Average Error: 31.6 → 17.5
Time: 4.3s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.810553704526183 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log \left(\frac{-1}{re}\right) \cdot -2}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \leq -6.923684505821139 \cdot 10^{-180}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \leq 6.406642900144384 \cdot 10^{-297}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(2 \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \leq 2.7773647014065815 \cdot 10^{+88}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\left(2 \cdot \log re\right) \cdot \left(\sqrt{0.5} \cdot {\left(\frac{1}{{\log 10}^{3}}\right)}^{0.25}\right)\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.810553704526183 \cdot 10^{+133}:\\
\;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log \left(\frac{-1}{re}\right) \cdot -2}{\sqrt{\log 10}}\right)\\

\mathbf{elif}\;re \leq -6.923684505821139 \cdot 10^{-180}:\\
\;\;\;\;\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 6.406642900144384 \cdot 10^{-297}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(2 \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\

\mathbf{elif}\;re \leq 2.7773647014065815 \cdot 10^{+88}:\\
\;\;\;\;\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{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\left(2 \cdot \log re\right) \cdot \left(\sqrt{0.5} \cdot {\left(\frac{1}{{\log 10}^{3}}\right)}^{0.25}\right)\right)\\

\end{array}
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -1.810553704526183e+133)
   (*
    (sqrt (/ 0.5 (sqrt (log 10.0))))
    (*
     (sqrt (/ 0.5 (sqrt (log 10.0))))
     (/ (* (log (/ -1.0 re)) -2.0) (sqrt (log 10.0)))))
   (if (<= re -6.923684505821139e-180)
     (*
      (/ 0.5 (sqrt (log 10.0)))
      (log (pow (+ (* re re) (* im im)) (/ 1.0 (sqrt (log 10.0))))))
     (if (<= re 6.406642900144384e-297)
       (*
        (/ 0.5 (sqrt (log 10.0)))
        (* 2.0 (* (log im) (sqrt (/ 1.0 (log 10.0))))))
       (if (<= re 2.7773647014065815e+88)
         (*
          (/ 0.5 (sqrt (log 10.0)))
          (log (pow (+ (* re re) (* im im)) (/ 1.0 (sqrt (log 10.0))))))
         (*
          (sqrt (/ 0.5 (sqrt (log 10.0))))
          (*
           (* 2.0 (log re))
           (* (sqrt 0.5) (pow (/ 1.0 (pow (log 10.0) 3.0)) 0.25)))))))))
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 <= -1.810553704526183e+133) {
		tmp = sqrt(0.5 / sqrt(log(10.0))) * (sqrt(0.5 / sqrt(log(10.0))) * ((log(-1.0 / re) * -2.0) / sqrt(log(10.0))));
	} else if (re <= -6.923684505821139e-180) {
		tmp = (0.5 / sqrt(log(10.0))) * log(pow(((re * re) + (im * im)), (1.0 / sqrt(log(10.0)))));
	} else if (re <= 6.406642900144384e-297) {
		tmp = (0.5 / sqrt(log(10.0))) * (2.0 * (log(im) * sqrt(1.0 / log(10.0))));
	} else if (re <= 2.7773647014065815e+88) {
		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(log(10.0))) * ((2.0 * log(re)) * (sqrt(0.5) * pow((1.0 / pow(log(10.0), 3.0)), 0.25)));
	}
	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

Derivation

  1. Split input into 4 regimes

  2. if re < -1.8105537045261831e133

    1. Initial program 58.1

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

    3. Applied add-sqr-sqrt_binary6458.1

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

    4. Applied pow1/2_binary6458.1

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

    5. Applied log-pow_binary6458.1

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

    6. Applied times-frac_binary6458.1

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

    8. Applied add-sqr-sqrt_binary6458.1

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

    9. Applied associate-*l*_binary6458.1

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

    10. Simplified58.1

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

    11. Taylor expanded around -inf 7.6

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

    12. Simplified7.6

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

    if -1.8105537045261831e133 < re < -6.9236845058211389e-180 or 6.40664290014438439e-297 < re < 2.7773647014065815e88

    1. Initial program 20.0

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

    3. Applied add-sqr-sqrt_binary6420.0

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

    4. Applied pow1/2_binary6420.0

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

    5. Applied log-pow_binary6420.0

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

    6. Applied times-frac_binary6419.9

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

    8. Applied add-log-exp_binary6419.9

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

    9. Simplified19.7

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

    if -6.9236845058211389e-180 < re < 6.40664290014438439e-297

    1. Initial program 30.4

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

    3. Applied add-sqr-sqrt_binary6430.4

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

    4. Applied pow1/2_binary6430.4

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

    5. Applied log-pow_binary6430.4

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

    6. Applied times-frac_binary6430.4

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

    7. Taylor expanded around 0 33.7

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

    if 2.7773647014065815e88 < re

    1. Initial program 48.0

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

    3. Applied add-sqr-sqrt_binary6448.0

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

    4. Applied pow1/2_binary6448.0

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

    5. Applied log-pow_binary6448.0

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

    6. Applied times-frac_binary6448.0

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

    8. Applied add-sqr-sqrt_binary6448.0

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

    9. Applied associate-*l*_binary6448.0

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

    10. Simplified48.0

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

    11. Taylor expanded around inf 8.6

      \[\leadsto \sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \color{blue}{\left(-2 \cdot \left(\left(\log \left(\frac{1}{re}\right) \cdot \sqrt{0.5}\right) \cdot {\left(\frac{1}{{\log 10}^{3}}\right)}^{0.25}\right)\right)}\]

    12. Simplified8.5

      \[\leadsto \sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \color{blue}{\left(\left(2 \cdot \log re\right) \cdot \left(\sqrt{0.5} \cdot {\left(\frac{1}{{\log 10}^{3}}\right)}^{0.25}\right)\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.810553704526183 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log \left(\frac{-1}{re}\right) \cdot -2}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \leq -6.923684505821139 \cdot 10^{-180}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \leq 6.406642900144384 \cdot 10^{-297}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(2 \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \leq 2.7773647014065815 \cdot 10^{+88}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\left(2 \cdot \log re\right) \cdot \left(\sqrt{0.5} \cdot {\left(\frac{1}{{\log 10}^{3}}\right)}^{0.25}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))