Average Error: 32.6 → 17.7
Time: 1.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.371313124618756687613664414505037173977 \cdot 10^{141}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 1.424552610128290672525771016182451075361 \cdot 10^{133}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.371313124618756687613664414505037173977 \cdot 10^{141}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 1.424552610128290672525771016182451075361 \cdot 10^{133}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r76541 = re;
        double r76542 = r76541 * r76541;
        double r76543 = im;
        double r76544 = r76543 * r76543;
        double r76545 = r76542 + r76544;
        double r76546 = sqrt(r76545);
        double r76547 = log(r76546);
        return r76547;
}
↓
double f(double re, double im) {
        double r76548 = re;
        double r76549 = -1.3713131246187567e+141;
        bool r76550 = r76548 <= r76549;
        double r76551 = -1.0;
        double r76552 = r76551 * r76548;
        double r76553 = log(r76552);
        double r76554 = 1.4245526101282907e+133;
        bool r76555 = r76548 <= r76554;
        double r76556 = r76548 * r76548;
        double r76557 = im;
        double r76558 = r76557 * r76557;
        double r76559 = r76556 + r76558;
        double r76560 = sqrt(r76559);
        double r76561 = log(r76560);
        double r76562 = log(r76548);
        double r76563 = r76555 ? r76561 : r76562;
        double r76564 = r76550 ? r76553 : r76563;
        return r76564;
}

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 < -1.3713131246187567e+141

    1. Initial program 61.4

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

    2. Taylor expanded around -inf 6.7

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

    if -1.3713131246187567e+141 < re < 1.4245526101282907e+133

    1. Initial program 21.9

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

    if 1.4245526101282907e+133 < re

    1. Initial program 58.1

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

    2. Taylor expanded around inf 7.7

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.371313124618756687613664414505037173977 \cdot 10^{141}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 1.424552610128290672525771016182451075361 \cdot 10^{133}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))