Average Error: 30.8 → 6.8
Time: 1.1s
Precision: binary64
[re im]: =sort([re im])
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq 5.693914703104481 \cdot 10^{-171}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;im \leq 6.435392623137265 \cdot 10^{+84}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;im \leq 5.693914703104481 \cdot 10^{-171}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;im \leq 6.435392623137265 \cdot 10^{+84}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= im 5.693914703104481e-171)
   (log (- re))
   (if (<= im 6.435392623137265e+84)
     (log (sqrt (+ (* re re) (* im im))))
     (log im))))
double code(double re, double im) {
	return log(sqrt((re * re) + (im * im)));
}
↓
double code(double re, double im) {
	double tmp;
	if (im <= 5.693914703104481e-171) {
		tmp = log(-re);
	} else if (im <= 6.435392623137265e+84) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else {
		tmp = log(im);
	}
	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 im < 5.69391470310448116e-171

    1. Initial program 31.6

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

    2. Taylor expanded around -inf 3.8

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

    3. Simplified3.8

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

    if 5.69391470310448116e-171 < im < 6.4353926231372653e84

    1. Initial program 12.2

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

    if 6.4353926231372653e84 < im

    1. Initial program 47.8

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

    2. Taylor expanded around 0 5.3

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.693914703104481 \cdot 10^{-171}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;im \leq 6.435392623137265 \cdot 10^{+84}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array}\]

Reproduce

herbie shell --seed 2020358 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))