math.log10 on complex, real part

Average Error: 32.5 → 19.4

Time: 14.9s

Precision: binary64

Cost: 28230

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -6.0797067422693544 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \leq -3.466869228507893 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \leq -3.214783349564626 \cdot 10^{-236}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \leq 2.0246090950680006 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{im}\right)}}\\ \mathbf{elif}\;re \leq 4.017317388691866 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log im \cdot 2}}\\ \mathbf{elif}\;re \leq 246.62146382329686:\\ \;\;\;\;\log \left(e^{\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{2 \cdot \log re}}\\ \end{array}\]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -6.0797067422693544e+85)
   (* (sqrt 0.5) (/ (sqrt 0.5) (/ (log 10.0) (* -2.0 (log (/ -1.0 re))))))
   (if (<= re -3.466869228507893e-172)
     (* (sqrt 0.5) (/ (sqrt 0.5) (/ (log 10.0) (log (+ (* re re) (* im im))))))
     (if (<= re -3.214783349564626e-236)
       (/ (log im) (log 10.0))
       (if (<= re 2.0246090950680006e-222)
         (*
          (sqrt 0.5)
          (/ (sqrt 0.5) (/ (log 10.0) (* -2.0 (log (/ -1.0 im))))))
         (if (<= re 4.017317388691866e-131)
           (* (sqrt 0.5) (/ (sqrt 0.5) (/ (log 10.0) (* (log im) 2.0))))
           (if (<= re 246.62146382329686)
             (log (exp (/ 0.5 (/ (log 10.0) (log (+ (* re re) (* im im)))))))
             (*
              (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 <= -6.0797067422693544e+85) {
		tmp = sqrt(0.5) * (sqrt(0.5) / (log(10.0) / (-2.0 * log(-1.0 / re))));
	} else if (re <= -3.466869228507893e-172) {
		tmp = sqrt(0.5) * (sqrt(0.5) / (log(10.0) / log((re * re) + (im * im))));
	} else if (re <= -3.214783349564626e-236) {
		tmp = log(im) / log(10.0);
	} else if (re <= 2.0246090950680006e-222) {
		tmp = sqrt(0.5) * (sqrt(0.5) / (log(10.0) / (-2.0 * log(-1.0 / im))));
	} else if (re <= 4.017317388691866e-131) {
		tmp = sqrt(0.5) * (sqrt(0.5) / (log(10.0) / (log(im) * 2.0)));
	} else if (re <= 246.62146382329686) {
		tmp = log(exp(0.5 / (log(10.0) / log((re * re) + (im * im)))));
	} 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

Alternatives

Alternative 1
Error	32.8
Cost	97856

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

Alternative 2
Error	32.8
Cost	84928

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

Alternative 3
Error	32.8
Cost	78656

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

Alternative 4
Error	32.8
Cost	65600

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

Alternative 5
Error	53.3
Cost	65472

\[\frac{\log \left(\sqrt{{re}^{6} + {im}^{6}}\right)}{\log 10} - \frac{\log \left(\sqrt{{re}^{4} + \left({im}^{4} - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)}\right)}{\log 10}\]

Alternative 6
Error	44.9
Cost	65216

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

Alternative 7
Error	32.8
Cost	59840

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

Alternative 8
Error	44.9
Cost	59200

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

Alternative 9
Error	32.5
Cost	52800

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

Alternative 10
Error	44.8
Cost	52672

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

Alternative 11
Error	53.2
Cost	52544

\[\frac{\log \left(\frac{\sqrt{{re}^{6} + {im}^{6}}}{\sqrt{{re}^{4} + \left({im}^{4} - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)}}\right)}{\log 10}\]

Alternative 12
Error	44.8
Cost	52416

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

Alternative 13
Error	53.3
Cost	46272

\[\frac{0.5}{\frac{\log 10}{\log \left({re}^{6} + {im}^{6}\right) - \log \left({re}^{4} + \left({im}^{4} - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)}}\]

Alternative 14
Error	54.6
Cost	46016

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

Alternative 15
Error	44.8
Cost	45888

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

Alternative 16
Error	44.8
Cost	45888

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

Alternative 17
Error	44.8
Cost	39872

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

Alternative 18
Error	32.5
Cost	39488

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

Alternative 19
Error	32.6
Cost	39360

\[\frac{0.3333333333333333}{\sqrt{\log 10}} \cdot \frac{3 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\]

Alternative 20
Error	32.5
Cost	33344

\[\frac{0.5}{\frac{\log 10}{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) + 2 \cdot \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}\]

