math.square on complex, real part

Percentage Accurate: 93.9% → 98.4%

Time: 3.3s

Alternatives: 4

Speedup: 0.4×

• 44.2% 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: 93.9% 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: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} \mathbf{if}\;re\_m \leq 1.85 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(re\_m, re\_m, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m + im\right) \cdot \left(re\_m + im\right)\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (if (<= re_m 1.85e+253)
   (fma re_m re_m (* im (- im)))
   (* (+ re_m im) (+ re_m im))))

C

re_m = fabs(re);
double re_sqr(double re_m, double im) {
	double tmp;
	if (re_m <= 1.85e+253) {
		tmp = fma(re_m, re_m, (im * -im));
	} else {
		tmp = (re_m + im) * (re_m + im);
	}
	return tmp;
}

Julia

re_m = abs(re)
function re_sqr(re_m, im)
	tmp = 0.0
	if (re_m <= 1.85e+253)
		tmp = fma(re_m, re_m, Float64(im * Float64(-im)));
	else
		tmp = Float64(Float64(re_m + im) * Float64(re_m + im));
	end
	return tmp
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
re$95$sqr[re$95$m_, im_] := If[LessEqual[re$95$m, 1.85e+253], N[(re$95$m * re$95$m + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m + im), $MachinePrecision] * N[(re$95$m + im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.85000000000000014e253

   1. Initial program 92.2%

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

      1. sqr-neg 92.2%

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

      2. cancel-sign-sub 92.2%

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

      3. fma-define 97.5%

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

   3. Simplified 97.5%

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

   if 1.85000000000000014e253 < re

   1. Initial program 66.7%

      \[re \cdot re - im \cdot im \]
   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 33.3%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-prod 66.7%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.85 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+271}:\\ \;\;\;\;re\_m \cdot re\_m - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;-{im}^{2}\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (if (<= (* im im) 1e+271) (- (* re_m re_m) (* im im)) (- (pow im 2.0))))

C

re_m = fabs(re);
double re_sqr(double re_m, double im) {
	double tmp;
	if ((im * im) <= 1e+271) {
		tmp = (re_m * re_m) - (im * im);
	} else {
		tmp = -pow(im, 2.0);
	}
	return tmp;
}

Fortran

re_m = abs(re)
real(8) function re_sqr(re_m, im)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im * im) <= 1d+271) then
        tmp = (re_m * re_m) - (im * im)
    else
        tmp = -(im ** 2.0d0)
    end if
    re_sqr = tmp
end function

Java

re_m = Math.abs(re);
public static double re_sqr(double re_m, double im) {
	double tmp;
	if ((im * im) <= 1e+271) {
		tmp = (re_m * re_m) - (im * im);
	} else {
		tmp = -Math.pow(im, 2.0);
	}
	return tmp;
}

Python

re_m = math.fabs(re)
def re_sqr(re_m, im):
	tmp = 0
	if (im * im) <= 1e+271:
		tmp = (re_m * re_m) - (im * im)
	else:
		tmp = -math.pow(im, 2.0)
	return tmp

Julia

re_m = abs(re)
function re_sqr(re_m, im)
	tmp = 0.0
	if (Float64(im * im) <= 1e+271)
		tmp = Float64(Float64(re_m * re_m) - Float64(im * im));
	else
		tmp = Float64(-(im ^ 2.0));
	end
	return tmp
end

MATLAB

re_m = abs(re);
function tmp_2 = re_sqr(re_m, im)
	tmp = 0.0;
	if ((im * im) <= 1e+271)
		tmp = (re_m * re_m) - (im * im);
	else
		tmp = -(im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
re$95$sqr[re$95$m_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 1e+271], N[(N[(re$95$m * re$95$m), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], (-N[Power[im, 2.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 9.99999999999999953e270

   1. Initial program 100.0%

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

   if 9.99999999999999953e270 < (*.f64 im im)

