Average Error: 31.6 → 6.9
Time: 2.4s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq 3.5598012421053094 \cdot 10^{-159}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;im \leq 3.0994021608132516 \cdot 10^{+110}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\right)\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;im \leq 3.5598012421053094 \cdot 10^{-159}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;im \leq 3.0994021608132516 \cdot 10^{+110}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\right)\\

\end{array}
(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= im 3.5598012421053094e-159)
   (log (- re))
   (if (<= im 3.0994021608132516e+110)
     (log (sqrt (+ (* re re) (* im im))))
     (log (+ im (* 0.5 (* re (/ re im))))))))
double code(double re, double im) {
	return log(sqrt((re * re) + (im * im)));
}
↓
double code(double re, double im) {
	double tmp;
	if (im <= 3.5598012421053094e-159) {
		tmp = log(-re);
	} else if (im <= 3.0994021608132516e+110) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else {
		tmp = log(im + (0.5 * (re * (re / 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 < 3.55980124210530944e-159

    1. Initial program 32.1

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

    2. Taylor expanded around -inf 4.5

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

    if 3.55980124210530944e-159 < im < 3.0994021608132516e110

    1. Initial program 11.5

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

    if 3.0994021608132516e110 < im

    1. Initial program 52.7

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

    2. Taylor expanded around 0 12.9

      \[\leadsto \log \color{blue}{\left(0.5 \cdot \frac{{re}^{2}}{im} + im\right)}\]

    3. Simplified12.9

      \[\leadsto \log \color{blue}{\left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity_binary6412.9

      \[\leadsto \log \left(im + 0.5 \cdot \frac{re \cdot re}{\color{blue}{1 \cdot im}}\right)\]

    6. Applied times-frac_binary645.4

      \[\leadsto \log \left(im + 0.5 \cdot \color{blue}{\left(\frac{re}{1} \cdot \frac{re}{im}\right)}\right)\]

    7. Simplified5.4

      \[\leadsto \log \left(im + 0.5 \cdot \left(\color{blue}{re} \cdot \frac{re}{im}\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.5598012421053094 \cdot 10^{-159}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;im \leq 3.0994021608132516 \cdot 10^{+110}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))