math.abs on complex

Percentage Accurate: 53.6% → 99.6%
Time: 3.5s
Alternatives: 4
Speedup: 1.1×

Specification

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

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

public static double modulus(double re, double im) {
	return Math.sqrt(((re * re) + (im * im)));
}

def modulus(re, im):
	return math.sqrt(((re * re) + (im * im)))

function modulus(re, im)
	return sqrt(Float64(Float64(re * re) + Float64(im * im)))
end

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

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

\begin{array}{l}

\\
\sqrt{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 4 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: 53.6% accurate, 1.0× speedup?

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

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

public static double modulus(double re, double im) {
	return Math.sqrt(((re * re) + (im * im)));
}

def modulus(re, im):
	return math.sqrt(((re * re) + (im * im)))

function modulus(re, im)
	return sqrt(Float64(Float64(re * re) + Float64(im * im)))
end

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

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

\begin{array}{l}

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

Alternative 1: 99.6% accurate, 1.0× speedup?

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \mathsf{fma}\left(re\_m, \frac{re\_m \cdot 0.5}{im\_m}, im\_m\right) \end{array} \]
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 (re_m im_m)
 :precision binary64
 (fma re_m (/ (* re_m 0.5) im_m) im_m))
im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double modulus(double re_m, double im_m) {
	return fma(re_m, ((re_m * 0.5) / im_m), im_m);
}

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function modulus(re_m, im_m)
	return fma(re_m, Float64(Float64(re_m * 0.5) / 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[re$95$m_, im$95$m_] := N[(re$95$m * N[(N[(re$95$m * 0.5), $MachinePrecision] / im$95$m), $MachinePrecision] + 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])\\
\\
\mathsf{fma}\left(re\_m, \frac{re\_m \cdot 0.5}{im\_m}, im\_m\right)
\end{array}
Derivation

  1. Initial program 54.9%

    \[\sqrt{re \cdot re + im \cdot im} \]
  2. Add Preprocessing

  3. Taylor expanded in re around 0

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

    1. +-commutativeN/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{im} + im} \]

    2. *-lft-identityN/A

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{1 \cdot {re}^{2}}}{im} + im \]

    3. associate-*l/N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{im} \cdot {re}^{2}\right)} + im \]

    4. associate-*l*N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}} + im \]

    5. unpow2N/A

      \[\leadsto \left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot \color{blue}{\left(re \cdot re\right)} + im \]

    6. associate-*r*N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot re\right) \cdot re} + im \]

    7. *-commutativeN/A

      \[\leadsto \color{blue}{re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot re\right)} + im \]

    8. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot re, im\right)} \]

    9. associate-*r/N/A

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{\frac{1}{2} \cdot 1}{im}} \cdot re, im\right) \]

    10. metadata-evalN/A

      \[\leadsto \mathsf{fma}\left(re, \frac{\color{blue}{\frac{1}{2}}}{im} \cdot re, im\right) \]

    11. associate-*l/N/A

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{\frac{1}{2} \cdot re}{im}}, im\right) \]

    12. lower-/.f64N/A

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{\frac{1}{2} \cdot re}{im}}, im\right) \]

    13. *-commutativeN/A

      \[\leadsto \mathsf{fma}\left(re, \frac{\color{blue}{re \cdot \frac{1}{2}}}{im}, im\right) \]

    14. lower-*.f64100.0

      \[\leadsto \mathsf{fma}\left(re, \frac{\color{blue}{re \cdot 0.5}}{im}, im\right) \]

  5. Applied rewrites100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{re \cdot 0.5}{im}, im\right)} \]
  6. Add Preprocessing

Alternative 2: 53.6% accurate, 1.1× speedup?

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \sqrt{\mathsf{fma}\left(re\_m, re\_m, im\_m \cdot im\_m\right)} \end{array} \]
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 (re_m im_m)
 :precision binary64
 (sqrt (fma re_m re_m (* im_m im_m))))
im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double modulus(double re_m, double im_m) {
	return sqrt(fma(re_m, re_m, (im_m * im_m)));
}

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function modulus(re_m, im_m)
	return sqrt(fma(re_m, re_m, Float64(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[re$95$m_, im$95$m_] := N[Sqrt[N[(re$95$m * re$95$m + N[(im$95$m * im$95$m), $MachinePrecision]), $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])\\
\\
\sqrt{\mathsf{fma}\left(re\_m, re\_m, im\_m \cdot im\_m\right)}
\end{array}
Derivation

  1. Initial program 53.6%

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

    1. lift-+.f64N/A

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

    2. lift-*.f64N/A

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

    3. lower-fma.f6453.6

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

  4. Applied rewrites53.6%

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

Reproduce

?
herbie shell --seed 2024228 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))