math.square on complex, real part

Percentage Accurate: 93.5% → 96.8%

Time: 2.8s

Alternatives: 4

Speedup: 0.4×

• 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: 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 94.5%

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

   1. sqr-neg 94.5%

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

   2. cancel-sign-sub 94.5%

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

   3. fma-define 97.7%

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

3. Simplified 97.7%

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

5. Final simplification 97.7%

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

Alternative 2: 96.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+308}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;-{im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 1e+308) (- (* re re) (* im im)) (- (pow im 2.0))))

C

double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1e+308) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = -pow(im, 2.0);
	}
	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) <= 1d+308) then
        tmp = (re * re) - (im * im)
    else
        tmp = -(im ** 2.0d0)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1e+308) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = -Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 1e+308:
		tmp = (re * re) - (im * im)
	else:
		tmp = -math.pow(im, 2.0)
	return tmp

Julia

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 1e+308)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(-(im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 1e+308)
		tmp = (re * re) - (im * im);
	else
		tmp = -(im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 1e+308], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], (-N[Power[im, 2.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 1e308

   1. Initial program 100.0%

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

   if 1e308 < (*.f64 im im)

   1. Initial program 79.1%

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

   3. Taylor expanded in re around 0 91.0%

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

      1. mul-1-neg 91.0%

         \[\leadsto \color{blue}{-{im}^{2}} \]

   5. Simplified 91.0%

      \[\leadsto \color{blue}{-{im}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+308}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;-{im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot re - im \cdot im\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore re_sqr (re im)
 :precision binary64
 (let* ((t_0 (- (* re re) (* im im))))
   (if (<= t_0 4e-20) t_0 (* (+ re im) (+ re im)))))

C

double re_sqr(double re, double im) {
	double t_0 = (re * re) - (im * im);
	double tmp;
	if (t_0 <= 4e-20) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (re * re) - (im * im)
    if (t_0 <= 4d-20) then
        tmp = t_0
    else
        tmp = (re + im) * (re + im)
    end if
    re_sqr = tmp
end function

Java

public static double re_sqr(double re, double im) {
	double t_0 = (re * re) - (im * im);
	double tmp;
	if (t_0 <= 4e-20) {
		tmp = t_0;
	} else {
		tmp = (re + im) * (re + im);
	}
	return tmp;
}

Python

def re_sqr(re, im):
	t_0 = (re * re) - (im * im)
	tmp = 0
	if t_0 <= 4e-20:
		tmp = t_0
	else:
		tmp = (re + im) * (re + im)
	return tmp

Julia

function re_sqr(re, im)
	t_0 = Float64(Float64(re * re) - Float64(im * im))
	tmp = 0.0
	if (t_0 <= 4e-20)
		tmp = t_0;
	else
		tmp = Float64(Float64(re + im) * Float64(re + im));
	end
	return tmp
end

MATLAB

function tmp_2 = re_sqr(re, im)
	t_0 = (re * re) - (im * im);
	tmp = 0.0;
	if (t_0 <= 4e-20)
		tmp = t_0;
	else
		tmp = (re + im) * (re + im);
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-20], t$95$0, N[(N[(re + im), $MachinePrecision] * N[(re + im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 re re) (*.f64 im im)) < 3.99999999999999978e-20

   1. Initial program 100.0%

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

   if 3.99999999999999978e-20 < (-.f64 (*.f64 re re) (*.f64 im im))

   1. Initial program 86.9%

      \[re \cdot re - im \cdot im \]
   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 51.4%

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

      4. sqrt-unprod 92.5%

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

      5. sqr-neg 92.5%

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

      6. sqrt-prod 44.9%

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

      7. add-sqr-sqrt 92.5%

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

   4. Applied egg-rr 92.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re - im \cdot im \leq 4 \cdot 10^{-20}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.7% 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 94.5%

   \[re \cdot re - im \cdot im \]
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 49.9%

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

   4. sqrt-unprod 75.0%

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

   5. sqr-neg 75.0%

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

   6. sqrt-prod 26.5%

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

   7. add-sqr-sqrt 54.1%

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

4. Applied egg-rr 54.1%

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

5. Final simplification 54.1%

   \[\leadsto \left(re + im\right) \cdot \left(re + im\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))