Average Error: 31.7 → 6.9
Time: 1.5s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq 2.198863241148998 \cdot 10^{-140}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;im \leq 8.764625990352157 \cdot 10^{+64}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;im \leq 2.198863241148998 \cdot 10^{-140}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;im \leq 8.764625990352157 \cdot 10^{+64}:\\
\;\;\;\;\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 2.198863241148998e-140)
   (log (- re))
   (if (<= im 8.764625990352157e+64)
     (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 <= 2.198863241148998e-140) {
		tmp = log(-re);
	} else if (im <= 8.764625990352157e+64) {
		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 < 2.19886324114899816e-140

    1. Initial program 32.1

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

    2. Taylor expanded around -inf 5.7

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

    3. Simplified5.7

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

    if 2.19886324114899816e-140 < im < 8.7646259903521571e64

    1. Initial program 10.7

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

    if 8.7646259903521571e64 < im

    1. Initial program 45.5

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

    2. Taylor expanded around 0 5.7

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.198863241148998 \cdot 10^{-140}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;im \leq 8.764625990352157 \cdot 10^{+64}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array}\]

Reproduce

herbie shell --seed 2021032 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))