math.log/1 on complex, real part

Percentage Accurate: 51.8% → 100.0%

Time: 1.6s

Alternatives: 2

Speedup: 1.5×

• Could not converge on a ground truth

  1. re = 4.614906735260092e-21
  2. im = +nan.0

Specification

Math

\[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]

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)
use fmin_fmax_functions
    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: 51.8% accurate, 1.0× speedup

Math

\[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]

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)
use fmin_fmax_functions
    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, 0.8× speedup

Math

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

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 51.8%

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

   1. lift-sqrt.f64 N/A

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

   2. sqrt-fabs-rev N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. sqr-neg-rev N/A

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

   11. fp-cancel-sign-sub-inv N/A

       \[\leadsto \log \left(\sqrt{\color{blue}{re \cdot re - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(im\right)\right)}}\right) \]

   12. fp-cancel-sub-sign-inv N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-lft-neg-in N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. sqr-neg-rev N/A

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

   17. sqr-neg-rev N/A

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

   18. lower-hypot.f64 100.0%

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

3. Applied rewrites 100.0%

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

Alternative 2: 99.2% accurate, 1.5× speedup

Math

\[\log \left(\mathsf{max}\left(\left|re\right|, \left|im\right|\right)\right) \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(fmax(abs(re), abs(im)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]

2. Taylor expanded in im around inf

   \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\log \left(\frac{1}{im}\right)} \]

   2. lower-log.f64 N/A

      \[\leadsto -1 \cdot \log \left(\frac{1}{im}\right) \]

   3. lower-/.f64 28.2%

      \[\leadsto -1 \cdot \log \left(\frac{1}{im}\right) \]

4. Applied rewrites 28.2%

   \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\log \left(\frac{1}{im}\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right) \]

   3. lift-log.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right) \]

   5. log-rec N/A

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

   6. remove-double-neg N/A

      \[\leadsto \log im \]

   7. lower-log.f64 28.2%

      \[\leadsto \log im \]

6. Applied rewrites 28.2%

   \[\leadsto \log im \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))