math.square on complex, real part

Percentage Accurate: 93.8% → 98.4%

Time: 3.1s

Alternatives: 4

Speedup: 0.6×

• 43.8% 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: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} \mathbf{if}\;re\_m \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(re\_m, re\_m, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m + im\right) \cdot \left(re\_m + im\right)\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (if (<= re_m 5e+200)
   (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 <= 5e+200) {
		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 <= 5e+200)
		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, 5e+200], 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 < 5.00000000000000019e200

   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 97.0%

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

   3. Simplified 97.0%

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

   if 5.00000000000000019e200 < re

   1. Initial program 71.4%

      \[re \cdot re - im \cdot im \]
   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 52.4%

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

      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 47.6%

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

      7. add-sqr-sqrt 95.2%

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

   4. Applied egg-rr 95.2%

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

Alternative 2: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{-85}:\\ \;\;\;\;\left(re\_m + im\right) \cdot \left(re\_m + im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (if (<= (* im im) 1e-85) (* (+ re_m im) (+ re_m im)) (* im (- im))))

C

re_m = fabs(re);
double re_sqr(double re_m, double im) {
	double tmp;
	if ((im * im) <= 1e-85) {
		tmp = (re_m + im) * (re_m + 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) <= 1d-85) then
        tmp = (re_m + im) * (re_m + 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) <= 1e-85) {
		tmp = (re_m + im) * (re_m + im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

re_m = math.fabs(re)
def re_sqr(re_m, im):
	tmp = 0
	if (im * im) <= 1e-85:
		tmp = (re_m + im) * (re_m + 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) <= 1e-85)
		tmp = Float64(Float64(re_m + im) * Float64(re_m + im));
	else
		tmp = Float64(im * Float64(-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-85)
		tmp = (re_m + im) * (re_m + 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], 1e-85], N[(N[(re$95$m + im), $MachinePrecision] * N[(re$95$m + im), $MachinePrecision]), $MachinePrecision], N[(im * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 9.9999999999999998e-86

   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 43.6%

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

      4. sqrt-unprod 94.0%

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

      5. sqr-neg 94.0%

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

      6. sqrt-prod 50.4%

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

      7. add-sqr-sqrt 90.9%

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

   4. Applied egg-rr 90.9%

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

   if 9.9999999999999998e-86 < (*.f64 im im)

   1. Initial program 86.3%

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

   3. Taylor expanded in re around 0 71.3%

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

      1. mul-1-neg 71.3%

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

   5. Simplified 71.3%

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

      1. unpow2 71.3%

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

   7. Applied egg-rr 71.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{-85}:\\ \;\;\;\;\left(re + im\right) \cdot \left(re + im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ \begin{array}{l} \mathbf{if}\;re\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re\_m \cdot re\_m - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m + im\right) \cdot \left(re\_m + im\right)\\ \end{array} \end{array} \]

FPCore

re_m = (fabs.f64 re)
(FPCore re_sqr (re_m im)
 :precision binary64
 (if (<= re_m 1.35e+154)
   (- (* 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 <= 1.35e+154) {
		tmp = (re_m * re_m) - (im * im);
	} 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) :: tmp
    if (re_m <= 1.35d+154) then
        tmp = (re_m * re_m) - (im * im)
    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 tmp;
	if (re_m <= 1.35e+154) {
		tmp = (re_m * re_m) - (im * im);
	} else {
		tmp = (re_m + im) * (re_m + im);
	}
	return tmp;
}

Python

re_m = math.fabs(re)
def re_sqr(re_m, im):
	tmp = 0
	if re_m <= 1.35e+154:
		tmp = (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 <= 1.35e+154)
		tmp = Float64(Float64(re_m * re_m) - Float64(im * im));
	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)
	tmp = 0.0;
	if (re_m <= 1.35e+154)
		tmp = (re_m * re_m) - (im * im);
	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_] := If[LessEqual[re$95$m, 1.35e+154], N[(N[(re$95$m * re$95$m), $MachinePrecision] - 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 < 1.35000000000000003e154

   1. Initial program 95.5%

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

   if 1.35000000000000003e154 < re

   1. Initial program 73.5%

      \[re \cdot re - im \cdot im \]
   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 52.9%

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

      4. sqrt-unprod 97.1%

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

      5. sqr-neg 97.1%

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

      6. sqrt-prod 44.1%

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

      7. add-sqr-sqrt 88.2%

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

   4. Applied egg-rr 88.2%

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

Alternative 4: 53.7% accurate, 1.8× speedup

Math

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

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 * Float64(-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 92.6%

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

3. Taylor expanded in re around 0 49.2%

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

   1. mul-1-neg 49.2%

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

5. Simplified 49.2%

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

   1. unpow2 49.2%

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

7. Applied egg-rr 49.2%

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

8. Final simplification 49.2%

   \[\leadsto im \cdot \left(-im\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024106 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))