Average Error: 31.9 → 17.7
Time: 21.0s
Precision: binary64
Cost: 40580
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq -7.061285158928872 \cdot 10^{+113}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{im}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;im \leq -5.535169434540731 \cdot 10^{-301}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;im \leq 3.643016007571221 \cdot 10^{-180}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right)\right)\\ \mathbf{elif}\;im \leq 5.179540222444118 \cdot 10^{+90}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;im \leq -7.061285158928872 \cdot 10^{+113}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{im}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\

\mathbf{elif}\;im \leq -5.535169434540731 \cdot 10^{-301}:\\
\;\;\;\;\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}\;im \leq 3.643016007571221 \cdot 10^{-180}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right)\right)\\

\mathbf{elif}\;im \leq 5.179540222444118 \cdot 10^{+90}:\\
\;\;\;\;\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}:\\
\;\;\;\;\frac{\log im}{\log 10}\\

\end{array}
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (if (<= im -7.061285158928872e+113)
   (*
    (/ 0.5 (sqrt (log 10.0)))
    (* -2.0 (* (log (/ -1.0 im)) (sqrt (/ 1.0 (log 10.0))))))
   (if (<= im -5.535169434540731e-301)
     (*
      (/ 0.5 (sqrt (log 10.0)))
      (log (pow (+ (* re re) (* im im)) (/ 1.0 (sqrt (log 10.0))))))
     (if (<= im 3.643016007571221e-180)
       (*
        (/ 0.5 (sqrt (log 10.0)))
        (* -2.0 (* (sqrt (/ 1.0 (log 10.0))) (log (/ -1.0 re)))))
       (if (<= im 5.179540222444118e+90)
         (*
          (/ 0.5 (sqrt (log 10.0)))
          (log (pow (+ (* re re) (* im im)) (/ 1.0 (sqrt (log 10.0))))))
         (/ (log im) (log 10.0)))))))
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 (im <= -7.061285158928872e+113) {
		tmp = (0.5 / sqrt(log(10.0))) * (-2.0 * (log(-1.0 / im) * sqrt(1.0 / log(10.0))));
	} else if (im <= -5.535169434540731e-301) {
		tmp = (0.5 / sqrt(log(10.0))) * log(pow(((re * re) + (im * im)), (1.0 / sqrt(log(10.0)))));
	} else if (im <= 3.643016007571221e-180) {
		tmp = (0.5 / sqrt(log(10.0))) * (-2.0 * (sqrt(1.0 / log(10.0)) * log(-1.0 / re)));
	} else if (im <= 5.179540222444118e+90) {
		tmp = (0.5 / sqrt(log(10.0))) * log(pow(((re * re) + (im * im)), (1.0 / sqrt(log(10.0)))));
	} else {
		tmp = log(im) / log(10.0);
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error32.2
Cost97856
\[\frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt[3]{\log 10}}\]
Alternative 2
Error32.2
Cost84928
\[\frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10}}\]
Alternative 3
Error32.2
Cost78656
\[\sqrt[3]{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \cdot \left(\sqrt[3]{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \cdot \sqrt[3]{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}}\right)\]
Alternative 4
Error53.3
Cost78528
\[\frac{0.5}{\sqrt{\log 10}} \cdot \left(\frac{\log \left({re}^{6} + {im}^{6}\right)}{\sqrt{\log 10}} - \frac{\log \left({re}^{4} + \left({im}^{4} - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)}{\sqrt{\log 10}}\right)\]
Alternative 5
Error32.2
Cost65856
\[\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{\log \left(re \cdot re + im \cdot im\right)} \cdot \left(\sqrt[3]{\log \left(re \cdot re + im \cdot im\right)} \cdot \sqrt[3]{\log \left(re \cdot re + im \cdot im\right)}\right)}{\sqrt{\log 10}}\]
Alternative 6
Error53.3
Cost65600
\[\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left({re}^{6} + {im}^{6}\right) - \log \left({re}^{4} + \left({im}^{4} - \left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)}{\sqrt{\log 10}}\]
Alternative 7
Error32.2
Cost65600
\[\left(\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}\right) \cdot \frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10}\]
Alternative 8
Error44.2
Cost65472
\[\frac{0.5}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}} \cdot \sqrt{\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\right)\]
Alternative 9
Error32.2
Cost59840
\[\sqrt[3]{0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}} \cdot \left(\sqrt[3]{0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}} \cdot \sqrt[3]{0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}}\right)\]
Alternative 10
Error31.8
Cost58688
\[\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\]
Alternative 11
Error31.8
Cost52800
\[\frac{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\log 10}\]
