math.log/2 on complex, real part

Average Error: 32.0 → 17.9

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -2.1045401481442277 \cdot 10^{+128}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \leq -1.9933387775564622 \cdot 10^{-274}:\\ \;\;\;\;1.5 \cdot \frac{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{elif}\;re \leq 2.4998168956729745 \cdot 10^{-273}:\\ \;\;\;\;\frac{1.5 \cdot \log \left({im}^{0.6666666666666666}\right)}{\log base}\\ \mathbf{elif}\;re \leq 8.116903149692801 \cdot 10^{+57}:\\ \;\;\;\;1.5 \cdot \frac{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

FPCore

(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 (<= re -2.1045401481442277e+128)
   (/ (log (- re)) (log base))
   (if (<= re -1.9933387775564622e-274)
     (* 1.5 (/ (log (cbrt (+ (* re re) (* im im)))) (log base)))
     (if (<= re 2.4998168956729745e-273)
       (/ (* 1.5 (log (pow im 0.6666666666666666))) (log base))
       (if (<= re 8.116903149692801e+57)
         (* 1.5 (/ (log (cbrt (+ (* re re) (* im im)))) (log base)))
         (/ (log re) (log base)))))))

C

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 (re <= -2.1045401481442277e+128) {
		tmp = log(-re) / log(base);
	} else if (re <= -1.9933387775564622e-274) {
		tmp = 1.5 * (log(cbrt((re * re) + (im * im))) / log(base));
	} else if (re <= 2.4998168956729745e-273) {
		tmp = (1.5 * log(pow(im, 0.6666666666666666))) / log(base);
	} else if (re <= 8.116903149692801e+57) {
		tmp = 1.5 * (log(cbrt((re * re) + (im * im))) / log(base));
	} else {
		tmp = log(re) / log(base);
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 4 regimes

2. if re < -2.10454014814422774e128

   1. Initial program 56.1

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

   2. Simplified 56.0

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

   3. Taylor expanded around -inf 9.0

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

   4. Simplified 9.0

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

   if -2.10454014814422774e128 < re < -1.99333877755646224e-274 or 2.4998168956729745e-273 < re < 8.1169031496928011e57

   1. Initial program 20.8

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

   2. Simplified 20.7

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

   4. Applied pow1/2_binary64 20.7

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

   5. Applied log-pow_binary64 20.7

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

   6. Applied associate-/l*_binary64 20.8

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

   8. Applied add-cube-cbrt_binary64 20.8

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

   9. Applied log-prod_binary64 20.8

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

   10. Simplified 20.8

       \[\leadsto \frac{0.5}{\frac{\log base}{\color{blue}{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) \cdot 2} + \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity_binary64 20.8

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

   13. Applied pow1_binary64 20.8

       \[\leadsto \frac{0.5}{\frac{\log \color{blue}{\left({base}^{1}\right)}}{1 \cdot \left(\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) \cdot 2 + \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)\right)}}\]

   14. Applied log-pow_binary64 20.8

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

   15. Applied times-frac_binary64 20.8

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

   16. Applied *-un-lft-identity_binary64 20.8

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

   17. Applied times-frac_binary64 20.8

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

   18. Simplified 20.8

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

   19. Simplified 20.8

       \[\leadsto 1 \cdot \color{blue}{\frac{1.5 \cdot \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\log base}}\]
   20. Using strategy rm

   21. Applied pow1_binary64 20.8

       \[\leadsto 1 \cdot \frac{1.5 \cdot \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\log \color{blue}{\left({base}^{1}\right)}}\]

   22. Applied log-pow_binary64 20.8

       \[\leadsto 1 \cdot \frac{1.5 \cdot \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\color{blue}{1 \cdot \log base}}\]

   23. Applied times-frac_binary64 20.8

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{1.5}{1} \cdot \frac{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\log base}\right)}\]

   24. Simplified 20.8

       \[\leadsto 1 \cdot \left(\color{blue}{1.5} \cdot \frac{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\log base}\right)\]

   if -1.99333877755646224e-274 < re < 2.4998168956729745e-273

   1. Initial program 34.8

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

   2. Simplified 34.7

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

   4. Applied pow1/2_binary64 34.7

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

   5. Applied log-pow_binary64 34.7

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

   6. Applied associate-/l*_binary64 34.7

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

   8. Applied add-cube-cbrt_binary64 34.7

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

   9. Applied log-prod_binary64 34.7

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

   10. Simplified 34.7

       \[\leadsto \frac{0.5}{\frac{\log base}{\color{blue}{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) \cdot 2} + \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity_binary64 34.7

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

   13. Applied pow1_binary64 34.7

       \[\leadsto \frac{0.5}{\frac{\log \color{blue}{\left({base}^{1}\right)}}{1 \cdot \left(\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) \cdot 2 + \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)\right)}}\]

   14. Applied log-pow_binary64 34.7

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

   15. Applied times-frac_binary64 34.7

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

   16. Applied *-un-lft-identity_binary64 34.7

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

   17. Applied times-frac_binary64 34.7

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

   18. Simplified 34.7

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

   19. Simplified 34.7

       \[\leadsto 1 \cdot \color{blue}{\frac{1.5 \cdot \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\log base}}\]

   20. Taylor expanded around 0 34.5

       \[\leadsto 1 \cdot \frac{1.5 \cdot \color{blue}{\log \left({im}^{0.6666666666666666}\right)}}{\log base}\]

   if 8.1169031496928011e57 < re

   1. Initial program 46.7

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

   2. Simplified 46.6

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

   3. Taylor expanded around inf 10.8

      \[\leadsto \frac{\log \color{blue}{re}}{\log base}\]
3. Recombined 4 regimes into one program.

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.1045401481442277 \cdot 10^{+128}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \leq -1.9933387775564622 \cdot 10^{-274}:\\ \;\;\;\;1.5 \cdot \frac{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{elif}\;re \leq 2.4998168956729745 \cdot 10^{-273}:\\ \;\;\;\;\frac{1.5 \cdot \log \left({im}^{0.6666666666666666}\right)}{\log base}\\ \mathbf{elif}\;re \leq 8.116903149692801 \cdot 10^{+57}:\\ \;\;\;\;1.5 \cdot \frac{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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))))