math.square on complex, real part

Percentage Accurate: 93.8% → 98.2%

Time: 4.1s

Alternatives: 5

Speedup: 0.5×

• 43.7% 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: 93.8% 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: 98.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} \mathbf{if}\;re\_m \leq 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(re\_m, re\_m, -im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re\_m \cdot re\_m\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (if (<= re_m 1e+210) (fma re_m re_m (- (* im im))) (* re_m re_m)))

C

re_m = fabs(re);
double re_sqr(double re_m, double im) {
	double tmp;
	if (re_m <= 1e+210) {
		tmp = fma(re_m, re_m, -(im * im));
	} else {
		tmp = re_m * re_m;
	}
	return tmp;
}

Julia

re_m = abs(re)
function re_sqr(re_m, im)
	tmp = 0.0
	if (re_m <= 1e+210)
		tmp = fma(re_m, re_m, Float64(-Float64(im * im)));
	else
		tmp = Float64(re_m * re_m);
	end
	return tmp
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
re$95$sqr[re$95$m_, im_] := If[LessEqual[re$95$m, 1e+210], N[(re$95$m * re$95$m + (-N[(im * im), $MachinePrecision])), $MachinePrecision], N[(re$95$m * re$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.99999999999999927e209

   1. Initial program 94.6%

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

      1. sqr-neg 94.6%

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

      2. cancel-sign-sub 94.6%

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

      3. fma-define 98.3%

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

   3. Simplified 98.3%

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

   if 9.99999999999999927e209 < re

   1. Initial program 73.3%

      \[re \cdot re - im \cdot im \]
   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 60.0%

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

      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 40.0%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

   6. Taylor expanded in re around inf 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(re, re, -im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+288}:\\ \;\;\;\;re\_m \cdot re\_m - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;-im \cdot im\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (if (<= (* im im) 2e+288) (- (* re_m re_m) (* im im)) (- (* im im))))

C

re_m = fabs(re);
double re_sqr(double re_m, double im) {
	double tmp;
	if ((im * im) <= 2e+288) {
		tmp = (re_m * re_m) - (im * im);
	} else {
		tmp = -(im * 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) :: tmp
    if ((im * im) <= 2d+288) then
        tmp = (re_m * re_m) - (im * im)
    else
        tmp = -(im * 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 tmp;
	if ((im * im) <= 2e+288) {
		tmp = (re_m * re_m) - (im * im);
	} else {
		tmp = -(im * im);
	}
	return tmp;
}

Python

re_m = math.fabs(re)
def re_sqr(re_m, im):
	tmp = 0
	if (im * im) <= 2e+288:
		tmp = (re_m * re_m) - (im * im)
	else:
		tmp = -(im * im)
	return tmp

Julia

re_m = abs(re)
function re_sqr(re_m, im)
	tmp = 0.0
	if (Float64(im * im) <= 2e+288)
		tmp = Float64(Float64(re_m * re_m) - Float64(im * im));
	else
		tmp = Float64(-Float64(im * im));
	end
	return tmp
end

MATLAB

re_m = abs(re);
function tmp_2 = re_sqr(re_m, im)
	tmp = 0.0;
	if ((im * im) <= 2e+288)
		tmp = (re_m * re_m) - (im * im);
	else
		tmp = -(im * im);
	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], 2e+288], N[(N[(re$95$m * re$95$m), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], (-N[(im * im), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 2e288

   1. Initial program 100.0%

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

   if 2e288 < (*.f64 im im)

   1. Initial program 67.3%

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

   3. Taylor expanded in re around 0 84.6%

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

      1. neg-mul-1 84.6%

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

   5. Simplified 84.6%

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

      1. unpow2 84.6%

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

      2. distribute-lft-neg-in 84.6%

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

   7. Applied egg-rr 84.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+288}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;-im \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+18}:\\ \;\;\;\;re\_m \cdot re\_m\\ \mathbf{else}:\\ \;\;\;\;-im \cdot im\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (if (<= (* im im) 1e+18) (* re_m re_m) (- (* im im))))

C

re_m = fabs(re);
double re_sqr(double re_m, double im) {
	double tmp;
	if ((im * im) <= 1e+18) {
		tmp = re_m * re_m;
	} else {
		tmp = -(im * 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) :: tmp
    if ((im * im) <= 1d+18) then
        tmp = re_m * re_m
    else
        tmp = -(im * 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 tmp;
	if ((im * im) <= 1e+18) {
		tmp = re_m * re_m;
	} else {
		tmp = -(im * im);
	}
	return tmp;
}

Python

re_m = math.fabs(re)
def re_sqr(re_m, im):
	tmp = 0
	if (im * im) <= 1e+18:
		tmp = re_m * re_m
	else:
		tmp = -(im * im)
	return tmp

Julia

re_m = abs(re)
function re_sqr(re_m, im)
	tmp = 0.0
	if (Float64(im * im) <= 1e+18)
		tmp = Float64(re_m * re_m);
	else
		tmp = Float64(-Float64(im * im));
	end
	return tmp
end

MATLAB

re_m = abs(re);
function tmp_2 = re_sqr(re_m, im)
	tmp = 0.0;
	if ((im * im) <= 1e+18)
		tmp = re_m * re_m;
	else
		tmp = -(im * im);
	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+18], N[(re$95$m * re$95$m), $MachinePrecision], (-N[(im * im), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 1e18

   1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]
   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 57.9%

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

      4. sqrt-unprod 88.4%

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

      5. sqr-neg 88.4%

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

      6. sqrt-prod 30.5%

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

      7. add-sqr-sqrt 80.5%

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

   4. Applied egg-rr 80.5%

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

   5. Taylor expanded in re around inf 80.8%

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

   6. Taylor expanded in re around inf 81.3%

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

   if 1e18 < (*.f64 im im)

   1. Initial program 85.6%

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

   3. Taylor expanded in re around 0 72.3%

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

      1. neg-mul-1 72.3%

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

   5. Simplified 72.3%

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

      1. unpow2 72.3%

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

      2. distribute-lft-neg-in 72.3%

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

   7. Applied egg-rr 72.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+18}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;-im \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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 * re_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

   \[re \cdot re - im \cdot im \]
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 53.0%

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

   4. sqrt-unprod 74.8%

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

   5. sqr-neg 74.8%

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

   6. sqrt-prod 23.2%

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

   7. add-sqr-sqrt 55.3%

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

4. Applied egg-rr 55.3%

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

5. Taylor expanded in re around inf 58.5%

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

6. Taylor expanded in re around inf 56.3%

   \[\leadsto \color{blue}{re} \cdot re \]
7. Add Preprocessing

Alternative 5: 11.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

re_m = fabs(re);
double re_sqr(double re_m, double im) {
	return im * 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 = im * im
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in re around 0 49.0%

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

   1. neg-mul-1 49.0%

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

5. Simplified 49.0%

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

   1. add-sqr-sqrt 4.3%

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

   2. sqrt-unprod 14.0%

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

   3. sqr-neg 14.0%

      \[\leadsto \sqrt{\color{blue}{{im}^{2} \cdot {im}^{2}}} \]

   4. sqrt-unprod 10.0%

      \[\leadsto \color{blue}{\sqrt{{im}^{2}} \cdot \sqrt{{im}^{2}}} \]

   5. add-sqr-sqrt 10.0%

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

   6. unpow2 10.0%

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

7. Applied egg-rr 10.0%

   \[\leadsto \color{blue}{im \cdot im} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))