Average Error: 32.3 → 6.6
Time: 2.3s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -9.003746470323266 \cdot 10^{+103}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -2.7017925900492935 \cdot 10^{-149}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\right)\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -9.003746470323266 \cdot 10^{+103}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -2.7017925900492935 \cdot 10^{-149}:\\
\;\;\;\;\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 (<= re -9.003746470323266e+103)
   (log (- re))
   (if (<= re -2.7017925900492935e-149)
     (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 (re <= -9.003746470323266e+103) {
		tmp = log(-re);
	} else if (re <= -2.7017925900492935e-149) {
		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 re < -9.00374647032326561e103

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 4.4

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

    3. Simplified4.4

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

    if -9.00374647032326561e103 < re < -2.70179259004929345e-149

    1. Initial program 10.8

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

    if -2.70179259004929345e-149 < re

    1. Initial program 32.7

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

    2. Taylor expanded around 0 8.0

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

    3. Simplified8.0

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

    5. Applied *-un-lft-identity_binary648.0

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

    6. Applied times-frac_binary645.1

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

    7. Simplified5.1

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.003746470323266 \cdot 10^{+103}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -2.7017925900492935 \cdot 10^{-149}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(im + 0.5 \cdot \left(re \cdot \frac{re}{im}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))