math.log10 on complex, real part

Average Error: 32.4 → 17.6

Time: 6.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.1762247444222311 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt{0.5}}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot -2\right)\right)\\ \mathbf{elif}\;re \leq 1.3781415203379638 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt{0.5}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.1082609750789312 \cdot 10^{-166} \lor \neg \left(re \leq 1.4551030635736294 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{2 \cdot \log re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \end{array}\]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1762247444222311e+151)
   (*
    (/ (sqrt 0.5) (sqrt (log 10.0)))
    (* (/ (sqrt 0.5) (sqrt (log 10.0))) (* (log (/ -1.0 re)) -2.0)))
   (if (<= re 1.3781415203379638e-202)
     (*
      (/ (sqrt 0.5) (sqrt (log 10.0)))
      (* (/ (sqrt 0.5) (sqrt (log 10.0))) (log (+ (* re re) (* im im)))))
     (if (or (<= re 1.1082609750789312e-166)
             (not (<= re 1.4551030635736294e+82)))
       (*
        (/ (* (cbrt 0.5) (cbrt 0.5)) (* (cbrt (log 10.0)) (cbrt (log 10.0))))
        (/ (cbrt 0.5) (/ (cbrt (log 10.0)) (* 2.0 (log re)))))
       (*
        (/ (* (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))))))))))

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 <= -1.1762247444222311e+151) {
		tmp = (sqrt(0.5) / sqrt(log(10.0))) * ((sqrt(0.5) / sqrt(log(10.0))) * (log(-1.0 / re) * -2.0));
	} else if (re <= 1.3781415203379638e-202) {
		tmp = (sqrt(0.5) / sqrt(log(10.0))) * ((sqrt(0.5) / sqrt(log(10.0))) * log((re * re) + (im * im)));
	} else if ((re <= 1.1082609750789312e-166) || !(re <= 1.4551030635736294e+82)) {
		tmp = ((cbrt(0.5) * cbrt(0.5)) / (cbrt(log(10.0)) * cbrt(log(10.0)))) * (cbrt(0.5) / (cbrt(log(10.0)) / (2.0 * log(re))));
	} else {
		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))));
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.17622474442223108e151

   1. Initial program 63.2

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

   3. Applied pow1/2_binary64_828 63.2

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

   4. Applied log-pow_binary64_837 63.2

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

   5. Applied associate-/l*_binary64_698 63.2

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

   7. Applied pow1_binary64_809 63.2

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

   8. Applied log-pow_binary64_837 63.2

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

   9. Applied add-sqr-sqrt_binary64_772 63.2

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

   10. Applied times-frac_binary64_757 63.2

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

   11. Applied add-sqr-sqrt_binary64_772 63.2

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

   12. Applied times-frac_binary64_757 63.2

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

   13. Simplified 63.2

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

   14. Simplified 63.2

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

   15. Taylor expanded around -inf 6.9

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

   16. Simplified 6.9

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

   if -1.17622474442223108e151 < re < 1.37814152033796385e-202

   1. Initial program 22.9

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

   3. Applied pow1/2_binary64_828 22.9

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

   4. Applied log-pow_binary64_837 22.9

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

   5. Applied associate-/l*_binary64_698 22.9

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

   7. Applied pow1_binary64_809 22.9

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

   8. Applied log-pow_binary64_837 22.9

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

   9. Applied add-sqr-sqrt_binary64_772 22.9

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

   10. Applied times-frac_binary64_757 23.0

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

   11. Applied add-sqr-sqrt_binary64_772 22.9

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

   12. Applied times-frac_binary64_757 22.8

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

   13. Simplified 22.8

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

   14. Simplified 22.8

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

   if 1.37814152033796385e-202 < re < 1.1082609750789312e-166 or 1.4551030635736294e82 < re

   1. Initial program 47.1

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

   3. Applied pow1/2_binary64_828 47.1

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

   4. Applied log-pow_binary64_837 47.1

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

   5. Applied associate-/l*_binary64_698 47.1

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

   7. Applied pow1_binary64_809 47.1

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

   8. Applied log-pow_binary64_837 47.1

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

   9. Applied add-cube-cbrt_binary64_783 47.3

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

   10. Applied times-frac_binary64_757 47.3

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

   11. Applied add-cube-cbrt_binary64_783 47.1

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

   12. Applied times-frac_binary64_757 47.1

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

   13. Simplified 47.1

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

   14. Taylor expanded around inf 15.1

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

   15. Simplified 15.1

       \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{\color{blue}{2 \cdot \log re}}}\]

   if 1.1082609750789312e-166 < re < 1.4551030635736294e82

   1. Initial program 16.3

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

   3. Applied pow1/2_binary64_828 16.3

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

   4. Applied log-pow_binary64_837 16.3

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

   5. Applied associate-/l*_binary64_698 16.3

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

   7. Applied pow1_binary64_809 16.3

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

   8. Applied log-pow_binary64_837 16.3

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

   9. Applied add-cube-cbrt_binary64_783 16.9

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

   10. Applied times-frac_binary64_757 16.9

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

   11. Applied add-cube-cbrt_binary64_783 16.2

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

   12. Applied times-frac_binary64_757 16.2

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

   13. Simplified 16.2

       \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1762247444222311 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt{0.5}}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot -2\right)\right)\\ \mathbf{elif}\;re \leq 1.3781415203379638 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt{0.5}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1.1082609750789312 \cdot 10^{-166} \lor \neg \left(re \leq 1.4551030635736294 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{2 \cdot \log re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{0.5}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))