math.log10 on complex, real part

Percentage Accurate: 51.7% → 98.8%

Time: 10.8s

Alternatives: 4

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log \left(\mathsf{fma}\left(re\_m, re\_m \cdot \frac{0.5}{im\_m}, im\_m\right)\right)}{\log 10} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (/ (log (fma re_m (* re_m (/ 0.5 im_m)) im_m)) (log 10.0)))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(fma(re_m, (re_m * (0.5 / im_m)), im_m)) / log(10.0);
}

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(log(fma(re_m, Float64(re_m * Float64(0.5 / im_m)), im_m)) / log(10.0))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[Log[N[(re$95$m * N[(re$95$m * N[(0.5 / im$95$m), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \frac{\log \color{blue}{\left(im + \frac{1}{2} \cdot \frac{{re}^{2}}{im}\right)}}{\log 10} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\log \color{blue}{\left(\frac{1}{2} \cdot \frac{{re}^{2}}{im} + im\right)}}{\log 10} \]

   2. *-lft-identity N/A

      \[\leadsto \frac{\log \left(\frac{1}{2} \cdot \frac{\color{blue}{1 \cdot {re}^{2}}}{im} + im\right)}{\log 10} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\log \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{im} \cdot {re}^{2}\right)} + im\right)}{\log 10} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\log \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot {re}^{2}} + im\right)}{\log 10} \]

   5. unpow2 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot \color{blue}{\left(re \cdot re\right)} + im\right)}{\log 10} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\log \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot re\right) \cdot re} + im\right)}{\log 10} \]

   7. *-commutative N/A

      \[\leadsto \frac{\log \left(\color{blue}{re \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot re\right)} + im\right)}{\log 10} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{fma}\left(re, \left(\frac{1}{2} \cdot \frac{1}{im}\right) \cdot re, im\right)\right)}}{\log 10} \]

   9. *-commutative N/A

      \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)}, im\right)\right)}{\log 10} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} \cdot \frac{1}{im}\right)}, im\right)\right)}{\log 10} \]

   11. associate-*r/ N/A

       \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{im}}, im\right)\right)}{\log 10} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, re \cdot \frac{\color{blue}{\frac{1}{2}}}{im}, im\right)\right)}{\log 10} \]

   13. lower-/.f64 33.0

       \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\frac{0.5}{im}}, im\right)\right)}{\log 10} \]

5. Applied rewrites 33.0%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{fma}\left(re, re \cdot \frac{0.5}{im}, im\right)\right)}}{\log 10} \]
6. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{1}{\log 10 \cdot \frac{1}{\log im\_m}} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (/ 1.0 (* (log 10.0) (/ 1.0 (log im_m)))))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return 1.0 / (log(10.0) * (1.0 / log(im_m)));
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = 1.0d0 / (log(10.0d0) * (1.0d0 / log(im_m)))
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return 1.0 / (Math.log(10.0) * (1.0 / Math.log(im_m)));
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return 1.0 / (math.log(10.0) * (1.0 / math.log(im_m)))

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(1.0 / Float64(log(10.0) * Float64(1.0 / log(im_m))))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = 1.0 / (log(10.0) * (1.0 / log(im_m)));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(1.0 / N[(N[Log[10.0], $MachinePrecision] * N[(1.0 / N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   4. lower-/.f64 53.3

      \[\leadsto \frac{1}{\color{blue}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   5. lift-log.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   7. pow1/2 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}} \]

   8. log-pow N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}} \]

   10. lower-log.f64 53.3

       \[\leadsto \frac{1}{\frac{\log 10}{0.5 \cdot \color{blue}{\log \left(re \cdot re + im \cdot im\right)}}} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{1}{\frac{\log 10}{\frac{1}{2} \cdot \log \color{blue}{\left(re \cdot re + im \cdot im\right)}}} \]

   12. +-commutative N/A

       \[\leadsto \frac{1}{\frac{\log 10}{\frac{1}{2} \cdot \log \color{blue}{\left(im \cdot im + re \cdot re\right)}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\log 10}{\frac{1}{2} \cdot \log \left(\color{blue}{im \cdot im} + re \cdot re\right)}} \]

