Average Error: 32.5 → 17.8
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.1975508038006968 \cdot 10^{153}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -9.52817244882649108 \cdot 10^{-265}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.04745553524127593 \cdot 10^{-281}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.0421619395688348 \cdot 10^{92}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.1975508038006968 \cdot 10^{153}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -9.52817244882649108 \cdot 10^{-265}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 1.04745553524127593 \cdot 10^{-281}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 8.0421619395688348 \cdot 10^{92}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{else}:\\
\;\;\;\;\log re\\

\end{array}
double f(double re, double im) {
        double r38488 = re;
        double r38489 = r38488 * r38488;
        double r38490 = im;
        double r38491 = r38490 * r38490;
        double r38492 = r38489 + r38491;
        double r38493 = sqrt(r38492);
        double r38494 = log(r38493);
        return r38494;
}
↓
double f(double re, double im) {
        double r38495 = re;
        double r38496 = -4.197550803800697e+153;
        bool r38497 = r38495 <= r38496;
        double r38498 = -1.0;
        double r38499 = r38498 * r38495;
        double r38500 = log(r38499);
        double r38501 = -9.528172448826491e-265;
        bool r38502 = r38495 <= r38501;
        double r38503 = r38495 * r38495;
        double r38504 = im;
        double r38505 = r38504 * r38504;
        double r38506 = r38503 + r38505;
        double r38507 = sqrt(r38506);
        double r38508 = log(r38507);
        double r38509 = 1.047455535241276e-281;
        bool r38510 = r38495 <= r38509;
        double r38511 = log(r38504);
        double r38512 = 8.042161939568835e+92;
        bool r38513 = r38495 <= r38512;
        double r38514 = log(r38495);
        double r38515 = r38513 ? r38508 : r38514;
        double r38516 = r38510 ? r38511 : r38515;
        double r38517 = r38502 ? r38508 : r38516;
        double r38518 = r38497 ? r38500 : r38517;
        return r38518;
}

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 < -4.197550803800697e+153

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 6.5

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

    if -4.197550803800697e+153 < re < -9.528172448826491e-265 or 1.047455535241276e-281 < re < 8.042161939568835e+92

    1. Initial program 21.3

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

    if -9.528172448826491e-265 < re < 1.047455535241276e-281

    1. Initial program 31.7

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

    2. Taylor expanded around 0 33.5

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

    if 8.042161939568835e+92 < re

    1. Initial program 50.4

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

    2. Taylor expanded around inf 7.8

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.1975508038006968 \cdot 10^{153}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -9.52817244882649108 \cdot 10^{-265}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.04745553524127593 \cdot 10^{-281}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.0421619395688348 \cdot 10^{92}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))