math.square on complex, real part

Percentage Accurate: 93.8% → 96.8%

Time: 2.0s

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: 93.8% 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: 96.8% 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 92.6%

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

   1. fma-neg 96.9%

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

   2. distribute-rgt-neg-in 96.9%

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

3. Simplified 96.9%

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

4. Final simplification 96.9%

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

Alternative 2: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.2 \cdot 10^{+63} \lor \neg \left(im \leq -5.5 \cdot 10^{-47} \lor \neg \left(im \leq -2.8 \cdot 10^{-141}\right) \land \left(im \leq 5.2 \cdot 10^{-111} \lor \neg \left(im \leq 4.8 \cdot 10^{-36}\right) \land im \leq 2.35 \cdot 10^{+57}\right)\right):\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (or (<= im -5.2e+63)
         (not
          (or (<= im -5.5e-47)
              (and (not (<= im -2.8e-141))
                   (or (<= im 5.2e-111)
                       (and (not (<= im 4.8e-36)) (<= im 2.35e+57)))))))
   (* im (- im))
   (* re re)))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im <= -5.2e+63) || !((im <= -5.5e-47) || (!(im <= -2.8e-141) && ((im <= 5.2e-111) || (!(im <= 4.8e-36) && (im <= 2.35e+57)))))) {
		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 ((im <= (-5.2d+63)) .or. (.not. (im <= (-5.5d-47)) .or. (.not. (im <= (-2.8d-141))) .and. (im <= 5.2d-111) .or. (.not. (im <= 4.8d-36)) .and. (im <= 2.35d+57))) 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 ((im <= -5.2e+63) || !((im <= -5.5e-47) || (!(im <= -2.8e-141) && ((im <= 5.2e-111) || (!(im <= 4.8e-36) && (im <= 2.35e+57)))))) {
		tmp = im * -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im <= -5.2e+63) or not ((im <= -5.5e-47) or (not (im <= -2.8e-141) and ((im <= 5.2e-111) or (not (im <= 4.8e-36) and (im <= 2.35e+57))))):
		tmp = im * -im
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if ((im <= -5.2e+63) || !((im <= -5.5e-47) || (!(im <= -2.8e-141) && ((im <= 5.2e-111) || (!(im <= 4.8e-36) && (im <= 2.35e+57))))))
		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 ((im <= -5.2e+63) || ~(((im <= -5.5e-47) || (~((im <= -2.8e-141)) && ((im <= 5.2e-111) || (~((im <= 4.8e-36)) && (im <= 2.35e+57)))))))
		tmp = im * -im;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[Or[LessEqual[im, -5.2e+63], N[Not[Or[LessEqual[im, -5.5e-47], And[N[Not[LessEqual[im, -2.8e-141]], $MachinePrecision], Or[LessEqual[im, 5.2e-111], And[N[Not[LessEqual[im, 4.8e-36]], $MachinePrecision], LessEqual[im, 2.35e+57]]]]]], $MachinePrecision]], N[(im * (-im)), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -5.2000000000000002e63 or -5.5000000000000002e-47 < im < -2.80000000000000012e-141 or 5.19999999999999965e-111 < im < 4.8e-36 or 2.3500000000000001e57 < im

   1. Initial program 85.9%

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

   2. Taylor expanded in re around 0 80.0%

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

      1. unpow2 80.0%

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

      2. mul-1-neg 80.0%

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

      3. distribute-rgt-neg-in 80.0%

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

   4. Simplified 80.0%

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

   if -5.2000000000000002e63 < im < -5.5000000000000002e-47 or -2.80000000000000012e-141 < im < 5.19999999999999965e-111 or 4.8e-36 < im < 2.3500000000000001e57

   1. Initial program 100.0%

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

   2. Taylor expanded in re around inf 87.7%

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

      1. unpow2 87.7%

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

   4. Simplified 87.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.2 \cdot 10^{+63} \lor \neg \left(im \leq -5.5 \cdot 10^{-47} \lor \neg \left(im \leq -2.8 \cdot 10^{-141}\right) \land \left(im \leq 5.2 \cdot 10^{-111} \lor \neg \left(im \leq 4.8 \cdot 10^{-36}\right) \land im \leq 2.35 \cdot 10^{+57}\right)\right):\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 3: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 4 \cdot 10^{+300}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 4e+300) (- (* re re) (* im im)) (* im (- im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 4e+300) {
		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) <= 4d+300) 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) <= 4e+300) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 4e+300:
		tmp = (re * re) - (im * im)
	else:
		tmp = im * -im
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 4e+300)
		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) <= 4e+300)
		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], 4e+300], 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) < 4.0000000000000002e300

   1. Initial program 100.0%

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

   if 4.0000000000000002e300 < (*.f64 im im)

   1. Initial program 70.8%

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

   2. Taylor expanded in re around 0 87.7%

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

      1. unpow2 87.7%

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

      2. mul-1-neg 87.7%

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

      3. distribute-rgt-neg-in 87.7%

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

   4. Simplified 87.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 4 \cdot 10^{+300}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternative 4: 54.3% 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 92.6%

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

2. Taylor expanded in re around inf 52.5%

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

   1. unpow2 52.5%

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

4. Simplified 52.5%

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

5. Final simplification 52.5%

   \[\leadsto re \cdot re \]

Reproduce

herbie shell --seed 2023173 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))