math.log10 on complex, real part

Percentage Accurate: 51.7% → 98.9%

Time: 7.8s

Alternatives: 5

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: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ {\left(\sqrt{\frac{1}{\log 10}}\right)}^{2} \cdot \log im\_m \end{array} \]

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
 (* (pow (sqrt (/ 1.0 (log 10.0))) 2.0) (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 pow(sqrt((1.0 / log(10.0))), 2.0) * 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 = (sqrt((1.0d0 / log(10.0d0))) ** 2.0d0) * 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 Math.pow(Math.sqrt((1.0 / Math.log(10.0))), 2.0) * 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 math.pow(math.sqrt((1.0 / math.log(10.0))), 2.0) * 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((sqrt(Float64(1.0 / log(10.0))) ^ 2.0) * 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 = (sqrt((1.0 / log(10.0))) ^ 2.0) * 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[(N[Power[N[Sqrt[N[(1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.9%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-log.f64 N/A

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

   4. neg-log N/A

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

   5. metadata-eval N/A

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

   6. lift-log.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 98.6

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

7. Applied rewrites 98.6%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-mul-1 N/A

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

   4. associate-*l/ N/A

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

   5. lift-/.f64 N/A

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

   6. lower-*.f64 98.5

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

   7. lift-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. lift-log.f64 N/A

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

   10. metadata-eval N/A

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

   11. neg-log N/A

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

   12. lift-log.f64 N/A

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

   13. frac-2neg N/A

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

   14. lower-/.f64 98.0

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

9. Applied rewrites 98.0%

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

    1. lift-/.f64 N/A

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

    2. frac-2neg N/A

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

    3. metadata-eval N/A

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

    4. lift-log.f64 N/A

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

    5. neg-log N/A

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

    6. metadata-eval N/A

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

    7. lift-log.f64 N/A

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

    8. lift-/.f64 98.5

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

    9. rem-square-sqrt N/A

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

    10. lift-sqrt.f64 N/A

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

    11. lift-sqrt.f64 N/A

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

    12. pow2 N/A

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

    13. lower-pow.f64 99.0

        \[\leadsto \color{blue}{{\left(\sqrt{\frac{-1}{\log 0.1}}\right)}^{2}} \cdot \log im \]

    14. lift-/.f64 N/A

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

    15. frac-2neg N/A

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

    16. metadata-eval N/A

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

    17. lift-log.f64 N/A

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

    18. neg-log N/A

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

    19. metadata-eval N/A

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

    20. lift-log.f64 N/A

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

    21. lift-/.f64 99.0

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

11. Applied rewrites 99.0%

    \[\leadsto \color{blue}{{\left(\sqrt{\frac{1}{\log 10}}\right)}^{2}} \cdot \log im \]
12. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log \left(\mathsf{fma}\left(re\_m, re\_m \cdot \frac{0.5}{im\_m}, im\_m\right)\right)}{-\log 0.1} \end{array} \]

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 0.1))))

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(0.1);
}

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)) / Float64(-log(0.1)))
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[0.1], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 98.8

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

5. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-log.f64 N/A

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

   4. neg-log N/A

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

   5. metadata-eval N/A

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

   6. lift-log.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 98.7

      \[\leadsto \frac{\color{blue}{-\log \left(\mathsf{fma}\left(re, re \cdot \frac{0.5}{im}, im\right)\right)}}{\log 0.1} \]

7. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{-\log \left(\mathsf{fma}\left(re, re \cdot \frac{0.5}{im}, im\right)\right)}{\log 0.1}} \]

8. Final simplification 98.7%

   \[\leadsto \frac{\log \left(\mathsf{fma}\left(re, re \cdot \frac{0.5}{im}, im\right)\right)}{-\log 0.1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))