Alternative 12
Error31.8
Cost52672
\[\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) + 2 \cdot \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\]
Alternative 13
Error44.2
Cost52544
\[\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\sqrt{\log \left(re \cdot re + im \cdot im\right)} \cdot \sqrt{\log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10}}\]
Alternative 14
Error44.2
Cost52416
\[\sqrt{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \cdot \sqrt{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}}\]
Alternative 15
Error31.8
Cost46272
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right) + 2 \cdot \log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}\]
Alternative 16
Error44.2
Cost45888
\[\frac{\sqrt{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\frac{\log 10}{\sqrt{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
Alternative 17
Error44.2
Cost45888
\[\sqrt{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \frac{\sqrt{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10}\]
Alternative 18
Error32.2
Cost45760
\[\frac{0.5}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt[3]{\log 10}}\]
Alternative 19
Error31.9
Cost45696
\[\frac{0.5}{\sqrt{\log 10}} \cdot \sqrt[3]{{\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)}^{3}}\]
Alternative 20
Error31.9
Cost45696
\[\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{{\log \left(re \cdot re + im \cdot im\right)}^{3}}}{\sqrt{\log 10}}\]
Alternative 21
Error44.4
Cost45632
\[\frac{0.5}{\sqrt{\log 10}} \cdot e^{\log \left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)}\]
Alternative 22
Error44.4
Cost45632
\[\frac{0.5}{\sqrt{\log 10}} \cdot \frac{e^{\log \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10}}\]
Alternative 23
Error44.2
Cost39872
\[\sqrt{0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}} \cdot \sqrt{0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}}\]
Alternative 24
Error31.9
Cost39296
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\sqrt[3]{{\log \left(re \cdot re + im \cdot im\right)}^{3}}}}\]
Alternative 25
Error31.7
Cost39296
\[\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]
Alternative 26
Error44.3
Cost39232
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{e^{\log \left(\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}\right)}}\]
Alternative 27
Error51.2
Cost39168
\[\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{im}\right)}^{\left(-2 \cdot \sqrt{\frac{1}{\log 10}}\right)}\right)\]
Alternative 28
Error50.7
Cost39168
\[\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{1}{im}\right)}^{\left(-2 \cdot \sqrt{\frac{1}{\log 10}}\right)}\right)\]
Alternative 29
Error50.5
Cost39168
\[\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-2 \cdot \sqrt{\frac{1}{\log 10}}\right)}\right)\]
Alternative 30
Error50.6
Cost39168
\[\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{1}{re}\right)}^{\left(-2 \cdot \sqrt{\frac{1}{\log 10}}\right)}\right)\]
Alternative 31
Error46.1
Cost39040
\[\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)\]
Alternative 32
Error32.0
Cost32960
\[\left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right) \cdot \frac{\sqrt[3]{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\]
Alternative 33
Error31.7
Cost32960
\[\frac{0.5}{\sqrt{\log 10}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\]
Alternative 34
Error31.8
Cost32960
\[\frac{0.5}{\sqrt{\log 10}} \cdot \frac{1}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]
Alternative 35
Error47.2
Cost32832
\[\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{im}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\]
Alternative 36
Error46.3
Cost32832
\[\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right)\right)\]
Alternative 37
Error31.8
Cost32832
\[\frac{\log \left(re \cdot re + im \cdot im\right) \cdot \frac{0.5}{\sqrt{\log 10}}}{\sqrt{\log 10}}\]
Alternative 38
Error31.8
Cost32832
\[\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]
Alternative 39
Error47.2
Cost32768
\[\sqrt{\frac{1}{\log 10}} \cdot \left(-\log \left(\frac{-1}{im}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\]
Alternative 40
Error46.1
Cost32704
\[\frac{0.5}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(re \cdot re\right)\right)\]
Alternative 41
Error47.2
Cost32704
\[\frac{0.5}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{im}\right)}{\sqrt{\log 10}}\]
Alternative 42
Error46.3
Cost32704
