math.log10 on complex, real part

Average Error: 31.5 → 17.5

Time: 9.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.2818911259175852 \cdot 10^{+112}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \leq -8.924391194312242 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \leq -1.4464803866898927 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \leq 4.4402811843754925 \cdot 10^{-292}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \leq 7.847390610675132 \cdot 10^{-242}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{im}\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \leq 4.7489781145347514 \cdot 10^{-151}:\\ \;\;\;\;\frac{-1}{\frac{\log 10}{-\log im}}\\ \mathbf{elif}\;re \leq 1.1192013375802831 \cdot 10^{+108}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{-2 \cdot \left(-\log re\right)}}\\ \end{array}\]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -1.2818911259175852e+112)
   (*
    (/ 0.5 (sqrt (log 10.0)))
    (* -2.0 (* (log (/ -1.0 re)) (sqrt (/ 1.0 (log 10.0))))))
   (if (<= re -8.924391194312242e-179)
     (/
      (* (/ 0.5 (sqrt (log 10.0))) (log (+ (* re re) (* im im))))
      (sqrt (log 10.0)))
     (if (<= re -1.4464803866898927e-220)
       (* (sqrt 0.5) (/ (sqrt 0.5) (/ (log 10.0) (* -2.0 (log (/ -1.0 re))))))
       (if (<= re 4.4402811843754925e-292)
         (*
          (/ 0.5 (sqrt (log 10.0)))
          (log (pow (+ (* re re) (* im im)) (/ 1.0 (sqrt (log 10.0))))))
         (if (<= re 7.847390610675132e-242)
           (*
            (/ 0.5 (sqrt (log 10.0)))
            (/ (* -2.0 (log (/ -1.0 im))) (sqrt (log 10.0))))
           (if (<= re 4.7489781145347514e-151)
             (/ -1.0 (/ (log 10.0) (- (log im))))
             (if (<= re 1.1192013375802831e+108)
               (*
                (/ 0.5 (sqrt (log 10.0)))
                (log (pow (+ (* re re) (* im im)) (/ 1.0 (sqrt (log 10.0))))))
               (*
                (sqrt 0.5)
                (/ (sqrt 0.5) (/ (log 10.0) (* -2.0 (- (log re))))))))))))))

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 <= -1.2818911259175852e+112) {
		tmp = (0.5 / sqrt(log(10.0))) * (-2.0 * (log(-1.0 / re) * sqrt(1.0 / log(10.0))));
	} else if (re <= -8.924391194312242e-179) {
		tmp = ((0.5 / sqrt(log(10.0))) * log((re * re) + (im * im))) / sqrt(log(10.0));
	} else if (re <= -1.4464803866898927e-220) {
		tmp = sqrt(0.5) * (sqrt(0.5) / (log(10.0) / (-2.0 * log(-1.0 / re))));
	} else if (re <= 4.4402811843754925e-292) {
		tmp = (0.5 / sqrt(log(10.0))) * log(pow(((re * re) + (im * im)), (1.0 / sqrt(log(10.0)))));
	} else if (re <= 7.847390610675132e-242) {
		tmp = (0.5 / sqrt(log(10.0))) * ((-2.0 * log(-1.0 / im)) / sqrt(log(10.0)));
	} else if (re <= 4.7489781145347514e-151) {
		tmp = -1.0 / (log(10.0) / -log(im));
	} else if (re <= 1.1192013375802831e+108) {
		tmp = (0.5 / sqrt(log(10.0))) * log(pow(((re * re) + (im * im)), (1.0 / sqrt(log(10.0)))));
	} else {
		tmp = sqrt(0.5) * (sqrt(0.5) / (log(10.0) / (-2.0 * -log(re))));
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 7 regimes