Alternative 21
Error	44.8
Cost	33216

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

Alternative 22
Error	32.5
Cost	32960

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

Alternative 23
Error	32.6
Cost	32960

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

Alternative 24
Error	32.4
Cost	32832

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

Alternative 25
Error	53.2
Cost	32640

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

Alternative 26
Error	32.5
Cost	32640

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

Alternative 27
Error	44.9
Cost	32576

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

Alternative 28
Error	62.5
Cost	32576

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

Alternative 29
Error	32.5
Cost	26560

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

Alternative 30
Error	32.4
Cost	26432

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

Alternative 31
Error	32.5
Cost	26432

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

Alternative 32
Error	32.4
Cost	26432

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

Alternative 33
Error	32.5
Cost	26368

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

Alternative 34
Error	32.5
Cost	26368

\[\sqrt[3]{\frac{0.125}{{\left(\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}\right)}^{3}}}\]

Alternative 35
Error	32.5
Cost	26368

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

Alternative 36
Error	32.5
Cost	26368

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

Alternative 37
Error	32.5
Cost	26304

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

Alternative 38
Error	32.5
Cost	26304

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

Alternative 39
Error	47.0
Cost	26304

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

Alternative 40
Error	46.3
Cost	26304

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

Alternative 41
Error	46.9
Cost	26304

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

Alternative 42
Error	46.3
Cost	26304

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

Alternative 43
Error	44.9
Cost	26304

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

Alternative 44
Error	32.5
Cost	26304

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

Alternative 45
Error	44.9
Cost	26304

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

Alternative 46
Error	44.9
Cost	26304

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

Alternative 47
Error	46.1
Cost	26176

\[\sqrt{0.5} \cdot \left(2 \cdot \left(\frac{\sqrt{0.5}}{\log 10} \cdot \log im\right)\right)\]

Alternative 48
Error	46.8
Cost	26176

\[\sqrt{0.5} \cdot \left(\frac{\sqrt{0.5}}{\log 10} \cdot \log \left(re \cdot re\right)\right)\]

Alternative 49
Error	47.0
Cost	26176

\[\sqrt{0.5} \cdot \left(2 \cdot \left(\frac{\sqrt{0.5}}{\log 10} \cdot \log re\right)\right)\]

Alternative 50
Error	46.6
Cost	26176

\[\sqrt{0.5} \cdot \left(\frac{\sqrt{0.5}}{\log 10} \cdot \log \left(im \cdot im\right)\right)\]

Alternative 51
Error	46.1
Cost	26176

\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log im \cdot 2}}\]

Alternative 52
Error	47.0
Cost	26176

\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log re \cdot 2}}\]

Alternative 53
Error	46.6
Cost	26176

\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(im \cdot im\right)}}\]

Alternative 54
Error	46.8
Cost	26176

\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re\right)}}\]

Alternative 55
Error	32.6
Cost	20032

\[\frac{0.3333333333333333}{\frac{\log 10}{3 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

Alternative 56
Error	32.6
Cost	20032

\[0.3333333333333333 \cdot \frac{3 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

Alternative 57
Error	32.5
Cost	19904

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

Alternative 58
Error	32.5
Cost	19776

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

Alternative 59
Error	32.5
Cost	13632

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

Alternative 60
Error	32.5
Cost	13632

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

Alternative 61
Error	32.5
Cost	13504

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

Alternative 62
Error	32.5
Cost	13504

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

Alternative 63
Error	46.3
Cost	13376

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

Alternative 64
Error	46.9
Cost	13376

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

Alternative 65
Error	46.8
Cost	13248

\[\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re\right)}}\]

Alternative 66
Error	47.0
Cost	13248

\[\frac{0.5}{\frac{\log 10}{\log re \cdot 2}}\]

Alternative 67
Error	46.6
Cost	13248

\[\frac{0.5}{\frac{\log 10}{\log \left(im \cdot im\right)}}\]

Alternative 68
Error	46.9
Cost	13056

\[\frac{\log \left(-im\right)}{\log 10}\]

Alternative 69
Error	46.3
Cost	13056

\[\frac{\log \left(-re\right)}{\log 10}\]

Alternative 70
Error	47.0
Cost	12992

\[\frac{\log re}{\log 10}\]

Alternative 71
Error	46.1
Cost	12992

\[\frac{\log im}{\log 10}\]

Alternative 72
Error	56.9
Cost	64

\[1\]

Alternative 73
Error	62.0
Cost	64

\[0\]

Alternative 74
Error	60.9
Cost	64

\[-1\]

