Average Error: 31.3 → 6.4
Time: 1.5s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -3.58923264790877 \cdot 10^{+85}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -2.5396447653359938 \cdot 10^{-165}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -3.58923264790877 \cdot 10^{+85}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -2.5396447653359938 \cdot 10^{-165}:\\
\;\;\;\;\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 -3.58923264790877e+85)
   (log (- re))
   (if (<= re -2.5396447653359938e-165)
     (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 <= -3.58923264790877e+85) {
		tmp = log(-re);
	} else if (re <= -2.5396447653359938e-165) {
		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 < -3.5892326479087698e85

    1. Initial program 47.8

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

    2. Taylor expanded around -inf 4.8

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

    3. Simplified4.8

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

    if -3.5892326479087698e85 < re < -2.53964476533599376e-165

    1. Initial program 11.1

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

    if -2.53964476533599376e-165 < re

    1. Initial program 32.4

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

    2. Taylor expanded around 0 4.3

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.58923264790877 \cdot 10^{+85}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -2.5396447653359938 \cdot 10^{-165}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))