math.log10 on complex, real part

Average Error: 31.9 → 17.0

Time: 5.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.6754129982365756 \cdot 10^{119}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \mathbf{elif}\;re \le 6.13718247824505413 \cdot 10^{128}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\log re \cdot 2\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \end{array}\]

C

double code(double re, double im) {
	return (log(sqrt(((re * re) + (im * im)))) / log(10.0));
}

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -2.6754129982365756e+119)) {
		VAR = pow((-(2.0 * log((-1.0 / re))) * ((0.5 / sqrt(log(10.0))) / sqrt(log(10.0)))), 1.0);
	} else {
		double VAR_1;
		if ((re <= 6.137182478245054e+128)) {
			VAR_1 = ((0.5 / sqrt(log(10.0))) * log(pow(((re * re) + (im * im)), (1.0 / sqrt(log(10.0))))));
		} else {
			VAR_1 = pow(((log(re) * 2.0) * ((0.5 / sqrt(log(10.0))) / sqrt(log(10.0)))), 1.0);
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -2.6754129982365756e+119

   1. Initial program 55.5

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

   3. Applied add-sqr-sqrt 55.5

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

   4. Applied pow1/2 55.5

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

   5. Applied log-pow 55.5

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

   6. Applied times-frac 55.5

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp 55.5

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

   9. Simplified 55.5

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}\]
   10. Using strategy rm

   11. Applied pow1 55.5

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

   12. Applied pow1 55.5

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

   13. Applied pow-prod-down 55.5

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

   14. Simplified 55.5

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

   15. Taylor expanded around -inf 7.8

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

   if -2.6754129982365756e+119 < re < 6.137182478245054e+128

   1. Initial program 21.1

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

   3. Applied add-sqr-sqrt 21.1

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

   4. Applied pow1/2 21.1

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

   5. Applied log-pow 21.1

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

   6. Applied times-frac 21.1

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp 21.1

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

   9. Simplified 20.9

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

   if 6.137182478245054e+128 < re

   1. Initial program 57.5

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

   3. Applied add-sqr-sqrt 57.5

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

   4. Applied pow1/2 57.5

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

   5. Applied log-pow 57.5

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

   6. Applied times-frac 57.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp 57.4

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

   9. Simplified 57.4

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}\]
   10. Using strategy rm

   11. Applied pow1 57.4

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

   12. Applied pow1 57.4

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

   13. Applied pow-prod-down 57.4

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

   14. Simplified 57.4

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

   15. Taylor expanded around inf 8.7

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

   16. Simplified 8.7

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

4. Final simplification 17.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.6754129982365756 \cdot 10^{119}:\\ \;\;\;\;{\left(\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \mathbf{elif}\;re \le 6.13718247824505413 \cdot 10^{128}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\log re \cdot 2\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))