   14. lower-fma.f64 53.3

       \[\leadsto \frac{1}{\frac{\log 10}{0.5 \cdot \log \color{blue}{\left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}} \]

4. Applied rewrites 53.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{0.5 \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{\log 10}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{\log 10}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{\log 10}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}\right)}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{\log 10}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}\right)} \]

   6. div-inv N/A

      \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\log 10 \cdot \frac{1}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}\right)} \]

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

      \[\leadsto \frac{-1}{\color{blue}{\left(\mathsf{neg}\left(\log 10\right)\right) \cdot \frac{1}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\left(\mathsf{neg}\left(\log 10\right)\right) \cdot \frac{1}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}} \]

   9. lift-log.f64 N/A

      \[\leadsto \frac{-1}{\left(\mathsf{neg}\left(\color{blue}{\log 10}\right)\right) \cdot \frac{1}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}} \]

   10. neg-log N/A

       \[\leadsto \frac{-1}{\color{blue}{\log \left(\frac{1}{10}\right)} \cdot \frac{1}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}} \]

   11. lower-log.f64 N/A

       \[\leadsto \frac{-1}{\color{blue}{\log \left(\frac{1}{10}\right)} \cdot \frac{1}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{-1}{\log \color{blue}{\frac{1}{10}} \cdot \frac{1}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \frac{1}{\color{blue}{\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}} \]

   14. associate-/r* N/A

       \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \color{blue}{\frac{\frac{1}{\frac{1}{2}}}{\log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}} \]

   15. metadata-eval N/A

       \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \frac{\color{blue}{2}}{\log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}} \]

   16. lower-/.f64 53.3

       \[\leadsto \frac{-1}{\log 0.1 \cdot \color{blue}{\frac{2}{\log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}} \]

   17. lift-fma.f64 N/A

       \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \frac{2}{\log \color{blue}{\left(im \cdot im + re \cdot re\right)}}} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \frac{2}{\log \left(\color{blue}{im \cdot im} + re \cdot re\right)}} \]

   19. +-commutative N/A

       \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \frac{2}{\log \color{blue}{\left(re \cdot re + im \cdot im\right)}}} \]

   20. lift-*.f64 N/A

       \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \frac{2}{\log \left(\color{blue}{re \cdot re} + im \cdot im\right)}} \]

   21. lower-fma.f64 53.3

       \[\leadsto \frac{-1}{\log 0.1 \cdot \frac{2}{\log \color{blue}{\left(\mathsf{fma}\left(re, re, im \cdot im\right)\right)}}} \]

6. Applied rewrites 53.3%

   \[\leadsto \color{blue}{\frac{-1}{\log 0.1 \cdot \frac{2}{\log \left(\mathsf{fma}\left(re, re, im \cdot im\right)\right)}}} \]

7. Taylor expanded in im around inf

   \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \color{blue}{\frac{-1}{\log \left(\frac{1}{im}\right)}}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{\log \left(\frac{1}{im}\right)}} \]

   2. distribute-neg-frac N/A

      \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\log \left(\frac{1}{im}\right)}\right)\right)}} \]

   3. distribute-neg-frac2 N/A

      \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)}}} \]

   4. log-rec N/A

      \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log im\right)\right)}\right)}} \]

   5. remove-double-neg N/A

      \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \frac{1}{\color{blue}{\log im}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\log \frac{1}{10} \cdot \color{blue}{\frac{1}{\log im}}} \]

   7. lower-log.f64 33.1

      \[\leadsto \frac{-1}{\log 0.1 \cdot \frac{1}{\color{blue}{\log im}}} \]

9. Applied rewrites 33.1%

   \[\leadsto \frac{-1}{\log 0.1 \cdot \color{blue}{\frac{1}{\log im}}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{-1}{\log \frac{1}{10} \cdot \frac{1}{\log im}}} \]

    2. frac-2neg N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \frac{1}{10} \cdot \frac{1}{\log im}\right)}} \]

