Average Error: 31.8 → 17.7
Time: 1.2s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -6.842917597001959 \cdot 10^{+65}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 2.7721637282396777 \cdot 10^{+81}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -6.842917597001959 \cdot 10^{+65}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq 2.7721637282396777 \cdot 10^{+81}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{else}:\\
\;\;\;\;\log re\\

\end{array}
(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -6.842917597001959e+65)
   (log (- re))
   (if (<= re 2.7721637282396777e+81)
     (log (sqrt (+ (* re re) (* im im))))
     (log re))))
double code(double re, double im) {
	return log(sqrt((re * re) + (im * im)));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -6.842917597001959e+65) {
		tmp = log(-re);
	} else if (re <= 2.7721637282396777e+81) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else {
		tmp = log(re);
	}
	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 < -6.842917597001959e65

    1. Initial program 47.6

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

    2. Taylor expanded around -inf 11.0

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

    3. Simplified11.0

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

    if -6.842917597001959e65 < re < 2.77216372823967773e81

    1. Initial program 22.1

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

    if 2.77216372823967773e81 < re

    1. Initial program 47.7

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

    2. Taylor expanded around inf 9.8

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.842917597001959 \cdot 10^{+65}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 2.7721637282396777 \cdot 10^{+81}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020295 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))