math.abs on complex (squared)

Percentage Accurate: 100.0% → 100.0%
Time: 1.3s
Alternatives: 3
Speedup: 1.0×

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 3 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, 0.1× speedup?

\[\begin{array}{l} im = |im|\\ [re, im] = \mathsf{sort}([re, im])\\ \\ \mathsf{fma}\left(im, im, re \cdot re\right) \end{array} \]
NOTE: im should be positive before calling this function
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)))
im = abs(im);
assert(re < im);
double modulus_sqr(double re, double im) {
	return fma(im, im, (re * re));
}

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

NOTE: im should be positive before calling this function
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}
im = |im|\\
[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. Taylor expanded in re around 0 100.0%

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

    1. unpow2100.0%

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

    2. unpow2100.0%

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

    3. +-commutative100.0%

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

    4. fma-udef100.0%

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

  4. Simplified100.0%

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

  5. Final simplification100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup?

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

NOTE: im should be positive before calling this function
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 = (re * re) + (im * im)
end function

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

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

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

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

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

\begin{array}{l}
im = |im|\\
[re, im] = \mathsf{sort}([re, im])\\
\\
re \cdot re + im \cdot im
\end{array}
Derivation

  1. Initial program 100.0%

    \[re \cdot re + im \cdot im \]

  2. Final simplification100.0%

    \[\leadsto re \cdot re + im \cdot im \]

Alternative 3: 77.7% accurate, 2.3× speedup?

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

NOTE: im should be positive before calling this function
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

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

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

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

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

NOTE: im should be positive before calling this function
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}
im = |im|\\
[re, im] = \mathsf{sort}([re, im])\\
\\
im \cdot im
\end{array}
Derivation

  1. Initial program 100.0%

    \[re \cdot re + im \cdot im \]

  2. Taylor expanded in re around 0 55.3%

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

    1. unpow255.3%

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

  4. Simplified55.3%

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

  5. Final simplification55.3%

    \[\leadsto im \cdot im \]

Reproduce

?
herbie shell --seed 2023189 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))