Average Error: 32.1 → 18.7
Time: 6.1s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -2.6955531565480706 \cdot 10^{+118}:\\ \;\;\;\;\log \left(-re\right) \cdot \frac{1}{\log base}\\ \mathbf{elif}\;re \leq -1.801774026465648 \cdot 10^{-285}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{elif}\;re \leq 3.5257614717372405 \cdot 10^{-306}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \leq 3447503664711254:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{elif}\;re \leq 5.687874650979834 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log im\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -2.6955531565480706 \cdot 10^{+118}:\\
\;\;\;\;\log \left(-re\right) \cdot \frac{1}{\log base}\\

\mathbf{elif}\;re \leq -1.801774026465648 \cdot 10^{-285}:\\
\;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\

\mathbf{elif}\;re \leq 3.5257614717372405 \cdot 10^{-306}:\\
\;\;\;\;\frac{\log im}{\log base}\\

\mathbf{elif}\;re \leq 3447503664711254:\\
\;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\

\mathbf{elif}\;re \leq 5.687874650979834 \cdot 10^{+56}:\\
\;\;\;\;\frac{1}{\log base} \cdot \log im\\

\mathbf{else}:\\
\;\;\;\;\frac{\log re}{\log base}\\

\end{array}
(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))
↓
(FPCore (re im base)
 :precision binary64
 (if (<= re -2.6955531565480706e+118)
   (* (log (- re)) (/ 1.0 (log base)))
   (if (<= re -1.801774026465648e-285)
     (/ (log (sqrt (+ (* re re) (* im im)))) (log base))
     (if (<= re 3.5257614717372405e-306)
       (/ (log im) (log base))
       (if (<= re 3447503664711254.0)
         (/ (log (sqrt (+ (* re re) (* im im)))) (log base))
         (if (<= re 5.687874650979834e+56)
           (* (/ 1.0 (log base)) (log im))
           (/ (log re) (log base))))))))
double code(double re, double im, double base) {
	return ((log(sqrt((re * re) + (im * im))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}
↓
double code(double re, double im, double base) {
	double tmp;
	if (re <= -2.6955531565480706e+118) {
		tmp = log(-re) * (1.0 / log(base));
	} else if (re <= -1.801774026465648e-285) {
		tmp = log(sqrt((re * re) + (im * im))) / log(base);
	} else if (re <= 3.5257614717372405e-306) {
		tmp = log(im) / log(base);
	} else if (re <= 3447503664711254.0) {
		tmp = log(sqrt((re * re) + (im * im))) / log(base);
	} else if (re <= 5.687874650979834e+56) {
		tmp = (1.0 / log(base)) * log(im);
	} else {
		tmp = log(re) / log(base);
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if re < -2.6955531565480706e118

    1. Initial program 55.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified55.5

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

    4. Applied div-inv_binary64_42055.5

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

    5. Taylor expanded around -inf 8.7

      \[\leadsto \log \color{blue}{\left(-1 \cdot re\right)} \cdot \frac{1}{\log base}\]

    6. Simplified8.7

      \[\leadsto \log \color{blue}{\left(-re\right)} \cdot \frac{1}{\log base}\]

    if -2.6955531565480706e118 < re < -1.801774026465648e-285 or 3.52576147173724054e-306 < re < 3447503664711254

    1. Initial program 21.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified21.6

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

    if -1.801774026465648e-285 < re < 3.52576147173724054e-306

    1. Initial program 36.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified36.6

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

    3. Taylor expanded around 0 32.8

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

    if 3447503664711254 < re < 5.68787465097983412e56

    1. Initial program 18.2

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified18.1

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

    4. Applied div-inv_binary64_42018.1

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

    5. Taylor expanded around 0 47.8

      \[\leadsto \log \color{blue}{im} \cdot \frac{1}{\log base}\]

    if 5.68787465097983412e56 < re

    1. Initial program 46.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified46.3

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

    3. Taylor expanded around inf 11.3

      \[\leadsto \frac{\log \color{blue}{re}}{\log base}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification18.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.6955531565480706 \cdot 10^{+118}:\\ \;\;\;\;\log \left(-re\right) \cdot \frac{1}{\log base}\\ \mathbf{elif}\;re \leq -1.801774026465648 \cdot 10^{-285}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{elif}\;re \leq 3.5257614717372405 \cdot 10^{-306}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \leq 3447503664711254:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{elif}\;re \leq 5.687874650979834 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log im\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2020280 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))