Average Error: 31.1 → 17.5
Time: 20.3s
Precision: binary64
Cost: 41735
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq -2.081558982118848 \cdot 10^{+59}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{elif}\;im \leq -1.7324615588834855 \cdot 10^{-199}:\\ \;\;\;\;\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log base}\\ \mathbf{elif}\;im \leq 5.57434679198083 \cdot 10^{-283}:\\ \;\;\;\;\frac{0.5}{\frac{\log base}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;im \leq 2.5433593506319202 \cdot 10^{-253}:\\ \;\;\;\;\frac{\log \left(\left(re + 0.5 \cdot \frac{im \cdot im}{re}\right) - 0.125 \cdot \left(im \cdot {\left(\frac{im}{re}\right)}^{3}\right)\right)}{\log base}\\ \mathbf{elif}\;im \leq 1.4446922132637966 \cdot 10^{-229}:\\ \;\;\;\;\frac{0.5}{\frac{\log base}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;im \leq 9.790932882769397 \cdot 10^{-163}:\\ \;\;\;\;\frac{\log \left(\left(re + 0.5 \cdot \frac{im \cdot im}{re}\right) - 0.125 \cdot \left(im \cdot {\left(\frac{im}{re}\right)}^{3}\right)\right)}{\log base}\\ \mathbf{elif}\;im \leq 1.0066570199945735 \cdot 10^{+83}:\\ \;\;\;\;\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}
↓
\begin{array}{l}
\mathbf{if}\;im \leq -2.081558982118848 \cdot 10^{+59}:\\
\;\;\;\;\frac{\log \left(-im\right)}{\log base}\\

\mathbf{elif}\;im \leq -1.7324615588834855 \cdot 10^{-199}:\\
\;\;\;\;\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log base}\\

\mathbf{elif}\;im \leq 5.57434679198083 \cdot 10^{-283}:\\
\;\;\;\;\frac{0.5}{\frac{\log base}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\

\mathbf{elif}\;im \leq 2.5433593506319202 \cdot 10^{-253}:\\
\;\;\;\;\frac{\log \left(\left(re + 0.5 \cdot \frac{im \cdot im}{re}\right) - 0.125 \cdot \left(im \cdot {\left(\frac{im}{re}\right)}^{3}\right)\right)}{\log base}\\

\mathbf{elif}\;im \leq 1.4446922132637966 \cdot 10^{-229}:\\
\;\;\;\;\frac{0.5}{\frac{\log base}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\

\mathbf{elif}\;im \leq 9.790932882769397 \cdot 10^{-163}:\\
\;\;\;\;\frac{\log \left(\left(re + 0.5 \cdot \frac{im \cdot im}{re}\right) - 0.125 \cdot \left(im \cdot {\left(\frac{im}{re}\right)}^{3}\right)\right)}{\log base}\\

\mathbf{elif}\;im \leq 1.0066570199945735 \cdot 10^{+83}:\\
\;\;\;\;\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log base}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log im}{\log base}\\

