math.log/1 on complex, real part

Percentage Accurate: 52.4% → 100.0%

Time: 6.3s

Alternatives: 3

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \log \left(\sqrt{re \cdot re + im \cdot im}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\sqrt{re \cdot re + im \cdot im}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return log(hypot(re, im));
}

Java

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

Python

def code(re, im):
	return math.log(math.hypot(re, im))

Julia

function code(re, im)
	return log(hypot(re, im))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im));
end

Wolfram

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

Derivation

1. Initial program 50.8%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Step-by-step derivation

   1. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. accelerator-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

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

Alternative 2: 27.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \log \left(im + \frac{re}{im} \cdot \left(re \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log (+ im (* (/ re im) (* re 0.5)))))

C

double code(double re, double im) {
	return log((im + ((re / im) * (re * 0.5))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log((im + ((re / im) * (re * 0.5d0))))
end function

Java

public static double code(double re, double im) {
	return Math.log((im + ((re / im) * (re * 0.5))));
}

Python

def code(re, im):
	return math.log((im + ((re / im) * (re * 0.5))))

Julia

function code(re, im)
	return log(Float64(im + Float64(Float64(re / im) * Float64(re * 0.5))))
end

MATLAB

function tmp = code(re, im)
	tmp = log((im + ((re / im) * (re * 0.5))));
end

Wolfram

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

Derivation

1. Initial program 50.8%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Step-by-step derivation

   1. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. accelerator-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(im + \frac{1}{2} \cdot \frac{{re}^{2}}{im}\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + \frac{\frac{1}{2} \cdot {re}^{2}}{im}\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + \frac{\frac{1}{2}}{im} \cdot {re}^{2}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + \frac{\frac{1}{2} \cdot 1}{im} \cdot {re}^{2}\right)\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + \left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}\right)\right) \]

   5. *-rgt-identity N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + \left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}\right) \cdot 1\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + \left(\frac{\frac{1}{2} \cdot 1}{im} \cdot {re}^{2}\right) \cdot 1\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + \left(\frac{\frac{1}{2}}{im} \cdot {re}^{2}\right) \cdot 1\right)\right) \]

   8. associate-*l/ N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + \frac{\frac{1}{2} \cdot {re}^{2}}{im} \cdot 1\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{log.f64}\left(\left(im + \frac{\frac{1}{2} \cdot {re}^{2}}{im} \cdot \frac{im}{im}\right)\right) \]

   10. times-frac N/A

       \[\leadsto \mathsf{log.f64}\left(\left(im + \frac{\left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im}{im \cdot im}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{log.f64}\left(\left(im + \frac{\left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im}{{im}^{2}}\right)\right) \]

   12. associate-*l/ N/A

       \[\leadsto \mathsf{log.f64}\left(\left(im + \frac{\frac{1}{2} \cdot {re}^{2}}{{im}^{2}} \cdot im\right)\right) \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{log.f64}\left(\left(im + \left(\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}\right) \cdot im\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\left(\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}\right) \cdot im\right)\right)\right) \]

   15. associate-*r/ N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{\frac{1}{2} \cdot {re}^{2}}{{im}^{2}} \cdot im\right)\right)\right) \]

   16. associate-*l/ N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{\left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im}{{im}^{2}}\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{\left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im}{im \cdot im}\right)\right)\right) \]

   18. times-frac N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{\frac{1}{2} \cdot {re}^{2}}{im} \cdot \frac{im}{im}\right)\right)\right) \]

7. Simplified 24.0%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{\left(\frac{1}{2} \cdot re\right) \cdot re}{im}\right)\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\left(\frac{1}{2} \cdot re\right) \cdot \frac{re}{im}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\frac{re}{im} \cdot \left(\frac{1}{2} \cdot re\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\left(\frac{re}{im}\right), \left(\frac{1}{2} \cdot re\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{/.f64}\left(re, im\right), \left(\frac{1}{2} \cdot re\right)\right)\right)\right) \]

   6. *-lowering-*.f64 26.2%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(\mathsf{/.f64}\left(re, im\right), \mathsf{*.f64}\left(\frac{1}{2}, re\right)\right)\right)\right) \]

9. Applied egg-rr 26.2%

   \[\leadsto \log \left(im + \color{blue}{\frac{re}{im} \cdot \left(0.5 \cdot re\right)}\right) \]

10. Final simplification 26.2%

    \[\leadsto \log \left(im + \frac{re}{im} \cdot \left(re \cdot 0.5\right)\right) \]
11. Add Preprocessing

Alternative 3: 28.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log im))

C

double code(double re, double im) {
	return log(im);
}

Fortran

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

Java

public static double code(double re, double im) {
	return Math.log(im);
}

Python

def code(re, im):
	return math.log(im)

Julia

function code(re, im)
	return log(im)
end

MATLAB

function tmp = code(re, im)
	tmp = log(im);
end

Wolfram

code[re_, im_] := N[Log[im], $MachinePrecision]

Derivation

1. Initial program 50.8%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Step-by-step derivation

   1. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. accelerator-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\log im} \]
6. Step-by-step derivation

   1. log-lowering-log.f64 26.5%

      \[\leadsto \mathsf{log.f64}\left(im\right) \]

7. Simplified 26.5%

   \[\leadsto \color{blue}{\log im} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))