Derivation

1. Split input into 7 regimes

2. if re < -6.0797067422693544e85

   1. Initial program 50.1

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

   3. Applied pow1/2_binary64_817 50.1

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

   4. Applied log-pow_binary64_826 50.1

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

   5. Applied associate-/l*_binary64_682 50.1

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

   7. Applied pow1_binary64_798 50.1

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

   8. Applied log-pow_binary64_826 50.1

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

   9. Applied pow1_binary64_798 50.1

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

   10. Applied log-pow_binary64_826 50.1

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

   11. Applied times-frac_binary64_743 50.1

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

   12. Applied add-sqr-sqrt_binary64_759 50.1

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

   13. Applied times-frac_binary64_743 50.0

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

   14. Simplified 50.0

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

   15. Taylor expanded around -inf 10.4

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

   16. Simplified 10.4

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

   if -6.0797067422693544e85 < re < -3.4668692285078929e-172

   1. Initial program 17.2

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

   3. Applied pow1/2_binary64_817 17.2

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

   4. Applied log-pow_binary64_826 17.2

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

   5. Applied associate-/l*_binary64_682 17.2

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

   7. Applied pow1_binary64_798 17.2

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

   8. Applied log-pow_binary64_826 17.2

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

   9. Applied pow1_binary64_798 17.2

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

   10. Applied log-pow_binary64_826 17.2

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

   11. Applied times-frac_binary64_743 17.2

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

   12. Applied add-sqr-sqrt_binary64_759 17.3

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

   13. Applied times-frac_binary64_743 17.1

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

   14. Simplified 17.1

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

   15. Simplified 17.1

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

   if -3.4668692285078929e-172 < re < -3.2147833495646259e-236

   1. Initial program 31.6

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

   2. Taylor expanded around 0 36.1

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

   3. Simplified 36.1

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

   if -3.2147833495646259e-236 < re < 2.0246090950680006e-222

   1. Initial program 31.6

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

   3. Applied pow1/2_binary64_817 31.6

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

   4. Applied log-pow_binary64_826 31.6

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

   5. Applied associate-/l*_binary64_682 31.6

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

   7. Applied pow1_binary64_798 31.6

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

   8. Applied log-pow_binary64_826 31.6

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

   9. Applied pow1_binary64_798 31.6

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

   10. Applied log-pow_binary64_826 31.6

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

   11. Applied times-frac_binary64_743 31.6

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

   12. Applied add-sqr-sqrt_binary64_759 31.7

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

   13. Applied times-frac_binary64_743 31.5

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

   14. Simplified 31.5

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

   15. Taylor expanded around -inf 33.5

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

   16. Simplified 33.5

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

   if 2.0246090950680006e-222 < re < 4.01731738869186625e-131

   1. Initial program 27.6

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

   3. Applied pow1/2_binary64_817 27.6

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

   4. Applied log-pow_binary64_826 27.6

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

   5. Applied associate-/l*_binary64_682 27.6

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

   7. Applied pow1_binary64_798 27.6

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

   8. Applied log-pow_binary64_826 27.6

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

   9. Applied pow1_binary64_798 27.6

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

   10. Applied log-pow_binary64_826 27.6

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

   11. Applied times-frac_binary64_743 27.6

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

   12. Applied add-sqr-sqrt_binary64_759 27.7

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

   13. Applied times-frac_binary64_743 27.5

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

   14. Simplified 27.5

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

   15. Taylor expanded around inf 35.2

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

   16. Simplified 35.2

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

   17. Simplified 35.2

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

   if 4.01731738869186625e-131 < re < 246.621463823296864

   1. Initial program 16.7

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

   3. Applied pow1/2_binary64_817 16.7

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

   4. Applied log-pow_binary64_826 16.7

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

   5. Applied associate-/l*_binary64_682 16.7

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

   7. Applied add-log-exp_binary64_776 16.7

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

   8. Simplified 16.7

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

   if 246.621463823296864 < re

   1. Initial program 41.6

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

   3. Applied pow1/2_binary64_817 41.6

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

   4. Applied log-pow_binary64_826 41.6

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

   5. Applied associate-/l*_binary64_682 41.6

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

   7. Applied pow1_binary64_798 41.6

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

   8. Applied log-pow_binary64_826 41.6

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

   9. Applied pow1_binary64_798 41.6

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

   10. Applied log-pow_binary64_826 41.6

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

   11. Applied times-frac_binary64_743 41.6

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

   12. Applied add-sqr-sqrt_binary64_759 41.7

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

   13. Applied times-frac_binary64_743 41.6

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

   14. Simplified 41.6

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

   15. Taylor expanded around inf 13.2

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

   16. Simplified 13.2

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

   17. Simplified 13.2

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

4. Final simplification 19.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.0797067422693544 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \leq -3.466869228507893 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \leq -3.214783349564626 \cdot 10^{-236}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \leq 2.0246090950680006 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{im}\right)}}\\ \mathbf{elif}\;re \leq 4.017317388691866 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log im \cdot 2}}\\ \mathbf{elif}\;re \leq 246.62146382329686:\\ \;\;\;\;\log \left(e^{\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{2 \cdot \log re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))