    3. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{1}}{\mathsf{neg}\left(\log \frac{1}{10} \cdot \frac{1}{\log im}\right)} \]

    4. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(\log \frac{1}{10} \cdot \frac{1}{\log im}\right)}} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\log \frac{1}{10} \cdot \frac{1}{\log im}}\right)} \]

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

       \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(\log \frac{1}{10}\right)\right) \cdot \frac{1}{\log im}}} \]

    7. lift-log.f64 N/A

       \[\leadsto \frac{1}{\left(\mathsf{neg}\left(\color{blue}{\log \frac{1}{10}}\right)\right) \cdot \frac{1}{\log im}} \]

    8. neg-log N/A

       \[\leadsto \frac{1}{\color{blue}{\log \left(\frac{1}{\frac{1}{10}}\right)} \cdot \frac{1}{\log im}} \]

    9. metadata-eval N/A

       \[\leadsto \frac{1}{\log \color{blue}{10} \cdot \frac{1}{\log im}} \]

    10. lift-log.f64 N/A

        \[\leadsto \frac{1}{\color{blue}{\log 10} \cdot \frac{1}{\log im}} \]

    11. lower-*.f64 33.1

        \[\leadsto \frac{1}{\color{blue}{\log 10 \cdot \frac{1}{\log im}}} \]

11. Applied rewrites 33.1%

    \[\leadsto \color{blue}{\frac{1}{\log 10 \cdot \frac{1}{\log im}}} \]
12. Add Preprocessing

Alternative 3: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{1}{\frac{\log 10}{\log im\_m}} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (/ 1.0 (/ (log 10.0) (log im_m))))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return 1.0 / (log(10.0) / log(im_m));
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = 1.0d0 / (log(10.0d0) / log(im_m))
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return 1.0 / (Math.log(10.0) / Math.log(im_m));
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return 1.0 / (math.log(10.0) / math.log(im_m))

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(1.0 / Float64(log(10.0) / log(im_m)))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = 1.0 / (log(10.0) / log(im_m));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   4. lower-/.f64 53.3

      \[\leadsto \frac{1}{\color{blue}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   5. lift-log.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}} \]

   7. pow1/2 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}} \]

   8. log-pow N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}} \]

   10. lower-log.f64 53.3

       \[\leadsto \frac{1}{\frac{\log 10}{0.5 \cdot \color{blue}{\log \left(re \cdot re + im \cdot im\right)}}} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{1}{\frac{\log 10}{\frac{1}{2} \cdot \log \color{blue}{\left(re \cdot re + im \cdot im\right)}}} \]

   12. +-commutative N/A

       \[\leadsto \frac{1}{\frac{\log 10}{\frac{1}{2} \cdot \log \color{blue}{\left(im \cdot im + re \cdot re\right)}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\log 10}{\frac{1}{2} \cdot \log \left(\color{blue}{im \cdot im} + re \cdot re\right)}} \]

   14. lower-fma.f64 53.3

       \[\leadsto \frac{1}{\frac{\log 10}{0.5 \cdot \log \color{blue}{\left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}} \]

4. Applied rewrites 53.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{0.5 \cdot \log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right)}}} \]

5. Taylor expanded in im around inf

   \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)}}} \]

   2. log-rec N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log im\right)\right)}\right)}} \]

   3. remove-double-neg N/A

      \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\log im}}} \]

   4. lower-log.f64 33.1

      \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\log im}}} \]

7. Applied rewrites 33.1%

   \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\log im}}} \]
8. Add Preprocessing

Alternative 4: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log im\_m}{\log 10} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (/ (log im_m) (log 10.0)))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(im_m) / log(10.0);
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = log(im_m) / log(10.0d0)
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return Math.log(im_m) / Math.log(10.0);
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return math.log(im_m) / math.log(10.0)

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(log(im_m) / log(10.0))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = log(im_m) / log(10.0);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[Log[im$95$m], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
4. Step-by-step derivation

   1. lower-log.f64 33.1

      \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]

5. Applied rewrites 33.1%

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))