Average Error: 32.4 → 17.8
Time: 1.5s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -3.1151566707846637 \cdot 10^{+144}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -5.3392694545660306 \cdot 10^{-201}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.96179636989168 \cdot 10^{-232}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 1.9120082663705915 \cdot 10^{+130}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -3.1151566707846637 \cdot 10^{+144}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -5.3392694545660306 \cdot 10^{-201}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq 2.96179636989168 \cdot 10^{-232}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 1.9120082663705915 \cdot 10^{+130}:\\
\;\;\;\;\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 -3.1151566707846637e+144)
   (log (- re))
   (if (<= re -5.3392694545660306e-201)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re 2.96179636989168e-232)
       (log im)
       (if (<= re 1.9120082663705915e+130)
         (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 <= -3.1151566707846637e+144) {
		tmp = log(-re);
	} else if (re <= -5.3392694545660306e-201) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else if (re <= 2.96179636989168e-232) {
		tmp = log(im);
	} else if (re <= 1.9120082663705915e+130) {
		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 < -3.115156670784664e144

    1. Initial program 61.8

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

    2. Taylor expanded around -inf 7.0

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

    3. Simplified7.0

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

    if -3.115156670784664e144 < re < -5.3392694545660306e-201 or 2.9617963698916799e-232 < re < 1.91200826637059146e130

    1. Initial program 18.7

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

    if -5.3392694545660306e-201 < re < 2.9617963698916799e-232

    1. Initial program 32.5

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

    2. Taylor expanded around 0 33.1

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

    if 1.91200826637059146e130 < re

    1. Initial program 58.3

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

    2. Taylor expanded around inf 8.6

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1151566707846637 \cdot 10^{+144}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -5.3392694545660306 \cdot 10^{-201}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.96179636989168 \cdot 10^{-232}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 1.9120082663705915 \cdot 10^{+130}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))