math.log10 on complex, real part

Average Error: 31.9 → 7.0

Time: 11.8s

Precision: binary64

\[[re, im]=\mathsf{sort}([re, im])\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \leq 3.5598012421053094 \cdot 10^{-159}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;im \leq 2.907640811759891 \cdot 10^{+120}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({im}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= im 3.5598012421053094e-159)
   (* (/ 1.0 (sqrt (log 10.0))) (log (pow (- re) (/ 1.0 (sqrt (log 10.0))))))
   (if (<= im 2.907640811759891e+120)
     (*
      (* (/ 1.0 (sqrt (log 10.0))) (/ 1.0 (sqrt (log 10.0))))
      (log (sqrt (+ (* re re) (* im im)))))
     (* (/ 1.0 (sqrt (log 10.0))) (log (pow im (/ 1.0 (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 (im <= 3.5598012421053094e-159) {
		tmp = (1.0 / sqrt(log(10.0))) * log(pow(-re, (1.0 / sqrt(log(10.0)))));
	} else if (im <= 2.907640811759891e+120) {
		tmp = ((1.0 / sqrt(log(10.0))) * (1.0 / sqrt(log(10.0)))) * log(sqrt((re * re) + (im * im)));
	} else {
		tmp = (1.0 / sqrt(log(10.0))) * log(pow(im, (1.0 / sqrt(log(10.0)))));
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if im < 3.55980124210530944e-159

   1. Initial program 32.4

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

   3. Applied add-cbrt-cube_binary64 32.4

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

   4. Simplified 32.4

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt_binary64 32.4

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

   7. Applied pow1_binary64 32.4

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

   8. Applied log-pow_binary64 32.4

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

   9. Applied times-frac_binary64 32.4

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

   10. Applied unpow-prod-down_binary64 32.4

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

   11. Applied cbrt-prod_binary64 32.4

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

   12. Simplified 32.4

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

   13. Simplified 32.3

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

   15. Applied add-log-exp_binary64 32.3

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

   16. Simplified 32.2

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

   17. Taylor expanded around -inf 4.7

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

   18. Simplified 4.7

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

   if 3.55980124210530944e-159 < im < 2.9076408117598909e120

   1. Initial program 11.8

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

   3. Applied add-cbrt-cube_binary64 11.9

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

   4. Simplified 11.9

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt_binary64 11.9

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

   7. Applied pow1_binary64 11.9

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

   8. Applied log-pow_binary64 11.9

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

   9. Applied times-frac_binary64 11.9

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

   10. Applied unpow-prod-down_binary64 11.9

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

   11. Applied cbrt-prod_binary64 11.9

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

   12. Simplified 11.9

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

   13. Simplified 11.8

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

   15. Applied add-log-exp_binary64 11.8

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

   16. Simplified 11.6

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

   18. Applied log-pow_binary64 11.6

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

   19. Applied associate-*r*_binary64 11.6

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

   if 2.9076408117598909e120 < im

   1. Initial program 55.1

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

   3. Applied add-cbrt-cube_binary64 55.1

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

   4. Simplified 55.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt_binary64 55.1

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

   7. Applied pow1_binary64 55.1

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

   8. Applied log-pow_binary64 55.1

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

   9. Applied times-frac_binary64 55.1

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

   10. Applied unpow-prod-down_binary64 55.1

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

   11. Applied cbrt-prod_binary64 55.1

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

   12. Simplified 55.1

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

   13. Simplified 55.1

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

   15. Applied add-log-exp_binary64 55.1

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

   16. Simplified 55.1

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

   17. Taylor expanded around 0 5.0

       \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left({\color{blue}{im}}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 7.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.5598012421053094 \cdot 10^{-159}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;im \leq 2.907640811759891 \cdot 10^{+120}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({im}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))