math.log10 on complex, real part

Average Error: 32.4 → 18.3

Time: 5.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -6.5452086884385746 \cdot 10^{+75}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \leq -1.376243184252727 \cdot 10^{-175}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\\ \mathbf{elif}\;re \leq -1.199094603442337 \cdot 10^{-193}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{re}\right)}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \leq 7.7571798384429 \cdot 10^{+129}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{2 \cdot \log re}{\sqrt{\log 10}}\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -6.5452086884385746e+75)
   (*
    (/ 0.5 (sqrt (log 10.0)))
    (* -2.0 (* (log (/ -1.0 re)) (sqrt (/ 1.0 (log 10.0))))))
   (if (<= re -1.376243184252727e-175)
     (*
      (/ (* (cbrt 0.5) (cbrt 0.5)) (sqrt (sqrt (log 10.0))))
      (*
       (/ (log (+ (* re re) (* im im))) (sqrt (log 10.0)))
       (/ (cbrt 0.5) (sqrt (sqrt (log 10.0))))))
     (if (<= re -1.199094603442337e-193)
       (*
        (/ (* (cbrt 0.5) (cbrt 0.5)) (sqrt (sqrt (log 10.0))))
        (*
         (/ (cbrt 0.5) (sqrt (sqrt (log 10.0))))
         (/ (* -2.0 (log (/ -1.0 re))) (sqrt (log 10.0)))))
       (if (<= re 7.7571798384429e+129)
         (*
          (/ 0.5 (sqrt (log 10.0)))
          (log (pow (+ (* re re) (* im im)) (/ 1.0 (sqrt (log 10.0))))))
         (*
          (/ (* (cbrt 0.5) (cbrt 0.5)) (sqrt (sqrt (log 10.0))))
          (*
           (/ (cbrt 0.5) (sqrt (sqrt (log 10.0))))
           (/ (* 2.0 (log re)) (sqrt (log 10.0))))))))))

C

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 <= -6.5452086884385746e+75) {
		tmp = (0.5 / sqrt(log(10.0))) * (-2.0 * (log(-1.0 / re) * sqrt(1.0 / log(10.0))));
	} else if (re <= -1.376243184252727e-175) {
		tmp = ((cbrt(0.5) * cbrt(0.5)) / sqrt(sqrt(log(10.0)))) * ((log((re * re) + (im * im)) / sqrt(log(10.0))) * (cbrt(0.5) / sqrt(sqrt(log(10.0)))));
	} else if (re <= -1.199094603442337e-193) {
		tmp = ((cbrt(0.5) * cbrt(0.5)) / sqrt(sqrt(log(10.0)))) * ((cbrt(0.5) / sqrt(sqrt(log(10.0)))) * ((-2.0 * log(-1.0 / re)) / sqrt(log(10.0))));
	} else if (re <= 7.7571798384429e+129) {
		tmp = (0.5 / sqrt(log(10.0))) * log(pow(((re * re) + (im * im)), (1.0 / sqrt(log(10.0)))));
	} else {
		tmp = ((cbrt(0.5) * cbrt(0.5)) / sqrt(sqrt(log(10.0)))) * ((cbrt(0.5) / sqrt(sqrt(log(10.0)))) * ((2.0 * log(re)) / sqrt(log(10.0))));
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -6.54520868843857457e75

   1. Initial program 48.1

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt_binary64_98 48.1

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

   4. Applied pow1/2_binary64_154 48.1

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

   5. Applied log-pow_binary64_163 48.1

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

   6. Applied times-frac_binary64_83 48.1

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

   7. Taylor expanded around -inf 10.0

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \color{blue}{\left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]

   if -6.54520868843857457e75 < re < -1.376243184252727e-175

   1. Initial program 18.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt_binary64_98 18.0

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

   4. Applied pow1/2_binary64_154 18.0

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

   5. Applied log-pow_binary64_163 18.0

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

   6. Applied times-frac_binary64_83 18.0

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp_binary64_113 18.0

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

   9. Simplified 17.8

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt_binary64_98 18.4

       \[\leadsto \frac{0.5}{\color{blue}{\sqrt{\sqrt{\log 10}} \cdot \sqrt{\sqrt{\log 10}}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