\end{array}
(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))
↓
(FPCore (re im base)
 :precision binary64
 (if (<= im -2.081558982118848e+59)
   (/ (log (- im)) (log base))
   (if (<= im -1.7324615588834855e-199)
     (/
      (log
       (*
        (fabs (cbrt (+ (* re re) (* im im))))
        (sqrt (cbrt (+ (* re re) (* im im))))))
      (log base))
     (if (<= im 5.57434679198083e-283)
       (/ 0.5 (/ (log base) (* -2.0 (log (/ -1.0 re)))))
       (if (<= im 2.5433593506319202e-253)
         (/
          (log
           (-
            (+ re (* 0.5 (/ (* im im) re)))
            (* 0.125 (* im (pow (/ im re) 3.0)))))
          (log base))
         (if (<= im 1.4446922132637966e-229)
           (/ 0.5 (/ (log base) (* -2.0 (log (/ -1.0 re)))))
           (if (<= im 9.790932882769397e-163)
             (/
              (log
               (-
                (+ re (* 0.5 (/ (* im im) re)))
                (* 0.125 (* im (pow (/ im re) 3.0)))))
              (log base))
             (if (<= im 1.0066570199945735e+83)
               (/
                (log
                 (*
                  (fabs (cbrt (+ (* re re) (* im im))))
                  (sqrt (cbrt (+ (* re re) (* im im))))))
                (log base))
               (/ (log im) (log base))))))))))
double code(double re, double im, double base) {
	return ((log(sqrt((re * re) + (im * im))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}
↓
double code(double re, double im, double base) {
	double tmp;
	if (im <= -2.081558982118848e+59) {
		tmp = log(-im) / log(base);
	} else if (im <= -1.7324615588834855e-199) {
		tmp = log(fabs(cbrt((re * re) + (im * im))) * sqrt(cbrt((re * re) + (im * im)))) / log(base);
	} else if (im <= 5.57434679198083e-283) {
		tmp = 0.5 / (log(base) / (-2.0 * log(-1.0 / re)));
	} else if (im <= 2.5433593506319202e-253) {
		tmp = log((re + (0.5 * ((im * im) / re))) - (0.125 * (im * pow((im / re), 3.0)))) / log(base);
	} else if (im <= 1.4446922132637966e-229) {
		tmp = 0.5 / (log(base) / (-2.0 * log(-1.0 / re)));
	} else if (im <= 9.790932882769397e-163) {
		tmp = log((re + (0.5 * ((im * im) / re))) - (0.125 * (im * pow((im / re), 3.0)))) / log(base);
	} else if (im <= 1.0066570199945735e+83) {
		tmp = log(fabs(cbrt((re * re) + (im * im))) * sqrt(cbrt((re * re) + (im * im)))) / log(base);
	} else {
		tmp = log(im) / log(base);
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error31.7
Cost97856
\[\frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt[3]{\log base} \cdot \sqrt[3]{\log base}} \cdot \frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt[3]{\log base}}\]
Alternative 2
Error31.6
Cost78656
\[\sqrt[3]{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}} \cdot \left(\sqrt[3]{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}} \cdot \sqrt[3]{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\right)\]
Alternative 3
Error43.9
Cost78144
\[\frac{\sqrt{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt[3]{\log base} \cdot \sqrt[3]{\log base}} \cdot \frac{\sqrt{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt[3]{\log base}}\]
Alternative 4
Error31.6
Cost65600
\[\left(\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}\right) \cdot \frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log base}\]
Alternative 5
Error31.6
Cost59840
\[\sqrt[3]{0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log base}} \cdot \left(\sqrt[3]{0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log base}} \cdot \sqrt[3]{0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log base}}\right)\]
Alternative 6
Error47.6
Cost52416
\[\sqrt{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}} \cdot \sqrt{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\]
