Average Error: 32.2 → 18.3

Time: 17.3s

Precision: binary64

Cost: 66500

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -7.957190502393604 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\ \mathbf{elif}\;re \leq -3.396417318103129 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \leq 1.5808915944341867 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)}}\\ \mathbf{elif}\;re \leq 3.5515048939018026 \cdot 10^{+132}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.5}}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \left(-\log re\right)}}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;re \leq -7.957190502393604 \cdot 10^{+89}:\\
\;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\

\mathbf{elif}\;re \leq -3.396417318103129 \cdot 10^{-304}:\\
\;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\

\mathbf{elif}\;re \leq 1.5808915944341867 \cdot 10^{-181}:\\
\;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)}}\\

\mathbf{elif}\;re \leq 3.5515048939018026 \cdot 10^{+132}:\\
\;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{0.5}}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \left(-\log re\right)}}\\

\end{array}

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -7.957190502393604e+89)
   (/ (sqrt 0.5) (/ (log 10.0) (* (sqrt 0.5) (* -2.0 (log (/ -1.0 re))))))
   (if (<= re -3.396417318103129e-304)
     (* (sqrt 0.5) (/ (sqrt 0.5) (/ (log 10.0) (log (+ (* re re) (* im im))))))
     (if (<= re 1.5808915944341867e-181)
       (/ (sqrt 0.5) (/ (log 10.0) (* (sqrt 0.5) (* -2.0 (log (/ -1.0 im))))))
       (if (<= re 3.5515048939018026e+132)
         (*
          (/ (* (cbrt 0.5) (cbrt 0.5)) (* (cbrt (log 10.0)) (cbrt (log 10.0))))
          (/ (cbrt 0.5) (/ (cbrt (log 10.0)) (log (+ (* re re) (* im im))))))
         (/
          (sqrt 0.5)
          (* -0.5 (/ (log 10.0) (* (sqrt 0.5) (- (log re)))))))))))

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

↓

double code(double re, double im) {
	double tmp;
	if (re <= -7.957190502393604e+89) {
		tmp = sqrt(0.5) / (log(10.0) / (sqrt(0.5) * (-2.0 * log(-1.0 / re))));
	} else if (re <= -3.396417318103129e-304) {
		tmp = sqrt(0.5) * (sqrt(0.5) / (log(10.0) / log((re * re) + (im * im))));
	} else if (re <= 1.5808915944341867e-181) {
		tmp = sqrt(0.5) / (log(10.0) / (sqrt(0.5) * (-2.0 * log(-1.0 / im))));
	} else if (re <= 3.5515048939018026e+132) {
		tmp = ((cbrt(0.5) * cbrt(0.5)) / (cbrt(log(10.0)) * cbrt(log(10.0)))) * (cbrt(0.5) / (cbrt(log(10.0)) / log((re * re) + (im * im))));
	} else {
		tmp = sqrt(0.5) / (-0.5 * (log(10.0) / (sqrt(0.5) * -log(re))));
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `re`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	18.3
Cost	27716

\[\begin{array}{l} \mathbf{if}\;re \leq -3.8230971066908273 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\ \mathbf{elif}\;re \leq -2.0587136117511207 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \leq 1.4612152070729814 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)}}\\ \mathbf{elif}\;re \leq 1.0813851646160545 \cdot 10^{+133}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.5}}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \left(-\log re\right)}}\\ \end{array}\]

Alternative 2
Error	18.3
Cost	27716

\[\begin{array}{l} \mathbf{if}\;re \leq -1.7092302464501153 \cdot 10^{+84}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\ \mathbf{elif}\;re \leq -1.341759796267171 \cdot 10^{-305}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}\\ \mathbf{elif}\;re \leq 1.5100632843712227 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)}}\\ \mathbf{elif}\;re \leq 1.748985785853791 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.5}}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \left(-\log re\right)}}\\ \end{array}\]

Alternative 3
Error	18.3
Cost	27524

\[\begin{array}{l} \mathbf{if}\;re \leq -3.1949520550424426 \cdot 10^{+84}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\ \mathbf{elif}\;re \leq -1.14882176169717 \cdot 10^{-305}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}\\ \mathbf{elif}\;re \leq 2.589049802161673 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)}}\\ \mathbf{elif}\;re \leq 2.3275519908472327 \cdot 10^{+131}:\\ \;\;\;\;\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.5}}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \left(-\log re\right)}}\\ \end{array}\]

Alternative 4
Error	18.4
Cost	27524

\[\begin{array}{l} \mathbf{if}\;re \leq -1.2541583614703662 \cdot 10^{+84}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\ \mathbf{elif}\;re \leq -6.64588331996408 \cdot 10^{-304}:\\ \;\;\;\;0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}\\ \mathbf{elif}\;re \leq 5.175353787195054 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)}}\\ \mathbf{elif}\;re \leq 2.1597781474088058 \cdot 10^{+132}:\\ \;\;\;\;\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.5}}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \left(-\log re\right)}}\\ \end{array}\]