\[\frac{0.5}{\sqrt{\log 10}} \cdot \frac{-2 \cdot \log \left(\frac{-1}{re}\right)}{\sqrt{\log 10}}\]
Alternative 43
Error46.5
Cost32704
\[\frac{0.5}{\sqrt{\log 10}} \cdot \left(2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\right)\]
Alternative 44
Error31.9
Cost32640
\[\sqrt[3]{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}\]
Alternative 45
Error44.3
Cost32576
\[e^{\log \left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}\]
Alternative 46
Error62.7
Cost32576
\[\frac{\log \log \left(e^{\sqrt{re \cdot re + im \cdot im}}\right)}{\log 10}\]
Alternative 47
Error31.8
Cost26560
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{1}{\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}}}\]
Alternative 48
Error31.9
Cost26560
\[\frac{\sqrt{0.5}}{\log 10} \cdot \frac{\sqrt{0.5}}{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}\]
Alternative 49
Error31.8
Cost26432
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\]
Alternative 50
Error31.9
Cost26432
\[\sqrt{0.5} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\sqrt{0.5}}{\log 10}\right)\]
Alternative 51
Error31.9
Cost26368
\[\sqrt[3]{{\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}\right)}^{3} \cdot 0.125}\]
Alternative 52
Error31.9
Cost26368
\[\sqrt[3]{{\left(\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\right)}^{3}}\]
Alternative 53
Error46.3
Cost26304
\[\sqrt{0.5} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \frac{\sqrt{0.5}}{\log 10}\right)\right)\]
Alternative 54
Error44.3
Cost26304
\[e^{\log \left(0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}\right)}\]
Alternative 55
Error47.2
Cost26304
\[\sqrt{0.5} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{im}\right) \cdot \frac{\sqrt{0.5}}{\log 10}\right)\right)\]
Alternative 56
Error47.2
Cost26304
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{im}\right)}}\]
Alternative 57
Error46.3
Cost26304
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\]
Alternative 58
Error46.2
Cost26176
\[\sqrt{0.5} \cdot \left(\log \left(re \cdot re\right) \cdot \frac{\sqrt{0.5}}{\log 10}\right)\]
Alternative 59
Error46.3
Cost26176
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{2 \cdot \log im}}\]
Alternative 60
Error46.5
Cost26176
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{2 \cdot \log re}}\]
Alternative 61
Error46.7
Cost26176
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(im \cdot im\right)}}\]
Alternative 62
Error46.6
Cost26176
\[\sqrt{0.5} \cdot \left(2 \cdot \left(\frac{\sqrt{0.5}}{\log 10} \cdot \log re\right)\right)\]
Alternative 63
Error46.7
Cost26176
\[\sqrt{0.5} \cdot \left(\frac{\sqrt{0.5}}{\log 10} \cdot \log \left(im \cdot im\right)\right)\]
Alternative 64
Error46.3
Cost26176
\[\sqrt{0.5} \cdot \left(2 \cdot \left(\frac{\sqrt{0.5}}{\log 10} \cdot \log im\right)\right)\]
Alternative 65
Error46.1
Cost26176
\[\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re\right)}}\]
Alternative 66
Error48.7
Cost20480
\[\frac{\log \left(\left(re + 0.5 \cdot \frac{im \cdot im}{re}\right) - 0.125 \cdot \left(im \cdot {\left(\frac{im}{re}\right)}^{3}\right)\right)}{\log 10}\]
Alternative 67
Error48.5
Cost20480
\[\frac{\log \left(\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right) - 0.125 \cdot \left(re \cdot {\left(\frac{re}{im}\right)}^{3}\right)\right)}{\log 10}\]
Alternative 68
Error31.9
Cost19904
\[\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]
Alternative 69
Error32.0
Cost19840
\[\log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{0.5}{\log 10}\right)}\right)\]
Alternative 70
Error31.9
Cost19776
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
Alternative 71
Error31.9
Cost13504
\[\frac{0.5}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\]
Alternative 72
Error31.9
Cost13504
\[0.5 \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log 10}\]
Alternative 73
Error46.3
Cost13248
\[\frac{-1}{\frac{\log 10}{\log \left(\frac{-1}{re}\right)}}\]
Alternative 74
Error47.2
Cost13248
\[\frac{-1}{\frac{\log 10}{\log \left(\frac{-1}{im}\right)}}\]
Alternative 75
Error46.3
Cost13184
\[\frac{-1}{\frac{\log 10}{-\log im}}\]
Alternative 76
Error46.5
Cost13184
\[\frac{-1}{\frac{\log 10}{-\log re}}\]
Alternative 77
Error47.2
Cost13056
\[\frac{\log \left(-im\right)}{\log 10}\]
Alternative 78
Error46.3
Cost13056
\[\frac{\log \left(-re\right)}{\log 10}\]
Alternative 79
Error46.5
Cost12992
\[\frac{\log re}{\log 10}\]
Alternative 80
Error46.3
Cost12992
\[\frac{\log im}{\log 10}\]
Alternative 81
Error56.9
Cost64
\[1\]
Alternative 82
Error62.0
Cost64
\[0\]
Alternative 83
Error60.8
Cost64
\[-1\]

