATLAB

re = abs(re)
function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if (im <= 0.000155)
		tmp = re * re;
	else
		tmp = im * -im;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: re should be positive before calling this function
re$95$sqr[re_, im_] := If[LessEqual[im, 0.000155], N[(re * re), $MachinePrecision], N[(im * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.55e-4

   1. Initial program 94.4%

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

   2. Taylor expanded in re around inf 67.2%

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

      1. unpow2 67.2%

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

   4. Simplified 67.2%

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

   if 1.55e-4 < im

   1. Initial program 69.5%

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

   2. Taylor expanded in re around 0 78.0%

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

      1. unpow2 78.0%

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

      2. mul-1-neg 78.0%

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

      3. distribute-rgt-neg-in 78.0%

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

   4. Simplified 78.0%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.000155:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternative 4: 53.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ re \cdot re \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im) :precision binary64 (* re re))

C

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

Fortran

NOTE: re should be positive before calling this function
real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = re * re
end function

Java

re = Math.abs(re);
public static double re_sqr(double re, double im) {
	return re * re;
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: re should be positive before calling this function
re$95$sqr[re_, im_] := N[(re * re), $MachinePrecision]

Derivation

1. Initial program 88.7%

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

2. Taylor expanded in re around inf 56.8%

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

   1. unpow2 56.8%

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

4. Simplified 56.8%

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

5. Final simplification 56.8%

   \[\leadsto re \cdot re \]

Reproduce

herbie shell --seed 2023238 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))