Average Error: 31.5 → 16.8
Time: 1.1s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.0607390914767552 \cdot 10^{+119}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 8.169445472099669 \cdot 10^{+99}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.0607390914767552 \cdot 10^{+119}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq 8.169445472099669 \cdot 10^{+99}:\\
\;\;\;\;\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 -1.0607390914767552e+119)
   (log (- re))
   (if (<= re 8.169445472099669e+99)
     (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 <= -1.0607390914767552e+119) {
		tmp = log(-re);
	} else if (re <= 8.169445472099669e+99) {
		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 < -1.0607390914767552e119

    1. Initial program 55.3

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

    2. Taylor expanded around -inf 7.1

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

    3. Simplified7.1

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

    if -1.0607390914767552e119 < re < 8.16944547209966887e99

    1. Initial program 21.2

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

    if 8.16944547209966887e99 < re

    1. Initial program 50.5

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

    2. Taylor expanded around inf 8.0

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.0607390914767552 \cdot 10^{+119}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 8.169445472099669 \cdot 10^{+99}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))