Alternative 7
Error31.3
Cost46208
\[\sqrt[3]{{\left(0.5 \cdot \frac{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) + 2 \cdot \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\log base}\right)}^{3}}\]
Alternative 8
Error43.7
Cost45888
\[\frac{\sqrt{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\frac{\log base}{\sqrt{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
Alternative 9
Error43.7
Cost45888
\[\sqrt{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \frac{\sqrt{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log base}\]
Alternative 10
Error31.5
Cost45760
\[\frac{0.5}{\sqrt[3]{\log base} \cdot \sqrt[3]{\log base}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt[3]{\log base}}\]
Alternative 11
Error47.6
Cost39872
\[\sqrt{0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log base}} \cdot \sqrt{0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log base}}\]
Alternative 12
Error47.6
Cost39744
\[\frac{\frac{0.5}{\sqrt{\frac{\log base}{\log \left(re \cdot re + im \cdot im\right)}}}}{\sqrt{\frac{\log base}{\log \left(re \cdot re + im \cdot im\right)}}}\]
Alternative 13
Error47.6
Cost39744
\[\frac{0.5}{\sqrt{\frac{\log base}{\log \left(re \cdot re + im \cdot im\right)}} \cdot \sqrt{\frac{\log base}{\log \left(re \cdot re + im \cdot im\right)}}}\]
Alternative 14
Error31.1
Cost39552
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base}\]
Alternative 15
Error31.1
Cost39488
\[\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log base}\]
Alternative 16
Error46.3
Cost39040
\[\sqrt[3]{3.375 \cdot \frac{{\log \left({\left(\frac{-1}{im}\right)}^{-0.6666666666666666}\right)}^{3}}{{\log base}^{3}}}\]
Alternative 17
Error47.5
Cost39040
\[\sqrt[3]{3.375 \cdot \frac{{\log \left({\left(\frac{-1}{re}\right)}^{-0.6666666666666666}\right)}^{3}}{{\log base}^{3}}}\]
Alternative 18
Error43.7
Cost33344
\[\frac{0.5}{\frac{\log base}{\sqrt{\log \left(re \cdot re + im \cdot im\right)}} \cdot \frac{1}{\sqrt{\log \left(re \cdot re + im \cdot im\right)}}}\]
Alternative 19
Error31.1
Cost33344
\[\frac{0.5}{\frac{\log base}{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) + 2 \cdot \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}\]
Alternative 20
Error31.1
Cost33344
\[0.5 \cdot \frac{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) + 2 \cdot \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\log base}\]
Alternative 21
Error43.7
Cost33216
\[\frac{0.5 \cdot \sqrt{\log \left(re \cdot re + im \cdot im\right)}}{\frac{\log base}{\sqrt{\log \left(re \cdot re + im \cdot im\right)}}}\]
Alternative 22
Error47.5
Cost32832
\[\frac{0.5}{\sqrt{\log base}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log base}}\]
Alternative 23
Error31.3
Cost32640
\[\sqrt[3]{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\right)}^{3}}\]
Alternative 24
Error46.6
Cost32576
\[\sqrt[3]{3.375 \cdot \frac{{\left(\log im \cdot 0.6666666666666666\right)}^{3}}{{\log base}^{3}}}\]
Alternative 25
Error46.2
Cost32576
\[\sqrt[3]{3.375 \cdot \frac{{\left(\log re \cdot 0.6666666666666666\right)}^{3}}{{\log base}^{3}}}\]
Alternative 26
Error31.4
Cost32576
\[\log \left(e^{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\right)\]
Alternative 27
Error47.5
Cost32576
\[e^{\log \left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\right)}\]
Alternative 28
Error46.3
Cost32512
\[\sqrt[3]{-\frac{{\log \left(\frac{-1}{im}\right)}^{3}}{{\log base}^{3}}}\]
Alternative 29
Error46.5
Cost32320
\[\sqrt[3]{\frac{{\log im}^{3}}{{\log base}^{3}}}\]
Alternative 30
Error46.2
Cost32320
\[\sqrt[3]{\frac{{\log re}^{3}}{{\log base}^{3}}}\]
Alternative 31
Error31.3
Cost26432
\[\frac{\sqrt{0.5}}{\log base} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{0.5}\right)\]
