math.log10 on complex, real part

Percentage Accurate: 50.9% → 98.6%

Time: 7.1s

Alternatives: 3

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: 50.9% 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: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \left(-{\log 0.1}^{-3}\right) \cdot \left(\log im\_m \cdot {\log 0.1}^{2}\right) \end{array} \]

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) -3.0)) (* (log im_m) (pow (log 0.1) 2.0))))

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), -3.0) * (log(im_m) * pow(log(0.1), 2.0));
}

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(0.1d0) ** (-3.0d0)) * (log(im_m) * (log(0.1d0) ** 2.0d0))
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.pow(Math.log(0.1), -3.0) * (Math.log(im_m) * Math.pow(Math.log(0.1), 2.0));
}

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.pow(math.log(0.1), -3.0) * (math.log(im_m) * math.pow(math.log(0.1), 2.0))

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(Float64(-(log(0.1) ^ -3.0)) * Float64(log(im_m) * (log(0.1) ^ 2.0)))
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(0.1) ^ -3.0) * (log(im_m) * (log(0.1) ^ 2.0));
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[Power[N[Log[0.1], $MachinePrecision], -3.0], $MachinePrecision]) * N[(N[Log[im$95$m], $MachinePrecision] * N[Power[N[Log[0.1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 30.7

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

5. Applied rewrites 30.7%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. metadata-eval N/A

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

   4. neg-log N/A

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

   5. lift-log.f64 N/A

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

   6. neg-sub0 N/A

      \[\leadsto \frac{\log im}{\color{blue}{0 - \log \frac{1}{10}}} \]

   7. flip3-- N/A

      \[\leadsto \frac{\log im}{\color{blue}{\frac{{0}^{3} - {\log \frac{1}{10}}^{3}}{0 \cdot 0 + \left(\log \frac{1}{10} \cdot \log \frac{1}{10} + 0 \cdot \log \frac{1}{10}\right)}}} \]

   8. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\log im}{{0}^{3} - {\log \frac{1}{10}}^{3}} \cdot \left(0 \cdot 0 + \left(\log \frac{1}{10} \cdot \log \frac{1}{10} + 0 \cdot \log \frac{1}{10}\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\log im}{{0}^{3} - {\log \frac{1}{10}}^{3}} \cdot \left(0 \cdot 0 + \left(\log \frac{1}{10} \cdot \log \frac{1}{10} + 0 \cdot \log \frac{1}{10}\right)\right)} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\log im}{{0}^{3} - {\log \frac{1}{10}}^{3}}} \cdot \left(0 \cdot 0 + \left(\log \frac{1}{10} \cdot \log \frac{1}{10} + 0 \cdot \log \frac{1}{10}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \frac{\log im}{\color{blue}{0} - {\log \frac{1}{10}}^{3}} \cdot \left(0 \cdot 0 + \left(\log \frac{1}{10} \cdot \log \frac{1}{10} + 0 \cdot \log \frac{1}{10}\right)\right) \]

   12. sub0-neg N/A

       \[\leadsto \frac{\log im}{\color{blue}{\mathsf{neg}\left({\log \frac{1}{10}}^{3}\right)}} \cdot \left(0 \cdot 0 + \left(\log \frac{1}{10} \cdot \log \frac{1}{10} + 0 \cdot \log \frac{1}{10}\right)\right) \]

   13. lower-neg.f64 N/A

       \[\leadsto \frac{\log im}{\color{blue}{-{\log \frac{1}{10}}^{3}}} \cdot \left(0 \cdot 0 + \left(\log \frac{1}{10} \cdot \log \frac{1}{10} + 0 \cdot \log \frac{1}{10}\right)\right) \]

   14. lower-pow.f64 N/A

       \[\leadsto \frac{\log im}{-\color{blue}{{\log \frac{1}{10}}^{3}}} \cdot \left(0 \cdot 0 + \left(\log \frac{1}{10} \cdot \log \frac{1}{10} + 0 \cdot \log \frac{1}{10}\right)\right) \]