Error

Derivation

  1. Split input into 4 regimes

  2. if im < -7.0612851589288723e113

    1. Initial program 53.3

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

    3. Applied add-sqr-sqrt_binary64_10053.3

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

    4. Applied pow1/2_binary64_15853.3

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

    5. Applied log-pow_binary64_16753.3

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

    6. Applied times-frac_binary64_8453.3

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

    7. Taylor expanded around -inf 8.8

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

    8. Simplified8.8

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

    9. Simplified8.8

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

    if -7.0612851589288723e113 < im < -5.53516943454073139e-301 or 3.643016007571221e-180 < im < 5.1795402224441176e90

    1. Initial program 19.8

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

    3. Applied add-sqr-sqrt_binary64_10019.8

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

    4. Applied pow1/2_binary64_15819.8

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

    5. Applied log-pow_binary64_16719.8

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

    6. Applied times-frac_binary64_8419.8

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

    8. Applied add-log-exp_binary64_11719.8

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

    9. Simplified19.6

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

    10. Simplified19.6

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

    if -5.53516943454073139e-301 < im < 3.643016007571221e-180

    1. Initial program 32.7

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

    3. Applied add-sqr-sqrt_binary64_10032.7

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

    4. Applied pow1/2_binary64_15832.7

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

    5. Applied log-pow_binary64_16732.7

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

    6. Applied times-frac_binary64_8432.7

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

    7. Taylor expanded around -inf 33.8

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

    8. Simplified33.8

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

    if 5.1795402224441176e90 < im

    1. Initial program 50.4

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

    2. Taylor expanded around 0 9.4

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

    3. Simplified9.4

      \[\leadsto \color{blue}{\frac{\log im}{\log 10}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.061285158928872 \cdot 10^{+113}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{im}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;im \leq -5.535169434540731 \cdot 10^{-301}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;im \leq 3.643016007571221 \cdot 10^{-180}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right)\right)\\ \mathbf{elif}\;im \leq 5.179540222444118 \cdot 10^{+90}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))