math.abs on complex (squared)

Percentage Accurate: 100.0% → 100.0%
Time: 2.5s
Alternatives: 2
Speedup: 1.2×

Specification

?
\[\begin{array}{l} \\ re \cdot re + im \cdot im \end{array} \]
(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))
double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}

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

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

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

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

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

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

\begin{array}{l}

\\
re \cdot re + im \cdot im
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 2 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ re \cdot re + im \cdot im \end{array} \]
(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))
double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}

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

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

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

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

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

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

\begin{array}{l}

\\
re \cdot re + im \cdot im
\end{array}

Alternative 1: 100.0% accurate, 1.2× speedup?

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \mathsf{fma}\left(im\_m, im\_m, re\_m \cdot re\_m\right) \end{array} \]
im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore modulus_sqr (re_m im_m)
 :precision binary64
 (fma im_m im_m (* re_m re_m)))
im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double modulus_sqr(double re_m, double im_m) {
	return fma(im_m, im_m, (re_m * re_m));
}

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function modulus_sqr(re_m, im_m)
	return fma(im_m, im_m, Float64(re_m * re_m))
end

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
modulus$95$sqr[re$95$m_, im$95$m_] := N[(im$95$m * im$95$m + N[(re$95$m * re$95$m), $MachinePrecision]), $MachinePrecision]

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

  1. Initial program 100.0%

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

    1. lift-+.f64N/A

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

    2. +-commutativeN/A

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

    3. lift-*.f64N/A

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

    4. lower-fma.f64100.0

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

  4. Applied rewrites100.0%

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

Alternative 2: 99.5% accurate, 2.3× speedup?

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ im\_m \cdot im\_m \end{array} \]
im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore modulus_sqr (re_m im_m) :precision binary64 (* im_m im_m))
im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double modulus_sqr(double re_m, double im_m) {
	return im_m * im_m;
}

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function modulus_sqr(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    modulus_sqr = im_m * im_m
end function

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double modulus_sqr(double re_m, double im_m) {
	return im_m * im_m;
}

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def modulus_sqr(re_m, im_m):
	return im_m * im_m

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function modulus_sqr(re_m, im_m)
	return Float64(im_m * im_m)
end

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = modulus_sqr(re_m, im_m)
	tmp = im_m * im_m;
end

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
modulus$95$sqr[re$95$m_, im$95$m_] := N[(im$95$m * im$95$m), $MachinePrecision]

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

  1. Initial program 100.0%

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

  3. Taylor expanded in re around 0

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

    1. unpow2N/A

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

    2. lower-*.f6455.4

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

  5. Applied rewrites55.4%

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

Reproduce

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herbie shell --seed 2024249 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))