Alternative 32
Error31.3
Cost26432
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log base}{\log \left(re \cdot re + im \cdot im\right)}}\]
Alternative 33
Error31.3
Cost26368
\[\sqrt[3]{{\left(0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log base}\right)}^{3}}\]
Alternative 34
Error31.3
Cost26368
\[\frac{0.5}{\sqrt[3]{{\left(\frac{\log base}{\log \left(re \cdot re + im \cdot im\right)}\right)}^{3}}}\]
Alternative 35
Error31.2
Cost26368
\[\frac{0.5}{\frac{\log base}{\sqrt[3]{{\log \left(re \cdot re + im \cdot im\right)}^{3}}}}\]
Alternative 36
Error47.5
Cost26304
\[e^{\log \left(0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log base}\right)}\]
Alternative 37
Error31.2
Cost26304
\[\frac{0.5}{\log \left(e^{\frac{\log base}{\log \left(re \cdot re + im \cdot im\right)}}\right)}\]
Alternative 38
Error47.5
Cost26304
\[\frac{0.5}{e^{\log \left(\frac{\log base}{\log \left(re \cdot re + im \cdot im\right)}\right)}}\]
Alternative 39
Error47.4
Cost26240
\[\sqrt[3]{{\left(0.5 \cdot \frac{-2 \cdot \log \left(\frac{-1}{re}\right)}{\log base}\right)}^{3}}\]
Alternative 40
Error46.0
Cost26112
\[\sqrt[3]{{\left(0.5 \cdot \frac{\log \left(im \cdot im\right)}{\log base}\right)}^{3}}\]
Alternative 41
Error48.4
Cost20480
\[\frac{\log \left(\left(re + 0.5 \cdot \frac{im \cdot im}{re}\right) - 0.125 \cdot \left(im \cdot {\left(\frac{im}{re}\right)}^{3}\right)\right)}{\log base}\]
Alternative 42
Error48.5
Cost20480
\[\frac{\log \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - 0.125 \cdot \left(re \cdot {\left(\frac{re}{im}\right)}^{3}\right)\right)}{\log base}\]
Alternative 43
Error31.4
Cost19904
\[\frac{\log \left(re \cdot re + im \cdot im\right)}{\log base} \cdot \sqrt[3]{0.125}\]
Alternative 44
Error31.1
Cost19904
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{1}{\log base}\]
Alternative 45
Error31.1
Cost19904
\[\frac{1}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]
Alternative 46
Error31.4
Cost19840
\[\log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{0.5}{\log base}\right)}\right)\]
Alternative 47
Error31.1
Cost19776
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\]
Alternative 48
Error31.1
Cost13632
\[\frac{1}{\log base} \cdot \left(0.5 \cdot \log \left(re \cdot re + im \cdot im\right)\right)\]
Alternative 49
Error31.1
Cost13632
\[\frac{0.5}{\log base \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}\]
Alternative 50
Error31.1
Cost13504
\[\log \left(re \cdot re + im \cdot im\right) \cdot \frac{0.5}{\log base}\]
Alternative 51
Error31.1
Cost13504
\[\frac{0.5}{\frac{\log base}{\log \left(re \cdot re + im \cdot im\right)}}\]
Alternative 52
Error31.1
Cost13504
\[0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log base}\]
Alternative 53
Error46.2
Cost13376
\[\frac{0.5}{\frac{\log base}{-2 \cdot \log \left(\frac{-1}{im}\right)}}\]
Alternative 54
Error47.4
Cost13376
\[\frac{0.5}{\frac{\log base}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\]
Alternative 55
Error46.2
Cost13312
\[\frac{0.5}{-0.5 \cdot \frac{\log base}{-\log re}}\]
Alternative 56
Error46.5
Cost13312
\[\frac{0.5}{-0.5 \cdot \frac{\log base}{-\log im}}\]
Alternative 57
Error46.2
Cost13248
\[\frac{0.5}{\frac{\log base}{\log \left(re \cdot re\right)}}\]
Alternative 58
Error45.9
Cost13248
\[\frac{0.5}{\frac{\log base}{\log \left(im \cdot im\right)}}\]
Alternative 59
Error46.2
Cost13056
\[\frac{\log \left(-im\right)}{\log base}\]
Alternative 60
Error47.4
Cost13056
\[\frac{\log \left(-re\right)}{\log base}\]
Alternative 61
Error46.1
Cost12992
\[\frac{\log re}{\log base}\]
Alternative 62
Error46.5
Cost12992
\[\frac{\log im}{\log base}\]
Alternative 63
Error57.3
Cost64
\[1\]
Alternative 64
Error62.0
Cost64
\[0\]
Alternative 65
Error57.4
Cost64
\[-1\]

