math.square on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 2.3s

Alternatives: 1

Speedup: 1.6×

Specification

Math

\[\begin{array}{l} \\ re \cdot im + im \cdot re \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ re \cdot im + im \cdot re \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ re\_s \cdot \left(im\_s \cdot \left(\left(re\_m + re\_m\right) \cdot im\_m\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore im_sqr (re_s im_s re_m im_m)
 :precision binary64
 (* re_s (* im_s (* (+ re_m re_m) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
re\_m = fabs(re);
re\_s = copysign(1.0, re);
assert(re_m < im_m);
double im_sqr(double re_s, double im_s, double re_m, double im_m) {
	return re_s * (im_s * ((re_m + re_m) * im_m));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
re\_m = abs(re)
re\_s = copysign(1.0d0, re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function im_sqr(re_s, im_s, re_m, im_m)
    real(8), intent (in) :: re_s
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    im_sqr = re_s * (im_s * ((re_m + re_m) * im_m))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
assert re_m < im_m;
public static double im_sqr(double re_s, double im_s, double re_m, double im_m) {
	return re_s * (im_s * ((re_m + re_m) * im_m));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
[re_m, im_m] = sort([re_m, im_m])
def im_sqr(re_s, im_s, re_m, im_m):
	return re_s * (im_s * ((re_m + re_m) * im_m))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
re\_m = abs(re)
re\_s = copysign(1.0, re)
re_m, im_m = sort([re_m, im_m])
function im_sqr(re_s, im_s, re_m, im_m)
	return Float64(re_s * Float64(im_s * Float64(Float64(re_m + re_m) * im_m)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = im_sqr(re_s, im_s, re_m, im_m)
	tmp = re_s * (im_s * ((re_m + re_m) * im_m));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
im$95$sqr[re$95$s_, im$95$s_, re$95$m_, im$95$m_] := N[(re$95$s * N[(im$95$s * N[(N[(re$95$m + re$95$m), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Reproduce

herbie shell --seed 2024244 
(FPCore im_sqr (re im)
  :name "math.square on complex, imaginary part"
  :precision binary64
  (+ (* re im) (* im re)))