Average Error: 32.2 → 7.4
Time: 1.9s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.0441193087838437 \cdot 10^{+50}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -7.456020271396426 \cdot 10^{-148}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.0441193087838437 \cdot 10^{+50}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -7.456020271396426 \cdot 10^{-148}:\\
\;\;\;\;\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 (<= re -1.0441193087838437e+50)
   (log (- re))
   (if (<= re -7.456020271396426e-148)
     (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 (re <= -1.0441193087838437e+50) {
		tmp = log(-re);
	} else if (re <= -7.456020271396426e-148) {
		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 re < -1.0441193087838437e50

    1. Initial program 44.7

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

    2. Taylor expanded around -inf 6.9

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

    if -1.0441193087838437e50 < re < -7.4560202713964259e-148

    1. Initial program 11.4

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

    if -7.4560202713964259e-148 < re

    1. Initial program 32.1

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

    2. Taylor expanded around 0 5.5

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.0441193087838437 \cdot 10^{+50}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -7.456020271396426 \cdot 10^{-148}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array}\]

Reproduce

herbie shell --seed 2021075 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))