   1. Initial program 67.6%

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

   3. Taylor expanded in re around 0 85.9%

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

      1. mul-1-neg 85.9%

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

   5. Simplified 85.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+271}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;-{im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} t_0 := re\_m \cdot re\_m - im \cdot im\\ \mathbf{if}\;t\_0 \leq 10^{+257}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m + im\right) \cdot \left(re\_m + im\right)\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (let* ((t_0 (- (* re_m re_m) (* im im))))
   (if (<= t_0 1e+257) t_0 (* (+ re_m im) (+ re_m im)))))

C

re_m = fabs(re);
double re_sqr(double re_m, double im) {
	double t_0 = (re_m * re_m) - (im * im);
	double tmp;
	if (t_0 <= 1e+257) {
		tmp = t_0;
	} else {
		tmp = (re_m + im) * (re_m + im);
	}
	return tmp;
}

Fortran

re_m = abs(re)
real(8) function re_sqr(re_m, im)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re_m * re_m) - (im * im)
    if (t_0 <= 1d+257) then
        tmp = t_0
    else
        tmp = (re_m + im) * (re_m + im)
    end if
    re_sqr = tmp
end function

Java

re_m = Math.abs(re);
public static double re_sqr(double re_m, double im) {
	double t_0 = (re_m * re_m) - (im * im);
	double tmp;
	if (t_0 <= 1e+257) {
		tmp = t_0;
	} else {
		tmp = (re_m + im) * (re_m + im);
	}
	return tmp;
}

Python

re_m = math.fabs(re)
def re_sqr(re_m, im):
	t_0 = (re_m * re_m) - (im * im)
	tmp = 0
	if t_0 <= 1e+257:
		tmp = t_0
	else:
		tmp = (re_m + im) * (re_m + im)
	return tmp

Julia

re_m = abs(re)
function re_sqr(re_m, im)
	t_0 = Float64(Float64(re_m * re_m) - Float64(im * im))
	tmp = 0.0
	if (t_0 <= 1e+257)
		tmp = t_0;
	else
		tmp = Float64(Float64(re_m + im) * Float64(re_m + im));
	end
	return tmp
end

MATLAB

re_m = abs(re);
function tmp_2 = re_sqr(re_m, im)
	t_0 = (re_m * re_m) - (im * im);
	tmp = 0.0;
	if (t_0 <= 1e+257)
		tmp = t_0;
	else
		tmp = (re_m + im) * (re_m + im);
	end
	tmp_2 = tmp;
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
re$95$sqr[re$95$m_, im_] := Block[{t$95$0 = N[(N[(re$95$m * re$95$m), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+257], t$95$0, N[(N[(re$95$m + im), $MachinePrecision] * N[(re$95$m + im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 re re) (*.f64 im im)) < 1.00000000000000003e257

   1. Initial program 100.0%

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

   if 1.00000000000000003e257 < (-.f64 (*.f64 re re) (*.f64 im im))

   1. Initial program 70.1%

      \[re \cdot re - im \cdot im \]
   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 48.1%

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

      4. sqrt-unprod 88.3%

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

      5. sqr-neg 88.3%

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

      6. sqrt-prod 44.2%

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

      7. add-sqr-sqrt 83.1%

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

   4. Applied egg-rr 83.1%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re - im \cdot im \leq 10^{+257}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.0% accurate, 1.0× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im) :precision binary64 (* (+ re_m im) (+ re_m im)))

C

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

Fortran

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

Java

re_m = Math.abs(re);
public static double re_sqr(double re_m, double im) {
	return (re_m + im) * (re_m + im);
}

Python

re_m = math.fabs(re)
def re_sqr(re_m, im):
	return (re_m + im) * (re_m + im)

Julia

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

MATLAB

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

Wolfram

re_m = N[Abs[re], $MachinePrecision]
re$95$sqr[re$95$m_, im_] := N[(N[(re$95$m + im), $MachinePrecision] * N[(re$95$m + im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

   \[re \cdot re - im \cdot im \]
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 77.5%

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

   5. sqr-neg 77.5%

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

   6. sqrt-prod 28.7%

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

   7. add-sqr-sqrt 54.4%

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

4. Applied egg-rr 54.4%

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

5. Final simplification 54.4%

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

Reproduce

herbie shell --seed 2024071 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))