math.log10 on complex, real part

Percentage Accurate: 51.3% → 99.5%

Time: 6.6s

Alternatives: 3

Speedup: 0.7×

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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left({\log 10}^{-0.5}\right)}^{2} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (pow (pow (log 10.0) -0.5) 2.0) (log (hypot im re))))

C

double code(double re, double im) {
	return pow(pow(log(10.0), -0.5), 2.0) * log(hypot(im, re));
}

Java

public static double code(double re, double im) {
	return Math.pow(Math.pow(Math.log(10.0), -0.5), 2.0) * Math.log(Math.hypot(im, re));
}

Python

def code(re, im):
	return math.pow(math.pow(math.log(10.0), -0.5), 2.0) * math.log(math.hypot(im, re))

Julia

function code(re, im)
	return Float64(((log(10.0) ^ -0.5) ^ 2.0) * log(hypot(im, re)))
end

MATLAB

function tmp = code(re, im)
	tmp = ((log(10.0) ^ -0.5) ^ 2.0) * log(hypot(im, re));
end

Wolfram

code[re_, im_] := N[(N[Power[N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision] * N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. lower-neg.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. neg-log N/A

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

   13. lower-log.f64 N/A

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

   14. metadata-eval 99.1

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

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\log 0.1}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. unpow-1 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. neg-sub0 N/A

      \[\leadsto {\log \frac{1}{10}}^{-1} \cdot \color{blue}{\left(0 - \log \left(\mathsf{hypot}\left(im, re\right)\right)\right)} \]

   8. flip-- N/A

      \[\leadsto {\log \frac{1}{10}}^{-1} \cdot \color{blue}{\frac{0 \cdot 0 - \log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)}{0 + \log \left(\mathsf{hypot}\left(im, re\right)\right)}} \]

   9. +-lft-identity N/A

      \[\leadsto {\log \frac{1}{10}}^{-1} \cdot \frac{0 \cdot 0 - \log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)}{\color{blue}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}} \]

   10. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{{\log \frac{1}{10}}^{-1} \cdot \left(0 \cdot 0 - \log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right)}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{{\log \frac{1}{10}}^{-1} \cdot \left(0 \cdot 0 - \log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right)\right)}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}} \]

6. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{{\log 0.1}^{-1} \cdot \left(-{\log \left(\mathsf{hypot}\left(re, im\right)\right)}^{2}\right)}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

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

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

   6. lift-pow.f64 N/A

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

   7. unpow-1 N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-frac2 N/A

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

   10. metadata-eval N/A

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

   11. lift-log.f64 N/A

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

   12. neg-log N/A

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

   13. metadata-eval N/A

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

   14. lift-log.f64 N/A

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

   15. associate-/l* N/A

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

   16. lift-pow.f64 N/A

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

8. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot {\left({\log 10}^{-0.5}\right)}^{2}} \]

9. Final simplification 99.5%

   \[\leadsto {\left({\log 10}^{-0.5}\right)}^{2} \cdot \log \left(\mathsf{hypot}\left(im, re\right)\right) \]
10. Add Preprocessing

Alternative 2: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(im, re\right)\right)}{-\log 0.1} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log (hypot im re)) (- (log 0.1))))

C

double code(double re, double im) {
	return log(hypot(im, re)) / -log(0.1);
}

Java

public static double code(double re, double im) {
	return Math.log(Math.hypot(im, re)) / -Math.log(0.1);
}

Python

def code(re, im):
	return math.log(math.hypot(im, re)) / -math.log(0.1)

Julia

function code(re, im)
	return Float64(log(hypot(im, re)) / Float64(-log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(im, re)) / -log(0.1);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision]], $MachinePrecision] / (-N[Log[0.1], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. lower-neg.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. neg-log N/A

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

   13. lower-log.f64 N/A

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

   14. metadata-eval 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(im, re\right)\right)}{-\log 0.1} \]
6. Add Preprocessing

Alternative 3: 27.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log im) (log 10.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 27.0

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

5. Applied rewrites 27.0%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))