7. Applied rewrites 30.7%

   \[\leadsto \color{blue}{\frac{\log im}{-{\log 0.1}^{3}} \cdot {\log 0.1}^{2}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\log im}{-{\log \frac{1}{10}}^{3}} \cdot {\log \frac{1}{10}}^{2}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log im}{-{\log \frac{1}{10}}^{3}}} \cdot {\log \frac{1}{10}}^{2} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\log im \cdot {\log \frac{1}{10}}^{2}}{-{\log \frac{1}{10}}^{3}}} \]

   4. div-inv N/A

      \[\leadsto \color{blue}{\left(\log im \cdot {\log \frac{1}{10}}^{2}\right) \cdot \frac{1}{-{\log \frac{1}{10}}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\log im \cdot {\log \frac{1}{10}}^{2}\right) \cdot \frac{1}{-{\log \frac{1}{10}}^{3}}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left({\log \frac{1}{10}}^{2} \cdot \log im\right)} \cdot \frac{1}{-{\log \frac{1}{10}}^{3}} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({\log \frac{1}{10}}^{2} \cdot \log im\right)} \cdot \frac{1}{-{\log \frac{1}{10}}^{3}} \]

   8. lift-neg.f64 N/A

      \[\leadsto \left({\log \frac{1}{10}}^{2} \cdot \log im\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left({\log \frac{1}{10}}^{3}\right)}} \]

   9. distribute-frac-neg2 N/A

      \[\leadsto \left({\log \frac{1}{10}}^{2} \cdot \log im\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\log \frac{1}{10}}^{3}}\right)\right)} \]

   10. lower-neg.f64 N/A

       \[\leadsto \left({\log \frac{1}{10}}^{2} \cdot \log im\right) \cdot \color{blue}{\left(-\frac{1}{{\log \frac{1}{10}}^{3}}\right)} \]

   11. lift-pow.f64 N/A

       \[\leadsto \left({\log \frac{1}{10}}^{2} \cdot \log im\right) \cdot \left(-\frac{1}{\color{blue}{{\log \frac{1}{10}}^{3}}}\right) \]

   12. pow-flip N/A

       \[\leadsto \left({\log \frac{1}{10}}^{2} \cdot \log im\right) \cdot \left(-\color{blue}{{\log \frac{1}{10}}^{\left(\mathsf{neg}\left(3\right)\right)}}\right) \]

   13. lower-pow.f64 N/A

       \[\leadsto \left({\log \frac{1}{10}}^{2} \cdot \log im\right) \cdot \left(-\color{blue}{{\log \frac{1}{10}}^{\left(\mathsf{neg}\left(3\right)\right)}}\right) \]

   14. metadata-eval 30.8

       \[\leadsto \left({\log 0.1}^{2} \cdot \log im\right) \cdot \left(-{\log 0.1}^{\color{blue}{-3}}\right) \]

9. Applied rewrites 30.8%

   \[\leadsto \color{blue}{\left({\log 0.1}^{2} \cdot \log im\right) \cdot \left(-{\log 0.1}^{-3}\right)} \]

10. Final simplification 30.8%

    \[\leadsto \left(-{\log 0.1}^{-3}\right) \cdot \left(\log im \cdot {\log 0.1}^{2}\right) \]
11. Add Preprocessing

Alternative 2: 99.0% accurate, 0.7× speedup

Math

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

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 (hypot im_m re_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(hypot(im_m, re_m)) / -log(0.1);
}

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(Math.hypot(im_m, re_m)) / -Math.log(0.1);
}

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(math.hypot(im_m, re_m)) / -math.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(hypot(im_m, re_m)) / Float64(-log(0.1)))
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(hypot(im_m, re_m)) / -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[Sqrt[im$95$m ^ 2 + re$95$m ^ 2], $MachinePrecision]], $MachinePrecision] / (-N[Log[0.1], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 99.0

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

4. Applied rewrites 99.0%

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

5. Final simplification 99.0%

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

Alternative 3: 98.3% 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 54.1%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 30.7

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

5. Applied rewrites 30.7%

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

Reproduce

herbie shell --seed 2024255 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))