math.abs on complex (squared)

Percentage Accurate: 100.0% → 100.0%
Time: 3.0s
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?

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

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

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

\begin{array}{l}
[re, im] = \mathsf{sort}([re, im])\\
\\
\mathsf{fma}\left(im, im, re \cdot re\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: 57.9% accurate, 2.3× speedup?

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

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

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

[re, im] = sort([re, im])
def modulus_sqr(re, im):
	return im * im

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

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

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

\begin{array}{l}
[re, im] = \mathsf{sort}([re, im])\\
\\
im \cdot im
\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-*.f6457.9

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

  5. Applied rewrites57.9%

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

Reproduce

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herbie shell --seed 2024230 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))