math.square on complex, real part

Percentage Accurate: 93.8% → 98.3%

Time: 1.9s

Alternatives: 4

Speedup: 0.8×

• 43.7% 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: 93.8% 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: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 1e+201) (fma re re (* im (- im))) (* re re)))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if (re <= 1e+201) {
		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 <= 1e+201)
		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, 1e+201], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.00000000000000004e201

   1. Initial program 96.3%

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

      1. fma-neg 98.3%

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

      2. distribute-rgt-neg-in 98.3%

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

   3. Simplified 98.3%

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

   if 1.00000000000000004e201 < re

   1. Initial program 80.0%

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

   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 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 2: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+153}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 4e+153) (- (* re re) (* im im)) (* re re)))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if (re <= 4e+153) {
		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 <= 4d+153) 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 <= 4e+153) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

re = abs(re)
def re_sqr(re, im):
	tmp = 0
	if re <= 4e+153:
		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 <= 4e+153)
		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 <= 4e+153)
		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, 4e+153], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4e153

   1. Initial program 96.6%

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

   if 4e153 < re

   1. Initial program 81.0%

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

   2. Taylor expanded in re around inf 95.2%

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

      1. unpow2 95.2%

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

   4. Simplified 95.2%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+153}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 3: 78.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 4.5 \cdot 10^{+21}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* re re) 4.5e+21) (* im (- im)) (* re re)))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 4.5e+21) {
		tmp = 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 * re) <= 4.5d+21) then
        tmp = 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 * re) <= 4.5e+21) {
		tmp = im * -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

re = abs(re)
def re_sqr(re, im):
	tmp = 0
	if (re * re) <= 4.5e+21:
		tmp = im * -im
	else:
		tmp = re * re
	return tmp

Julia

re = abs(re)
function re_sqr(re, im)
	tmp = 0.0
	if (Float64(re * re) <= 4.5e+21)
		tmp = Float64(im * Float64(-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 * re) <= 4.5e+21)
		tmp = 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[N[(re * re), $MachinePrecision], 4.5e+21], N[(im * (-im)), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 4.5e21

   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. 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%

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

   if 4.5e21 < (*.f64 re re)

   1. Initial program 90.2%

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

   2. Taylor expanded in re around inf 75.2%

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

      1. unpow2 75.2%

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

   4. Simplified 75.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 4.5 \cdot 10^{+21}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 4: 53.5% 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 95.3%

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

2. Taylor expanded in re around inf 50.0%

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

   1. unpow2 50.0%

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

4. Simplified 50.0%

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

5. Final simplification 50.0%

   \[\leadsto re \cdot re \]

Reproduce

herbie shell --seed 2023240 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))