2. if re < -1.2818911259175852e112

   1. Initial program 53.9

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

   3. Applied add-sqr-sqrt_binary64_100 53.9

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

   4. Applied pow1/2_binary64_158 53.9

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

   5. Applied log-pow_binary64_167 53.9

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

   6. Applied times-frac_binary64_84 53.9

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

   7. Taylor expanded around -inf 8.4

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

   if -1.2818911259175852e112 < re < -8.9243911943122418e-179

   1. Initial program 16.4

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

   3. Applied add-sqr-sqrt_binary64_100 16.4

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

   4. Applied pow1/2_binary64_158 16.4

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

   5. Applied log-pow_binary64_167 16.4

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

   6. Applied times-frac_binary64_84 16.4

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

   8. Applied associate-*l/_binary64_21 16.6

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

   9. Simplified 16.4

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

   if -8.9243911943122418e-179 < re < -1.4464803866898927e-220

   1. Initial program 35.3

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

   3. Applied add-sqr-sqrt_binary64_100 35.3

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

   4. Applied pow1/2_binary64_158 35.3

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

   5. Applied log-pow_binary64_167 35.3

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

   6. Applied times-frac_binary64_84 35.3

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

   8. Applied *-un-lft-identity_binary64_78 35.3

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

   9. Applied add-sqr-sqrt_binary64_100 35.2

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

   10. Applied times-frac_binary64_84 35.3

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

   11. Applied associate-*l*_binary64_19 35.2

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

   12. Simplified 35.1

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

   13. Taylor expanded around -inf 46.8

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

   if -1.4464803866898927e-220 < re < 4.44028118437549247e-292 or 4.7489781145347514e-151 < re < 1.119201337580283e108

   1. Initial program 19.4

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

   3. Applied add-sqr-sqrt_binary64_100 19.4

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

   4. Applied pow1/2_binary64_158 19.4

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

   5. Applied log-pow_binary64_167 19.4

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

   6. Applied times-frac_binary64_84 19.4

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

   8. Applied add-log-exp_binary64_117 19.4

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

   9. Simplified 19.2

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

   if 4.44028118437549247e-292 < re < 7.8473906106751324e-242

   1. Initial program 31.8

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

   3. Applied add-sqr-sqrt_binary64_100 31.8

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

   4. Applied pow1/2_binary64_158 31.8

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

   5. Applied log-pow_binary64_167 31.8

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

   6. Applied times-frac_binary64_84 31.7

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

   7. Taylor expanded around -inf 29.6

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

   if 7.8473906106751324e-242 < re < 4.7489781145347514e-151

   1. Initial program 29.1

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

   3. Applied add-sqr-sqrt_binary64_100 29.1

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

   4. Applied pow1/2_binary64_158 29.1

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

   5. Applied log-pow_binary64_167 29.1

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

   6. Applied times-frac_binary64_84 29.1

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

   8. Applied *-un-lft-identity_binary64_78 29.1

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

   9. Applied add-sqr-sqrt_binary64_100 29.1

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

   10. Applied times-frac_binary64_84 29.1

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

   11. Applied associate-*l*_binary64_19 29.1

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

   12. Simplified 29.1

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

   13. Taylor expanded around inf 36.0

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

   14. Simplified 35.9

       \[\leadsto \color{blue}{\frac{-1}{\frac{\log 10}{-\log im}}}\]

   if 1.119201337580283e108 < re

   1. Initial program 53.7

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

   3. Applied add-sqr-sqrt_binary64_100 53.7

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

   4. Applied pow1/2_binary64_158 53.7

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

   5. Applied log-pow_binary64_167 53.7

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

   6. Applied times-frac_binary64_84 53.7

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

   8. Applied *-un-lft-identity_binary64_78 53.7

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

   9. Applied add-sqr-sqrt_binary64_100 53.7

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

   10. Applied times-frac_binary64_84 53.7

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

   11. Applied associate-*l*_binary64_19 53.7

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

   12. Simplified 53.7

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

   13. Taylor expanded around inf 7.9

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

   14. Simplified 7.9

       \[\leadsto \frac{\sqrt{0.5}}{1} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\color{blue}{-2 \cdot \left(-\log re\right)}}}\]
3. Recombined 7 regimes into one program.

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.2818911259175852 \cdot 10^{+112}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \leq -8.924391194312242 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \leq -1.4464803866898927 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \leq 4.4402811843754925 \cdot 10^{-292}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \leq 7.847390610675132 \cdot 10^{-242}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{im}\right)}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \leq 4.7489781145347514 \cdot 10^{-151}:\\ \;\;\;\;\frac{-1}{\frac{\log 10}{-\log im}}\\ \mathbf{elif}\;re \leq 1.1192013375802831 \cdot 10^{+108}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{-2 \cdot \left(-\log re\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021009 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))