Average Error: 31.1 → 17.9
Time: 1.6s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.0802856481292018 \cdot 10^{+127}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -2.5199060035987696 \cdot 10^{-201}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.424827901233389 \cdot 10^{-276}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 4.161724496441469 \cdot 10^{+128}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.0802856481292018 \cdot 10^{+127}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -2.5199060035987696 \cdot 10^{-201}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq 2.424827901233389 \cdot 10^{-276}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 4.161724496441469 \cdot 10^{+128}:\\
\;\;\;\;\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 -1.0802856481292018e+127)
   (log (- re))
   (if (<= re -2.5199060035987696e-201)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re 2.424827901233389e-276)
       (log im)
       (if (<= re 4.161724496441469e+128)
         (log (sqrt (+ (* re re) (* im im))))
         (log re))))))
double code(double re, double im) {
	return ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
}
↓
double code(double re, double im) {
	double tmp;
	if ((re <= -1.0802856481292018e+127)) {
		tmp = ((double) log(((double) -(re))));
	} else {
		double tmp_1;
		if ((re <= -2.5199060035987696e-201)) {
			tmp_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double tmp_2;
			if ((re <= 2.424827901233389e-276)) {
				tmp_2 = ((double) log(im));
			} else {
				double tmp_3;
				if ((re <= 4.161724496441469e+128)) {
					tmp_3 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
				} else {
					tmp_3 = ((double) log(re));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	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 < -1.0802856481292018e127

    1. Initial program Error: 56.9 bits

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

    2. Taylor expanded around -inf Error: 7.3 bits

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

    3. SimplifiedError: 7.3 bits

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

    if -1.0802856481292018e127 < re < -2.51990600359876958e-201 or 2.4248279012333889e-276 < re < 4.1617244964414691e128

    1. Initial program Error: 19.3 bits

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

    if -2.51990600359876958e-201 < re < 2.4248279012333889e-276

    1. Initial program Error: 29.2 bits

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

    2. Taylor expanded around 0 Error: 34.3 bits

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

    if 4.1617244964414691e128 < re

    1. Initial program Error: 58.0 bits

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

    2. Taylor expanded around inf Error: 8.9 bits

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

  4. Final simplificationError: 17.9 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.0802856481292018 \cdot 10^{+127}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -2.5199060035987696 \cdot 10^{-201}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.424827901233389 \cdot 10^{-276}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 4.161724496441469 \cdot 10^{+128}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))