math.abs on complex

Percentage Accurate: 54.0% → 100.0%

Time: 4.6s

Alternatives: 6

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \sqrt{re \cdot re + im \cdot im} \end{array} \]

FPCore

(FPCore modulus (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))

C

double modulus(double re, double im) {
	return sqrt(((re * re) + (im * im)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{re \cdot re + im \cdot im} \end{array} \]

FPCore

(FPCore modulus (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))

C

double modulus(double re, double im) {
	return sqrt(((re * re) + (im * im)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(re, im\right) \end{array} \]

FPCore

(FPCore modulus (re im) :precision binary64 (hypot re im))

C

double modulus(double re, double im) {
	return hypot(re, im);
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 26.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{0.5}{im} \cdot re, re, -im\right)\\ t\_0 \cdot \frac{\mathsf{fma}\left(0.5 \cdot re, \frac{re}{im}, im\right)}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore modulus (re im)
 :precision binary64
 (let* ((t_0 (fma (* (/ 0.5 im) re) re (- im))))
   (* t_0 (/ (fma (* 0.5 re) (/ re im) im) t_0))))

C

double modulus(double re, double im) {
	double t_0 = fma(((0.5 / im) * re), re, -im);
	return t_0 * (fma((0.5 * re), (re / im), im) / t_0);
}

Julia

function modulus(re, im)
	t_0 = fma(Float64(Float64(0.5 / im) * re), re, Float64(-im))
	return Float64(t_0 * Float64(fma(Float64(0.5 * re), Float64(re / im), im) / t_0))
end

Wolfram

modulus[re_, im_] := Block[{t$95$0 = N[(N[(N[(0.5 / im), $MachinePrecision] * re), $MachinePrecision] * re + (-im)), $MachinePrecision]}, N[(t$95$0 * N[(N[(N[(0.5 * re), $MachinePrecision] * N[(re / im), $MachinePrecision] + im), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.0%

   \[\sqrt{re \cdot re + im \cdot im} \]
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. +-commutative N/A

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

   2. *-lft-identity N/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.f64 N/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-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 27.5

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

5. Applied rewrites 27.5%

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

7. Applied rewrites 28.3%

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

9. Applied rewrites 27.6%

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

11. Applied rewrites 27.6%

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

Alternative 3: 27.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.5}{im} \cdot re, re, im\right) \end{array} \]

FPCore

(FPCore modulus (re im) :precision binary64 (fma (* (/ 0.5 im) re) re im))

C

double modulus(double re, double im) {
	return fma(((0.5 / im) * re), re, im);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 56.0%

   \[\sqrt{re \cdot re + im \cdot im} \]
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. +-commutative N/A

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

   2. *-lft-identity N/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.f64 N/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-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 27.5

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

5. Applied rewrites 27.5%

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

7. Applied rewrites 28.3%

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

Alternative 4: 54.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} \end{array} \]

FPCore

(FPCore modulus (re im) :precision binary64 (sqrt (fma re re (* im im))))

C

double modulus(double re, double im) {
	return sqrt(fma(re, re, (im * im)));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 56.0

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

4. Applied rewrites 56.0%

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

Alternative 5: 29.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{im \cdot im} \end{array} \]

FPCore

(FPCore modulus (re im) :precision binary64 (sqrt (* im im)))

C

double modulus(double re, double im) {
	return sqrt((im * im));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in re around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 32.5

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

5. Applied rewrites 32.5%

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

Alternative 6: 26.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ -re \end{array} \]

FPCore

(FPCore modulus (re im) :precision binary64 (- re))

C

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

Fortran

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

Java

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

Python

def modulus(re, im):
	return -re

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in re around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 23.9

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

5. Applied rewrites 23.9%

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

Reproduce

herbie shell --seed 2024339 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))