   12. Applied add-cube-cbrt_binary64_109 17.8

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{0.5}}}{\sqrt{\sqrt{\log 10}} \cdot \sqrt{\sqrt{\log 10}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

   13. Applied times-frac_binary64_83 17.8

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

   14. Applied associate-*l*_binary64_20 17.9

       \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)}\]

   15. Simplified 17.9

       \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \color{blue}{\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)}\]

   if -1.376243184252727e-175 < re < -1.1990946034423369e-193

   1. Initial program 33.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt_binary64_98 33.3

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

   4. Applied pow1/2_binary64_154 33.3

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

   5. Applied log-pow_binary64_163 33.3

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

   6. Applied times-frac_binary64_83 33.3

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp_binary64_113 33.3

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

   9. Simplified 33.2

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt_binary64_98 33.6

       \[\leadsto \frac{0.5}{\color{blue}{\sqrt{\sqrt{\log 10}} \cdot \sqrt{\sqrt{\log 10}}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

   12. Applied add-cube-cbrt_binary64_109 33.2

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{0.5}}}{\sqrt{\sqrt{\log 10}} \cdot \sqrt{\sqrt{\log 10}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

   13. Applied times-frac_binary64_83 33.2

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

   14. Applied associate-*l*_binary64_20 33.2

       \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)}\]

   15. Simplified 33.2

       \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \color{blue}{\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)}\]

   16. Taylor expanded around -inf 44.3

       \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\color{blue}{-2 \cdot \log \left(\frac{-1}{re}\right)}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\]

   17. Simplified 44.3

       \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\color{blue}{\log \left(\frac{-1}{re}\right) \cdot -2}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\]

   if -1.1990946034423369e-193 < re < 7.7571798384428999e129

   1. Initial program 24.6

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt_binary64_98 24.6

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

   4. Applied pow1/2_binary64_154 24.6

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

   5. Applied log-pow_binary64_163 24.6

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

   6. Applied times-frac_binary64_83 24.5

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp_binary64_113 24.5

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

   9. Simplified 24.4

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}\]

   if 7.7571798384428999e129 < re

   1. Initial program 57.2

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt_binary64_98 57.2

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

   4. Applied pow1/2_binary64_154 57.2

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

   5. Applied log-pow_binary64_163 57.2

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

   6. Applied times-frac_binary64_83 57.2

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp_binary64_113 57.2

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

   9. Simplified 57.2

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt_binary64_98 57.2

       \[\leadsto \frac{0.5}{\color{blue}{\sqrt{\sqrt{\log 10}} \cdot \sqrt{\sqrt{\log 10}}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

   12. Applied add-cube-cbrt_binary64_109 57.2

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{0.5}}}{\sqrt{\sqrt{\log 10}} \cdot \sqrt{\sqrt{\log 10}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

   13. Applied times-frac_binary64_83 57.2

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

   14. Applied associate-*l*_binary64_20 57.2

       \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)}\]

   15. Simplified 57.2

       \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \color{blue}{\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)}\]

   16. Taylor expanded around inf 8.0

       \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\color{blue}{-2 \cdot \log \left(\frac{1}{re}\right)}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\]

   17. Simplified 8.0

       \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\color{blue}{2 \cdot \log re}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.5452086884385746 \cdot 10^{+75}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \leq -1.376243184252727 \cdot 10^{-175}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}}\right)\\ \mathbf{elif}\;re \leq -1.199094603442337 \cdot 10^{-193}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{re}\right)}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \leq 7.7571798384429 \cdot 10^{+129}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{0.5}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{2 \cdot \log re}{\sqrt{\log 10}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020299 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))