Alternative 5
Error	18.3
Cost	27524

\[\begin{array}{l} \mathbf{if}\;re \leq -4.9038143526604607 \cdot 10^{+89}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -3.749303227302988 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}\\ \mathbf{elif}\;re \leq 4.151622206787763 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)}}\\ \mathbf{elif}\;re \leq 1.5384294376496345 \cdot 10^{+132}:\\ \;\;\;\;\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.5}}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \left(-\log re\right)}}\\ \end{array}\]

Alternative 6
Error	18.3
Cost	27524

\[\begin{array}{l} \mathbf{if}\;re \leq -7.416831879408788 \cdot 10^{+89}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -5.256911607223242 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}\\ \mathbf{elif}\;re \leq 2.699719036635054 \cdot 10^{-183}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log 10}\\ \mathbf{elif}\;re \leq 5.421008368769625 \cdot 10^{+131}:\\ \;\;\;\;\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.5}}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \left(-\log re\right)}}\\ \end{array}\]

Alternative 7
Error	18.3
Cost	14788

\[\begin{array}{l} \mathbf{if}\;re \leq -5.795756010454339 \cdot 10^{+89}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -6.285914458263249 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}\\ \mathbf{elif}\;re \leq 7.9088512795762 \cdot 10^{-182}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log 10}\\ \mathbf{elif}\;re \leq 2.0686566951142447 \cdot 10^{+131}:\\ \;\;\;\;\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{\log 10}{-\log re}}\\ \end{array}\]

Alternative 8
Error	18.4
Cost	14788

\[\begin{array}{l} \mathbf{if}\;re \leq -5.255397387469522 \cdot 10^{+89}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \leq -1.1175447774881049 \cdot 10^{-304}:\\ \;\;\;\;\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \leq 1.3001277567494677 \cdot 10^{-182}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log 10}\\ \mathbf{elif}\;re \leq 9.26708196289371 \cdot 10^{+131}:\\ \;\;\;\;\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{\log 10}{-\log re}}\\ \end{array}\]

Alternative 9
Error	23.9
Cost	15816

\[\begin{array}{l} \mathbf{if}\;im \leq -1.2417036622411211 \cdot 10^{-56}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log 10}\\ \mathbf{elif}\;im \leq -3.767766824219734 \cdot 10^{-126}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq -4.29601264620126 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log 10}\\ \mathbf{elif}\;im \leq -6.887443022145891 \cdot 10^{-242}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{elif}\;im \leq -3.940313326110668 \cdot 10^{-269}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 2.3778016643234542 \cdot 10^{-284}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{elif}\;im \leq 1.0714898969958725 \cdot 10^{-273}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 8.652535386165096 \cdot 10^{-72}:\\ \;\;\;\;\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array}\]

Alternative 10
Error	24.7
Cost	15560

\[\begin{array}{l} \mathbf{if}\;im \leq -3.402873198778472 \cdot 10^{-57}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log 10}\\ \mathbf{elif}\;im \leq -2.8179583370122537 \cdot 10^{-127}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq -1.6925218008705358 \cdot 10^{-141}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log 10}\\ \mathbf{elif}\;im \leq -2.6620241004041015 \cdot 10^{-241}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{elif}\;im \leq -7.953281208305644 \cdot 10^{-269}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 1.9416406107261096 \cdot 10^{-284}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{elif}\;im \leq 5.8049964218187595 \cdot 10^{-272}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;im \leq 1.3066790705255886 \cdot 10^{-174}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array}\]

Alternative 11
Error	24.9
Cost	13634

\[\begin{array}{l} \mathbf{if}\;im \leq -6.319268992154543 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log 10}\\ \mathbf{elif}\;im \leq 2.5173148715042562 \cdot 10^{-174}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array}\]

Alternative 12
Error	36.0
Cost	13313

\[\begin{array}{l} \mathbf{if}\;re \leq 1.4452463921379706 \cdot 10^{-173}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Alternative 13
Error	43.0
Cost	13313

\[\begin{array}{l} \mathbf{if}\;re \leq 1.08511105960776 \cdot 10^{-309}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Alternative 14
Error	56.8
Cost	64

\[1\]

