Average Error: 29.7 → 18.0
Time: 3.3s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.4776996510629424 \cdot 10^{+161}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -3.999452158592024 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 7.873406083661657 \cdot 10^{-245}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.048893379706421 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.4776996510629424 \cdot 10^{+161}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -3.999452158592024 \cdot 10^{-169}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 7.873406083661657 \cdot 10^{-245}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 3.048893379706421 \cdot 10^{+133}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r1176544 = re;
        double r1176545 = r1176544 * r1176544;
        double r1176546 = im;
        double r1176547 = r1176546 * r1176546;
        double r1176548 = r1176545 + r1176547;
        double r1176549 = sqrt(r1176548);
        return r1176549;
}
↓
double f(double re, double im) {
        double r1176550 = re;
        double r1176551 = -1.4776996510629424e+161;
        bool r1176552 = r1176550 <= r1176551;
        double r1176553 = -r1176550;
        double r1176554 = -3.999452158592024e-169;
        bool r1176555 = r1176550 <= r1176554;
        double r1176556 = im;
        double r1176557 = r1176556 * r1176556;
        double r1176558 = r1176550 * r1176550;
        double r1176559 = r1176557 + r1176558;
        double r1176560 = sqrt(r1176559);
        double r1176561 = 7.873406083661657e-245;
        bool r1176562 = r1176550 <= r1176561;
        double r1176563 = 3.048893379706421e+133;
        bool r1176564 = r1176550 <= r1176563;
        double r1176565 = r1176564 ? r1176560 : r1176550;
        double r1176566 = r1176562 ? r1176556 : r1176565;
        double r1176567 = r1176555 ? r1176560 : r1176566;
        double r1176568 = r1176552 ? r1176553 : r1176567;
        return r1176568;
}

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 < -1.4776996510629424e+161

    1. Initial program 59.3

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around -inf 8.4

      \[\leadsto \color{blue}{-1 \cdot re}\]

    3. Simplified8.4

      \[\leadsto \color{blue}{-re}\]

    if -1.4776996510629424e+161 < re < -3.999452158592024e-169 or 7.873406083661657e-245 < re < 3.048893379706421e+133

    1. Initial program 17.7

      \[\sqrt{re \cdot re + im \cdot im}\]

    if -3.999452158592024e-169 < re < 7.873406083661657e-245

    1. Initial program 29.0

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around 0 33.6

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

    if 3.048893379706421e+133 < re

    1. Initial program 53.4

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around inf 8.9

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.4776996510629424 \cdot 10^{+161}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -3.999452158592024 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 7.873406083661657 \cdot 10^{-245}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.048893379706421 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))