math.log10 on complex, real part

Percentage Accurate: 50.9% → 98.9%

Time: 7.9s

Alternatives: 2

Speedup: 1.5×

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: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log \left(im\_m + re\_m \cdot \left(re\_m \cdot \left(\frac{-0.125 \cdot \frac{re\_m \cdot \frac{re\_m}{im\_m}}{im\_m}}{im\_m} + \frac{0.5}{im\_m}\right)\right)\right)}{\log 10} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
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
    (*
     re_m
     (*
      re_m
      (+ (/ (* -0.125 (/ (* re_m (/ re_m im_m)) im_m)) im_m) (/ 0.5 im_m))))))
  (log 10.0)))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log((im_m + (re_m * (re_m * (((-0.125 * ((re_m * (re_m / im_m)) / im_m)) / im_m) + (0.5 / im_m)))))) / log(10.0);
}

Fortran

re_m = abs(re)
im_m = abs(im)
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 + (re_m * (re_m * ((((-0.125d0) * ((re_m * (re_m / im_m)) / im_m)) / im_m) + (0.5d0 / im_m)))))) / log(10.0d0)
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return Math.log((im_m + (re_m * (re_m * (((-0.125 * ((re_m * (re_m / im_m)) / im_m)) / im_m) + (0.5 / im_m)))))) / Math.log(10.0);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return math.log((im_m + (re_m * (re_m * (((-0.125 * ((re_m * (re_m / im_m)) / im_m)) / im_m) + (0.5 / im_m)))))) / math.log(10.0)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(log(Float64(im_m + Float64(re_m * Float64(re_m * Float64(Float64(Float64(-0.125 * Float64(Float64(re_m * Float64(re_m / im_m)) / im_m)) / im_m) + Float64(0.5 / im_m)))))) / log(10.0))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = log((im_m + (re_m * (re_m * (((-0.125 * ((re_m * (re_m / im_m)) / im_m)) / im_m) + (0.5 / im_m)))))) / log(10.0);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $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[(im$95$m + N[(re$95$m * N[(re$95$m * N[(N[(N[(-0.125 * N[(N[(re$95$m * N[(re$95$m / im$95$m), $MachinePrecision]), $MachinePrecision] / im$95$m), $MachinePrecision]), $MachinePrecision] / im$95$m), $MachinePrecision] + N[(0.5 / im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
4. Add Preprocessing

5. Taylor expanded in re around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(im + {re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)}\right), \mathsf{log.f64}\left(10\right)\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left({re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(\left(re \cdot re\right) \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \left(re \cdot \left(re \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}}\right), \left(\frac{1}{2} \cdot \frac{1}{im}\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 22.3%

   \[\leadsto \frac{\log \color{blue}{\left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{\frac{re \cdot re}{im}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}}{\log 10} \]
8. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(re \cdot \frac{re}{im}\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(\frac{re}{im} \cdot re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{re}{im}\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

   4. /-lowering-/.f64 23.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(re, im\right), re\right), im\right)\right), im\right), \mathsf{/.f64}\left(\frac{1}{2}, im\right)\right)\right)\right)\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

9. Applied egg-rr 23.3%

   \[\leadsto \frac{\log \left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{\color{blue}{\frac{re}{im} \cdot re}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}{\log 10} \]

10. Final simplification 23.3%

    \[\leadsto \frac{\log \left(im + re \cdot \left(re \cdot \left(\frac{-0.125 \cdot \frac{re \cdot \frac{re}{im}}{im}}{im} + \frac{0.5}{im}\right)\right)\right)}{\log 10} \]
11. Add Preprocessing

Alternative 2: 98.4% accurate, 1.5× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
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

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

Fortran

re_m = abs(re)
im_m = abs(im)
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

re_m = Math.abs(re);
im_m = Math.abs(im);
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

re_m = math.fabs(re)
im_m = math.fabs(im)
[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

re_m = abs(re)
im_m = abs(im)
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

re_m = abs(re);
im_m = abs(im);
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

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $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 50.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right), \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right), \log \color{blue}{10}\right) \]

   3. hypot-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right), \log 10\right) \]

   4. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \log 10\right) \]

   5. log-lowering-log.f64 99.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right), \mathsf{log.f64}\left(10\right)\right) \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
4. Add Preprocessing

5. Taylor expanded in re around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\log im, \color{blue}{\log 10}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \log \color{blue}{10}\right) \]

   3. log-lowering-log.f64 25.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{log.f64}\left(10\right)\right) \]

7. Simplified 25.2%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))