Average Error: 32.1 → 17.3
Time: 1.3s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.1762247444222311 \cdot 10^{+151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 1.3781415203379638 \cdot 10^{-202} \lor \neg \left(re \leq 1.1082609750789312 \cdot 10^{-166}\right) \land re \leq 1.4551030635736294 \cdot 10^{+82}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.1762247444222311 \cdot 10^{+151}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq 1.3781415203379638 \cdot 10^{-202} \lor \neg \left(re \leq 1.1082609750789312 \cdot 10^{-166}\right) \land re \leq 1.4551030635736294 \cdot 10^{+82}:\\
\;\;\;\;\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 -1.1762247444222311e+151)
   (log (- re))
   (if (or (<= re 1.3781415203379638e-202)
           (and (not (<= re 1.1082609750789312e-166))
                (<= re 1.4551030635736294e+82)))
     (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 <= -1.1762247444222311e+151) {
		tmp = log(-re);
	} else if ((re <= 1.3781415203379638e-202) || (!(re <= 1.1082609750789312e-166) && (re <= 1.4551030635736294e+82))) {
		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 3 regimes

  2. if re < -1.17622474442223108e151

    1. Initial program 63.2

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

    2. Taylor expanded around -inf 6.6

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

    3. Simplified6.6

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

    if -1.17622474442223108e151 < re < 1.37814152033796385e-202 or 1.1082609750789312e-166 < re < 1.4551030635736294e82

    1. Initial program 20.4

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

    if 1.37814152033796385e-202 < re < 1.1082609750789312e-166 or 1.4551030635736294e82 < re

    1. Initial program 47.0

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

    2. Taylor expanded around inf 14.8

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1762247444222311 \cdot 10^{+151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 1.3781415203379638 \cdot 10^{-202} \lor \neg \left(re \leq 1.1082609750789312 \cdot 10^{-166}\right) \land re \leq 1.4551030635736294 \cdot 10^{+82}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))