Derivation

Split input into 5 regimes
if re < -7.957190502393604e89
1. Initial program 49.8
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied pow1/2_binary64_84049.8
  \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\log 10}\]
4. Applied log-pow_binary64_84949.8
  \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]
5. Applied associate-/l*_binary64_70549.8
  \[\leadsto \color{blue}{\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
6. Using strategy rm
7. Applied add-sqr-sqrt_binary64_78249.8
  \[\leadsto \frac{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\]
8. Applied associate-/l*_binary64_70549.8
  \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{0.5}}}}\]
9. Simplified49.8
  \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{0.5}}}}\]
10. Taylor expanded around -inf 10.4
  \[\leadsto \frac{\sqrt{0.5}}{\frac{\log 10}{\color{blue}{\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)} \cdot \sqrt{0.5}}}\]
11. Simplified10.4
  \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}}\]
if -7.957190502393604e89 < re < -3.39641731810312884e-304
1. Initial program 23.0
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied pow1/2_binary64_84023.0
  \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\log 10}\]
4. Applied log-pow_binary64_84923.0
  \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]
5. Applied associate-/l*_binary64_70523.0
  \[\leadsto \color{blue}{\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
6. Using strategy rm
7. Applied pow1_binary64_82123.0
  \[\leadsto \frac{0.5}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]
8. Applied log-pow_binary64_84923.0
  \[\leadsto \frac{0.5}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]
9. Applied pow1_binary64_82123.0
  \[\leadsto \frac{0.5}{\frac{\log \color{blue}{\left({10}^{1}\right)}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]
10. Applied log-pow_binary64_84923.0
  \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot \log 10}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]
11. Applied times-frac_binary64_76623.0
  \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{1} \cdot \frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
12. Applied add-sqr-sqrt_binary64_78223.1
  \[\leadsto \frac{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}}{\frac{1}{1} \cdot \frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\]
13. Applied times-frac_binary64_76622.9
  \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{1}{1}} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
14. Simplified22.9
  \[\leadsto \color{blue}{\sqrt{0.5}} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\]
15. Simplified22.9
  \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
if -3.39641731810312884e-304 < re < 1.5808915944341867e-181
1. Initial program 32.4
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied pow1/2_binary64_84032.4
  \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\log 10}\]
4. Applied log-pow_binary64_84932.4
  \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]
5. Applied associate-/l*_binary64_70532.4
  \[\leadsto \color{blue}{\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
6. Using strategy rm
7. Applied add-sqr-sqrt_binary64_78232.5
  \[\leadsto \frac{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\]
8. Applied associate-/l*_binary64_70532.3
  \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{0.5}}}}\]
9. Simplified32.3
  \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{0.5}}}}\]
10. Taylor expanded around -inf 33.8
  \[\leadsto \frac{\sqrt{0.5}}{\frac{\log 10}{\color{blue}{\left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)} \cdot \sqrt{0.5}}}\]
11. Simplified33.8
  \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)}}}\]
if 1.5808915944341867e-181 < re < 3.55150489390180262e132
1. Initial program 17.0
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied pow1/2_binary64_84017.0
  \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\log 10}\]
4. Applied log-pow_binary64_84917.0
  \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]
5. Applied associate-/l*_binary64_70517.0
  \[\leadsto \color{blue}{\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
6. Using strategy rm
7. Applied pow1_binary64_82117.0
  \[\leadsto \frac{0.5}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]
8. Applied log-pow_binary64_84917.0
  \[\leadsto \frac{0.5}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]
9. Applied add-cube-cbrt_binary64_79517.6
  \[\leadsto \frac{0.5}{\frac{\color{blue}{\left(\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}\right) \cdot \sqrt[3]{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]
10. Applied times-frac_binary64_76617.6
  \[\leadsto \frac{0.5}{\color{blue}{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1} \cdot \frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]
11. Applied add-cube-cbrt_binary64_79516.9
  \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{0.5}}}{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1} \cdot \frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]
12. Applied times-frac_binary64_76616.9
  \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]
13. Simplified16.9
  \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]
14. Simplified16.9
  \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]
if 3.55150489390180262e132 < re
1. Initial program 58.4
  \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm
3. Applied pow1/2_binary64_84058.4
  \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\log 10}\]
4. Applied log-pow_binary64_84958.4
  \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]
5. Applied associate-/l*_binary64_70558.4
  \[\leadsto \color{blue}{\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
6. Using strategy rm
7. Applied add-sqr-sqrt_binary64_78258.5
  \[\leadsto \frac{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\]
8. Applied associate-/l*_binary64_70558.5
  \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{0.5}}}}\]
9. Simplified58.4
  \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right) \cdot \sqrt{0.5}}}}\]
10. Taylor expanded around inf 8.1
  \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \log \left(\frac{1}{re}\right)}}}\]
11. Simplified8.1
  \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \left(-\log re\right)}}}\]
12. Simplified8.1
  \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \left(-\log re\right)}}}\]
Recombined 5 regimes into one program.
Final simplification18.3
\[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.957190502393604 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}}\\ \mathbf{elif}\;re \leq -3.396417318103129 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \leq 1.5808915944341867 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt{0.5} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)}}\\ \mathbf{elif}\;re \leq 3.5515048939018026 \cdot 10^{+132}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.5}}{-0.5 \cdot \frac{\log 10}{\sqrt{0.5} \cdot \left(-\log re\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

Error

Try it out

Alternatives

Error

Derivation

`if re < -7.957190502393604e89`

`if -7.957190502393604e89 < re < -3.39641731810312884e-304`

`if -3.39641731810312884e-304 < re < 1.5808915944341867e-181`

`if 1.5808915944341867e-181 < re < 3.55150489390180262e132`

`if 3.55150489390180262e132 < re`

Reproduce