Average Error: 32.1 → 9.6
Time: 13.3s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -5.60563964591766 \cdot 10^{+43}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \leq -1.3795563888914072 \cdot 10^{-05}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \leq -5.867956787373959 \cdot 10^{-35}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}\right) \cdot \frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log base}\\ \mathbf{elif}\;re \leq -3.3543382104116615 \cdot 10^{-69} \lor \neg \left(re \leq -2.1704890792695045 \cdot 10^{-147}\right):\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \end{array}\]
\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 -5.60563964591766 \cdot 10^{+43}:\\
\;\;\;\;\frac{\log \left(-re\right)}{\log base}\\

\mathbf{elif}\;re \leq -1.3795563888914072 \cdot 10^{-05}:\\
\;\;\;\;\frac{\log im}{\log base}\\

\mathbf{elif}\;re \leq -5.867956787373959 \cdot 10^{-35}:\\
\;\;\;\;\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 base}\\

\mathbf{elif}\;re \leq -3.3543382104116615 \cdot 10^{-69} \lor \neg \left(re \leq -2.1704890792695045 \cdot 10^{-147}\right):\\
\;\;\;\;\frac{\log im}{\log base}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\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 -5.60563964591766e+43)
   (/ (log (- re)) (log base))
   (if (<= re -1.3795563888914072e-05)
     (/ (log im) (log base))
     (if (<= re -5.867956787373959e-35)
       (*
        (*
         (cbrt (log (sqrt (+ (* re re) (* im im)))))
         (cbrt (log (sqrt (+ (* re re) (* im im))))))
        (/ (cbrt (log (sqrt (+ (* re re) (* im im))))) (log base)))
       (if (or (<= re -3.3543382104116615e-69)
               (not (<= re -2.1704890792695045e-147)))
         (/ (log im) (log base))
         (/ (log (sqrt (+ (* re re) (* im im)))) (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 <= -5.60563964591766e+43) {
		tmp = log(-re) / log(base);
	} else if (re <= -1.3795563888914072e-05) {
		tmp = log(im) / log(base);
	} else if (re <= -5.867956787373959e-35) {
		tmp = (cbrt(log(sqrt((re * re) + (im * im)))) * cbrt(log(sqrt((re * re) + (im * im))))) * (cbrt(log(sqrt((re * re) + (im * im)))) / log(base));
	} else if ((re <= -3.3543382104116615e-69) || !(re <= -2.1704890792695045e-147)) {
		tmp = log(im) / log(base);
	} else {
		tmp = log(sqrt((re * re) + (im * im))) / 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 4 regimes

  2. if re < -5.60563964591766037e43

    1. Initial program 43.2

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

    2. Simplified43.2

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

    3. Taylor expanded around -inf 6.4

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

    4. Simplified6.4

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

    if -5.60563964591766037e43 < re < -1.37955638889140722e-5 or -5.86795678737395936e-35 < re < -3.35433821041166152e-69 or -2.1704890792695045e-147 < re

    1. Initial program 28.4

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

    2. Simplified28.3

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

    3. Taylor expanded around 0 11.6

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

    if -1.37955638889140722e-5 < re < -5.86795678737395936e-35

    1. Initial program 12.6

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

    2. Simplified12.5

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

    4. Applied pow1_binary64_48012.5

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

    5. Applied log-pow_binary64_50812.5

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

    6. Applied add-cube-cbrt_binary64_45413.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}{1 \cdot \log base}\]

    7. Applied times-frac_binary64_42513.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{1} \cdot \frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log base}}\]

    if -3.35433821041166152e-69 < re < -2.1704890792695045e-147

    1. Initial program 10.4

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

    2. Simplified10.3

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.60563964591766 \cdot 10^{+43}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \leq -1.3795563888914072 \cdot 10^{-05}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \leq -5.867956787373959 \cdot 10^{-35}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}\right) \cdot \frac{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log base}\\ \mathbf{elif}\;re \leq -3.3543382104116615 \cdot 10^{-69} \lor \neg \left(re \leq -2.1704890792695045 \cdot 10^{-147}\right):\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2021093 
(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))))