Average Error: 31.9 → 18.7
Time: 2.4s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.2161323914063033 \cdot 10^{+76}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.0550493251927753 \cdot 10^{-114}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq -1.2654020355422494 \cdot 10^{-223}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 6.515608694411437 \cdot 10^{+115}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.2161323914063033 \cdot 10^{+76}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -1.0550493251927753 \cdot 10^{-114}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq -1.2654020355422494 \cdot 10^{-223}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 6.515608694411437 \cdot 10^{+115}:\\
\;\;\;\;\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 -1.2161323914063033e+76)
   (log (- re))
   (if (<= re -1.0550493251927753e-114)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re -1.2654020355422494e-223)
       (log im)
       (if (<= re 6.515608694411437e+115)
         (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 <= -1.2161323914063033e+76) {
		tmp = log(-re);
	} else if (re <= -1.0550493251927753e-114) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else if (re <= -1.2654020355422494e-223) {
		tmp = log(im);
	} else if (re <= 6.515608694411437e+115) {
		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 < -1.21614e76

    1. Initial program 47.0

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

    2. Taylor expanded around -inf 9.3

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

    3. Simplified9.3

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

    if -1.21614e76 < re < -1.05506e-114 or -1.26541e-223 < re < 6.51562e115

    1. Initial program 21.5

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

    if -1.05506e-114 < re < -1.26541e-223

    1. Initial program 27.7

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

    2. Taylor expanded around 0 39.3

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

    if 6.51562e115 < re

    1. Initial program 54.5

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

    2. Taylor expanded around inf 7.4

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

  4. Final simplification18.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.2161323914063033 \cdot 10^{+76}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.0550493251927753 \cdot 10^{-114}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq -1.2654020355422494 \cdot 10^{-223}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 6.515608694411437 \cdot 10^{+115}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))