Average Error: 31.5 → 17.6
Time: 2.1s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -6.278604197488825 \cdot 10^{+103}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.7268439068424185 \cdot 10^{-307}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 7.763683631167119 \cdot 10^{-258}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 1.0060942248814217 \cdot 10^{+49}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \leq -6.278604197488825 \cdot 10^{+103}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \leq -1.7268439068424185 \cdot 10^{-307}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \leq 7.763683631167119 \cdot 10^{-258}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \leq 1.0060942248814217 \cdot 10^{+49}:\\
\;\;\;\;\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 -6.278604197488825e+103)
   (log (- re))
   (if (<= re -1.7268439068424185e-307)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re 7.763683631167119e-258)
       (log im)
       (if (<= re 1.0060942248814217e+49)
         (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 <= -6.278604197488825e+103) {
		tmp = log(-re);
	} else if (re <= -1.7268439068424185e-307) {
		tmp = log(sqrt((re * re) + (im * im)));
	} else if (re <= 7.763683631167119e-258) {
		tmp = log(im);
	} else if (re <= 1.0060942248814217e+49) {
		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 < -6.2786041974888246e103

    1. Initial program 52.1

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

    2. Taylor expanded around -inf 8.4

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

    3. Simplified8.4

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

    if -6.2786041974888246e103 < re < -1.72684390684241845e-307 or 7.7636836311671192e-258 < re < 1.00609422488142175e49

    1. Initial program 20.7

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

    if -1.72684390684241845e-307 < re < 7.7636836311671192e-258

    1. Initial program 30.5

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

    2. Taylor expanded around 0 37.2

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

    if 1.00609422488142175e49 < re

    1. Initial program 46.3

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

    2. Taylor expanded around inf 12.0

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.278604197488825 \cdot 10^{+103}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.7268439068424185 \cdot 10^{-307}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 7.763683631167119 \cdot 10^{-258}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 1.0060942248814217 \cdot 10^{+49}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))