Average Error: 31.4 → 17.7
Time: 1.2s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -6.504888998661915 \cdot 10^{+102}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -8.510846260708714 \cdot 10^{-176}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq -3.735603583061054 \cdot 10^{-219}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 5.153255919931365 \cdot 10^{+39}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -6.504888998661915 \cdot 10^{+102}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -8.510846260708714 \cdot 10^{-176}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq -3.735603583061054 \cdot 10^{-219}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq 5.153255919931365 \cdot 10^{+39}:\\
\;\;\;\;\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 -6.504888998661915e+102)
   (log (- re))
   (if (<= re -8.510846260708714e-176)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re -3.735603583061054e-219)
       (log (- re))
       (if (<= re 5.153255919931365e+39)
         (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 <= -6.504888998661915e+102) {
		tmp = log(-re);
	} else if (re <= -8.510846260708714e-176) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else if (re <= -3.735603583061054e-219) {
		tmp = log(-re);
	} else if (re <= 5.153255919931365e+39) {
		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 < -6.504888998661915e102 or -8.5108462607087144e-176 < re < -3.7356035830610541e-219

    1. Initial program 48.4

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

    2. Taylor expanded around -inf 13.9

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

    3. Simplified13.9

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

    if -6.504888998661915e102 < re < -8.5108462607087144e-176 or -3.7356035830610541e-219 < re < 5.15325591993136483e39

    1. Initial program 21.6

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

    if 5.15325591993136483e39 < re

    1. Initial program 42.6

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

    2. Taylor expanded around inf 10.9

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.504888998661915 \cdot 10^{+102}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -8.510846260708714 \cdot 10^{-176}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq -3.735603583061054 \cdot 10^{-219}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq 5.153255919931365 \cdot 10^{+39}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020289 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))