Error

Derivation

  1. Split input into 5 regimes

  2. if im < -2.0815589821188481e59

    1. Initial program 45.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified45.4

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

    3. Taylor expanded around -inf 11.0

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

    4. Simplified11.0

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

    5. Simplified11.0

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

    if -2.0815589821188481e59 < im < -1.7324615588834855e-199 or 9.79093288276939673e-163 < im < 1.00665701999457348e83

    1. Initial program 15.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified15.8

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt_binary64_11315.8

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}}\right)}{\log base}\]

    5. Applied sqrt-prod_binary64_9415.8

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}}{\log base}\]

    6. Simplified15.8

      \[\leadsto \frac{\log \left(\color{blue}{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log base}\]

    7. Simplified15.8

      \[\leadsto \color{blue}{\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log base}}\]

    if -1.7324615588834855e-199 < im < 5.57434679198083037e-283 or 2.5433593506319202e-253 < im < 1.4446922132637966e-229

    1. Initial program 32.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified32.7

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\]
    3. Using strategy rm

    4. Applied pow1/2_binary64_15832.7

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\log base}\]

    5. Applied log-pow_binary64_16732.7

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

    6. Applied associate-/l*_binary64_2332.7

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

    7. Taylor expanded around -inf 34.9

      \[\leadsto \frac{0.5}{\frac{\log base}{\color{blue}{-2 \cdot \log \left(\frac{-1}{re}\right)}}}\]

    8. Simplified34.9

      \[\leadsto \color{blue}{\frac{0.5}{\frac{\log base}{-2 \cdot \log \left(\frac{-1}{re}\right)}}}\]

    if 5.57434679198083037e-283 < im < 2.5433593506319202e-253 or 1.4446922132637966e-229 < im < 9.79093288276939673e-163

    1. Initial program 31.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified31.3

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

    3. Taylor expanded around inf 40.0

      \[\leadsto \frac{\log \color{blue}{\left(\left(re + 0.5 \cdot \frac{{im}^{2}}{re}\right) - 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)}}{\log base}\]

    4. Simplified34.0

      \[\leadsto \frac{\log \color{blue}{\left(\left(re + 0.5 \cdot \frac{im \cdot im}{re}\right) - 0.125 \cdot \left({\left(\frac{im}{re}\right)}^{3} \cdot im\right)\right)}}{\log base}\]

    5. Simplified34.0

      \[\leadsto \color{blue}{\frac{\log \left(\left(re + 0.5 \cdot \frac{im \cdot im}{re}\right) - 0.125 \cdot \left(im \cdot {\left(\frac{im}{re}\right)}^{3}\right)\right)}{\log base}}\]

    if 1.00665701999457348e83 < im

    1. Initial program 48.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified48.0

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

    3. Taylor expanded around 0 9.0

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

    4. Simplified9.0

      \[\leadsto \color{blue}{\frac{\log im}{\log base}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.081558982118848 \cdot 10^{+59}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{elif}\;im \leq -1.7324615588834855 \cdot 10^{-199}:\\ \;\;\;\;\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log base}\\ \mathbf{elif}\;im \leq 5.57434679198083 \cdot 10^{-283}:\\ \;\;\;\;\frac{0.5}{\frac{\log base}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;im \leq 2.5433593506319202 \cdot 10^{-253}:\\ \;\;\;\;\frac{\log \left(\left(re + 0.5 \cdot \frac{im \cdot im}{re}\right) - 0.125 \cdot \left(im \cdot {\left(\frac{im}{re}\right)}^{3}\right)\right)}{\log base}\\ \mathbf{elif}\;im \leq 1.4446922132637966 \cdot 10^{-229}:\\ \;\;\;\;\frac{0.5}{\frac{\log base}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;im \leq 9.790932882769397 \cdot 10^{-163}:\\ \;\;\;\;\frac{\log \left(\left(re + 0.5 \cdot \frac{im \cdot im}{re}\right) - 0.125 \cdot \left(im \cdot {\left(\frac{im}{re}\right)}^{3}\right)\right)}{\log base}\\ \mathbf{elif}\;im \leq 1.0066570199945735 \cdot 10^{+83}:\\ \;\;\;\;\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))