math.abs on complex

Percentage Accurate: 53.2% → 100.0%
Time: 1.8s
Alternatives: 2
Speedup: 1.0×

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 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: 53.2% 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: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \mathsf{hypot}\left(re, im\right) \end{array} \]
(FPCore modulus (re im) :precision binary64 (hypot re im))
double modulus(double re, double im) {
	return hypot(re, im);
}

public static double modulus(double re, double im) {
	return Math.hypot(re, im);
}

def modulus(re, im):
	return math.hypot(re, im)

function modulus(re, im)
	return hypot(re, im)
end

function tmp = modulus(re, im)
	tmp = hypot(re, im);
end

modulus[re_, im_] := N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]

\begin{array}{l}

\\
\mathsf{hypot}\left(re, im\right)
\end{array}
Derivation

  1. Initial program 49.3%

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

    1. add-log-exp6.5%

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

    2. *-un-lft-identity6.5%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{re \cdot re + im \cdot im}}\right)} \]

    3. log-prod6.5%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{re \cdot re + im \cdot im}}\right)} \]

    4. metadata-eval6.5%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{re \cdot re + im \cdot im}}\right) \]

    5. add-log-exp49.3%

      \[\leadsto 0 + \color{blue}{\sqrt{re \cdot re + im \cdot im}} \]

    6. hypot-define100.0%

      \[\leadsto 0 + \color{blue}{\mathsf{hypot}\left(re, im\right)} \]

  4. Applied egg-rr100.0%

    \[\leadsto \color{blue}{0 + \mathsf{hypot}\left(re, im\right)} \]
  5. Step-by-step derivation

    1. +-lft-identity100.0%

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

  6. Simplified100.0%

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

  7. Final simplification100.0%

    \[\leadsto \mathsf{hypot}\left(re, im\right) \]
  8. Add Preprocessing

Alternative 2: 27.2% accurate, 107.0× speedup?

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

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

public static double modulus(double re, double im) {
	return im;
}

def modulus(re, im):
	return im

function modulus(re, im)
	return im
end

function tmp = modulus(re, im)
	tmp = im;
end

modulus[re_, im_] := im

\begin{array}{l}

\\
im
\end{array}
Derivation

  1. Initial program 49.3%

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

  3. Taylor expanded in re around 0 25.9%

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

  4. Final simplification25.9%

    \[\leadsto im \]
  5. Add Preprocessing

Reproduce

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herbie shell --seed 2024048 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))