math.log10 on complex, real part

Percentage Accurate: 51.3% → 98.7%

Time: 11.2s

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: 51.3% 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: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \left(\log 0.1 \cdot {\log 0.1}^{-2}\right) \cdot \left(-\log im\_m\right) \end{array} \]

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 0.1) (pow (log 0.1) -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 (log(0.1) * pow(log(0.1), -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 = (log(0.1d0) * (log(0.1d0) ** (-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.log(0.1) * Math.pow(Math.log(0.1), -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.log(0.1) * math.pow(math.log(0.1), -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(Float64(log(0.1) * (log(0.1) ^ -2.0)) * Float64(-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 = (log(0.1) * (log(0.1) ^ -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[(N[Log[0.1], $MachinePrecision] * N[Power[N[Log[0.1], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * (-N[Log[im$95$m], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 49.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 98.4

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

5. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lift-log.f64 N/A

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

   5. neg-log N/A

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

   6. metadata-eval N/A

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

   7. lift-log.f64 N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

   10. frac-2neg N/A

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

   11. metadata-eval N/A

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

   12. lift-log.f64 N/A

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

   13. neg-log N/A

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

   14. metadata-eval N/A

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

   15. lift-log.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-neg.f64 97.9

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

7. Applied rewrites 97.9%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-log.f64 N/A

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

   5. neg-log N/A

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

   6. metadata-eval N/A

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

   7. lift-log.f64 N/A

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

   8. inv-pow N/A

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

   9. metadata-eval N/A

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

   10. pow-plus N/A

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

   11. lift-pow.f64 N/A

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

   12. lower-*.f64 98.9

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

9. Applied rewrites 98.9%

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

10. Final simplification 98.9%

    \[\leadsto \left(\log 0.1 \cdot {\log 0.1}^{-2}\right) \cdot \left(-\log im\right) \]
11. Add Preprocessing

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ {\log 0.1}^{-2} \cdot \left(\log 0.1 \cdot \left(-\log im\_m\right)\right) \end{array} \]

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 (log 0.1) -2.0) (* (log 0.1) (- (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(log(0.1), -2.0) * (log(0.1) * -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 = (log(0.1d0) ** (-2.0d0)) * (log(0.1d0) * -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.log(0.1), -2.0) * (Math.log(0.1) * -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.log(0.1), -2.0) * (math.log(0.1) * -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((log(0.1) ^ -2.0) * Float64(log(0.1) * Float64(-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 = (log(0.1) ^ -2.0) * (log(0.1) * -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[Log[0.1], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[Log[0.1], $MachinePrecision] * (-N[Log[im$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.3%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. pow1/2 N/A

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

   7. log-pow N/A

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

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

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-log.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. lift-log.f64 N/A

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

   17. neg-log N/A

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

   18. lower-log.f64 N/A

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

   19. metadata-eval 51.3

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

4. Applied rewrites 51.3%

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

6. Applied rewrites 51.5%

   \[\leadsto \color{blue}{\left(\log 0.1 \cdot \left(-0.5 \cdot \log \left(\mathsf{fma}\left(re, re, im \cdot im\right)\right)\right)\right) \cdot {\log 0.1}^{-2}} \]

7. Taylor expanded in im around inf

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

   1. log-rec N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-log.f64 98.6

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

9. Applied rewrites 98.6%

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

10. Final simplification 98.6%

    \[\leadsto {\log 0.1}^{-2} \cdot \left(\log 0.1 \cdot \left(-\log im\right)\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))