Average Error: 31.5 → 17.5
Time: 1.2s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -3.8588398939432175 \cdot 10^{+132}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -4.2360820877371696 \cdot 10^{-235}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.1577752949736062 \cdot 10^{-187}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 1.2225237113435466 \cdot 10^{+30}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -3.8588398939432175 \cdot 10^{+132}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -4.2360820877371696 \cdot 10^{-235}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq 2.1577752949736062 \cdot 10^{-187}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 1.2225237113435466 \cdot 10^{+30}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -3.8588398939432175e+132)
   (log (- re))
   (if (<= re -4.2360820877371696e-235)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re 2.1577752949736062e-187)
       (log im)
       (if (<= re 1.2225237113435466e+30)
         (log (sqrt (+ (* re re) (* im im))))
         (log re))))))
double code(double re, double im) {
	return log(sqrt((re * re) + (im * im)));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -3.8588398939432175e+132) {
		tmp = log(-re);
	} else if (re <= -4.2360820877371696e-235) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else if (re <= 2.1577752949736062e-187) {
		tmp = log(im);
	} else if (re <= 1.2225237113435466e+30) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else {
		tmp = log(re);
	}
	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 < -3.8588398939432175e132

    1. Initial program 58.0

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

    2. Taylor expanded around -inf 7.4

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

    3. Simplified7.4

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

    if -3.8588398939432175e132 < re < -4.23608208773716957e-235 or 2.15777529497360624e-187 < re < 1.2225237113435466e30

    1. Initial program 18.2

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

    if -4.23608208773716957e-235 < re < 2.15777529497360624e-187

    1. Initial program 32.8

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

    2. Taylor expanded around 0 32.5

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

    if 1.2225237113435466e30 < re

    1. Initial program 42.7

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

    2. Taylor expanded around inf 11.0

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.8588398939432175 \cdot 10^{+132}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -4.2360820877371696 \cdot 10^{-235}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.1577752949736062 \cdot 10^{-187}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 1.2225237113435466 \cdot 10^{+30}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))