math.square on complex, real part

Percentage Accurate: 93.5% → 98.3%

Time: 2.0s

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: 93.5% 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

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 4e+224) (fma re re (* im (- im))) (* re re)))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if (re <= 4e+224) {
		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 <= 4e+224)
		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, 4e+224], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.99999999999999988e224

   1. Initial program 96.6%

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

      1. fma-neg 99.1%

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

      2. distribute-rgt-neg-in 99.1%

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

   3. Simplified 99.1%

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

   if 3.99999999999999988e224 < re

   1. Initial program 72.7%

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

   2. Taylor expanded in re around inf 95.5%

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

      1. unpow2 95.5%

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

   4. Simplified 95.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 2: 76.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 4.6 \cdot 10^{-173} \lor \neg \left(re \cdot re \leq 1.64 \cdot 10^{-108}\right) \land re \cdot re \leq 4.2 \cdot 10^{+45}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (or (<= (* re re) 4.6e-173)
         (and (not (<= (* re re) 1.64e-108)) (<= (* re re) 4.2e+45)))
   (* im (- im))
   (* re re)))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if (((re * re) <= 4.6e-173) || (!((re * re) <= 1.64e-108) && ((re * re) <= 4.2e+45))) {
		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) <= 4.6d-173) .or. (.not. ((re * re) <= 1.64d-108)) .and. ((re * re) <= 4.2d+45)) 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) <= 4.6e-173) || (!((re * re) <= 1.64e-108) && ((re * re) <= 4.2e+45))) {
		tmp = im * -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

re = abs(re)
def re_sqr(re, im):
	tmp = 0
	if ((re * re) <= 4.6e-173) or (not ((re * re) <= 1.64e-108) and ((re * re) <= 4.2e+45)):
		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) <= 4.6e-173) || (!(Float64(re * re) <= 1.64e-108) && (Float64(re * re) <= 4.2e+45)))
		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) <= 4.6e-173) || (~(((re * re) <= 1.64e-108)) && ((re * re) <= 4.2e+45)))
		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[Or[LessEqual[N[(re * re), $MachinePrecision], 4.6e-173], And[N[Not[LessEqual[N[(re * re), $MachinePrecision], 1.64e-108]], $MachinePrecision], LessEqual[N[(re * re), $MachinePrecision], 4.2e+45]]], N[(im * (-im)), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 4.59999999999999976e-173 or 1.6400000000000001e-108 < (*.f64 re re) < 4.1999999999999999e45

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 85.6%

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

      1. unpow2 85.6%

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

      2. mul-1-neg 85.6%

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

      3. distribute-rgt-neg-in 85.6%

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

   4. Simplified 85.6%

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

   if 4.59999999999999976e-173 < (*.f64 re re) < 1.6400000000000001e-108 or 4.1999999999999999e45 < (*.f64 re re)

   1. Initial program 89.6%

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

   2. Taylor expanded in re around inf 80.6%

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

      1. unpow2 80.6%

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

   4. Simplified 80.6%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 4.6 \cdot 10^{-173} \lor \neg \left(re \cdot re \leq 1.64 \cdot 10^{-108}\right) \land re \cdot re \leq 4.2 \cdot 10^{+45}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 3: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 7.4 \cdot 10^{+146}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 7.4e+146) (- (* re re) (* im im)) (* re re)))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if (re <= 7.4e+146) {
		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 <= 7.4d+146) 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 <= 7.4e+146) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

re = abs(re)
def re_sqr(re, im):
	tmp = 0
	if re <= 7.4e+146:
		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 <= 7.4e+146)
		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 <= 7.4e+146)
		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, 7.4e+146], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 7.40000000000000009e146

   1. Initial program 96.8%

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

   if 7.40000000000000009e146 < re

   1. Initial program 81.1%

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

   2. Taylor expanded in re around inf 94.6%

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

      1. unpow2 94.6%

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

   4. Simplified 94.6%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.4 \cdot 10^{+146}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 4: 53.0% 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 94.5%

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

2. Taylor expanded in re around inf 56.9%

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

   1. unpow2 56.9%

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

4. Simplified 56.9%

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

5. Final simplification 56.9%

   \[\leadsto re \cdot re \]

Reproduce

herbie shell --seed 2023217 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))