math.square on complex, real part

Percentage Accurate: 93.6% → 98.7%

Time: 4.0s

Alternatives: 4

Speedup: 0.6×

• 44.7% 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: 93.6% 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: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \cdot re - im \cdot im \leq -2 \cdot 10^{-239}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{re}^{2} \cdot \left(1 - \frac{im}{re} \cdot \frac{im}{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (- (* re re) (* im im)) -2e-239)
   (fma re re (* im (- im)))
   (* (pow re 2.0) (- 1.0 (* (/ im re) (/ im re))))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (((re * re) - (im * im)) <= -2e-239) {
		tmp = fma(re, re, (im * -im));
	} else {
		tmp = pow(re, 2.0) * (1.0 - ((im / re) * (im / re)));
	}
	return tmp;
}

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(Float64(re * re) - Float64(im * im)) <= -2e-239)
		tmp = fma(re, re, Float64(im * Float64(-im)));
	else
		tmp = Float64((re ^ 2.0) * Float64(1.0 - Float64(Float64(im / re) * Float64(im / re))));
	end
	return tmp
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], -2e-239], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(N[Power[re, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(im / re), $MachinePrecision] * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 re re) (*.f64 im im)) < -2.0000000000000002e-239

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   if -2.0000000000000002e-239 < (-.f64 (*.f64 re re) (*.f64 im im))

   1. Initial program 83.2%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf 73.8%

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(1 + -1 \cdot \frac{{im}^{2}}{{re}^{2}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 73.8%

         \[\leadsto {re}^{2} \cdot \left(1 + \color{blue}{\left(-\frac{{im}^{2}}{{re}^{2}}\right)}\right) \]

      2. unsub-neg 73.8%

         \[\leadsto {re}^{2} \cdot \color{blue}{\left(1 - \frac{{im}^{2}}{{re}^{2}}\right)} \]

   5. Simplified 73.8%

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(1 - \frac{{im}^{2}}{{re}^{2}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 73.8%

         \[\leadsto {re}^{2} \cdot \left(1 - \frac{\color{blue}{im \cdot im}}{{re}^{2}}\right) \]

      2. unpow2 73.8%

         \[\leadsto {re}^{2} \cdot \left(1 - \frac{im \cdot im}{\color{blue}{re \cdot re}}\right) \]

      3. times-frac 100.0%

         \[\leadsto {re}^{2} \cdot \left(1 - \color{blue}{\frac{im}{re} \cdot \frac{im}{re}}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto {re}^{2} \cdot \left(1 - \color{blue}{\frac{im}{re} \cdot \frac{im}{re}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.1 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 4.1e+250) (fma re re (* im (- im))) (* (+ re im) (+ re im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (re <= 4.1e+250) {
		tmp = fma(re, re, (im * -im));
	} else {
		tmp = (re + im) * (re + im);
	}
	return tmp;
}

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (re <= 4.1e+250)
		tmp = fma(re, re, Float64(im * Float64(-im)));
	else
		tmp = Float64(Float64(re + im) * Float64(re + im));
	end
	return tmp
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[re, 4.1e+250], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(N[(re + im), $MachinePrecision] * N[(re + im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.09999999999999999e250

   1. Initial program 91.7%

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

      1. sqr-neg 91.7%

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

      2. cancel-sign-sub 91.7%

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

      3. fma-define 96.3%

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

   3. Simplified 96.3%

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

   if 4.09999999999999999e250 < re

   1. Initial program 64.3%

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

      1. difference-of-squares 100.0%

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

      2. sub-neg 100.0%

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

      3. add-sqr-sqrt 50.0%

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

      4. sqrt-unprod 92.9%

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

      5. sqr-neg 92.9%

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

      6. sqrt-prod 42.9%

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

      7. add-sqr-sqrt 92.9%

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

   4. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\left(re + im\right) \cdot \left(re + im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8 \cdot 10^{+132}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 8e+132) (- (* re re) (* im im)) (* (+ re im) (+ re im))))

C

double re_sqr(double re, double im) {
	double tmp;
	if (re <= 8e+132) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = (re + im) * (re + 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 (re <= 8d+132) then
        tmp = (re * re) - (im * im)
    else
        tmp = (re + im) * (re + im)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if (re <= 8e+132) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = (re + im) * (re + im);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if re <= 8e+132:
		tmp = (re * re) - (im * im)
	else:
		tmp = (re + im) * (re + im)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (re <= 8e+132)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(Float64(re + im) * Float64(re + im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if (re <= 8e+132)
		tmp = (re * re) - (im * im);
	else
		tmp = (re + im) * (re + im);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[re, 8e+132], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(re + im), $MachinePrecision] * N[(re + im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 7.99999999999999993e132

   1. Initial program 94.4%

      \[re \cdot re - im \cdot im \]
   2. Add Preprocessing

   if 7.99999999999999993e132 < re

   1. Initial program 69.8%

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

      1. difference-of-squares 100.0%

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

      2. sub-neg 100.0%

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

      3. add-sqr-sqrt 58.1%

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

      4. sqrt-unprod 93.0%

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

      5. sqr-neg 93.0%

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

      6. sqrt-prod 34.9%

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

      7. add-sqr-sqrt 83.7%

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

   4. Applied egg-rr 83.7%

      \[\leadsto \color{blue}{\left(re + im\right) \cdot \left(re + im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(re + im\right) \cdot \left(re + im\right) \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (* (+ re im) (+ re im)))

C

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

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re + im) * (re + im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re + im) * (re + im);
}

Python

def re_sqr(re, im):
	return (re + im) * (re + im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re + im) * Float64(re + im))
end

MATLAB

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

Wolfram

re$95$sqr[re_, im_] := N[(N[(re + im), $MachinePrecision] * N[(re + im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

   1. difference-of-squares 100.0%

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

   2. sub-neg 100.0%

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

   3. add-sqr-sqrt 47.2%

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

   4. sqrt-unprod 72.2%

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

   5. sqr-neg 72.2%

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

   6. sqrt-prod 27.7%

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

   7. add-sqr-sqrt 52.8%

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

4. Applied egg-rr 52.8%

   \[\leadsto \color{blue}{\left(re + im\right) \cdot \left(re + im\right)} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))