math.abs on complex

Percentage Accurate: 54.2% → 100.0%
Time: 4.6s
Alternatives: 5
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 5 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: 54.2% 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, 0.2× 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 49.5%

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

    1. lift-sqrt.f64N/A

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

    2. lift-+.f64N/A

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

    3. lift-*.f64N/A

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

    4. lift-*.f64N/A

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

    5. lower-hypot.f64100.0

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

  4. Applied rewrites100.0%

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

Alternative 2: 26.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.5}{im} \cdot re, re, im\right) \end{array} \]
(FPCore modulus (re im) :precision binary64 (fma (* (/ 0.5 im) re) re im))
double modulus(double re, double im) {
	return fma(((0.5 / im) * re), re, im);
}

function modulus(re, im)
	return fma(Float64(Float64(0.5 / im) * re), re, im)
end

modulus[re_, im_] := N[(N[(N[(0.5 / im), $MachinePrecision] * re), $MachinePrecision] * re + im), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(\frac{0.5}{im} \cdot re, re, im\right)
\end{array}
Derivation

  1. Initial program 49.5%

    \[\sqrt{re \cdot re + im \cdot im} \]
  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. lower-fma.f64N/A

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

    6. associate-*r/N/A

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

    7. metadata-evalN/A

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

    8. lower-/.f64N/A

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

    9. unpow2N/A

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

    10. lower-*.f6426.9

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

  5. Applied rewrites26.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{im}, re \cdot re, im\right)} \]
  6. Step-by-step derivation

    1. Applied rewrites27.6%

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

    Alternative 3: 54.2% accurate, 1.1× speedup?

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

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

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

    \begin{array}{l}

\\
\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}
\end{array}
    Derivation

    1. Initial program 49.5%

      \[\sqrt{re \cdot re + im \cdot im} \]
    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.f6449.5

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

    4. Applied rewrites49.5%

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

    Alternative 4: 29.1% accurate, 1.5× speedup?

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

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

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

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

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

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

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

    \begin{array}{l}

\\
\sqrt{im \cdot im}
\end{array}
    Derivation

    1. Initial program 49.5%

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

    3. Taylor expanded in re around 0

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

      1. unpow2N/A

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

      2. lower-*.f6430.0

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

    5. Applied rewrites30.0%

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

    Alternative 5: 25.9% accurate, 8.0× speedup?

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

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

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

    def modulus(re, im):
	return -re

    function modulus(re, im)
	return Float64(-re)
end

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

    modulus[re_, im_] := (-re)

    \begin{array}{l}

\\
-re
\end{array}
    Derivation

    1. Initial program 49.5%

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

    3. Taylor expanded in re around -inf

      \[\leadsto \color{blue}{-1 \cdot re} \]
    4. Step-by-step derivation

      1. mul-1-negN/A

        \[\leadsto \color{blue}{\mathsf{neg}\left(re\right)} \]

      2. lower-neg.f6428.0

        \[\leadsto \color{blue}{-re} \]

    5. Applied rewrites28.0%

      \[\leadsto \color{blue}{-re} \]
    6. Add Preprocessing

    Reproduce

    ?
    herbie shell --seed 2024308 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))