Average Error: 31.3 → 17.3
Time: 1.0s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.810553704526183 \cdot 10^{+133}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -6.923684505821139 \cdot 10^{-180}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 6.146390503275677 \cdot 10^{-297}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 2.369953838316205 \cdot 10^{+88}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.810553704526183 \cdot 10^{+133}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -6.923684505821139 \cdot 10^{-180}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq 6.146390503275677 \cdot 10^{-297}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 2.369953838316205 \cdot 10^{+88}:\\
\;\;\;\;\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 -1.810553704526183e+133)
   (log (- re))
   (if (<= re -6.923684505821139e-180)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re 6.146390503275677e-297)
       (log im)
       (if (<= re 2.369953838316205e+88)
         (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 <= -1.810553704526183e+133) {
		tmp = log(-re);
	} else if (re <= -6.923684505821139e-180) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else if (re <= 6.146390503275677e-297) {
		tmp = log(im);
	} else if (re <= 2.369953838316205e+88) {
		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 4 regimes

  2. if re < -1.8105537045261831e133

    1. Initial program 58.1

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

    2. Taylor expanded around -inf 7.3

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

    3. Simplified7.3

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

    if -1.8105537045261831e133 < re < -6.9236845058211389e-180 or 6.14639050327567661e-297 < re < 2.36995383831620508e88

    1. Initial program 19.6

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

    if -6.9236845058211389e-180 < re < 6.14639050327567661e-297

    1. Initial program 30.1

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

    2. Taylor expanded around 0 33.6

      \[\leadsto \log \color{blue}{im}\]

    if 2.36995383831620508e88 < re

    1. Initial program 47.9

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

    2. Taylor expanded around inf 8.2

      \[\leadsto \log \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.810553704526183 \cdot 10^{+133}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -6.923684505821139 \cdot 10^{-180}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 6.146390503275677 \cdot 10^{-297}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 2.369953838316205 \cdot 10^{+88}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))