Average Error: 31.0 → 17.5
Time: 1.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.505752205836537605611230467447200313868 \cdot 10^{136}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.200563398436491693418328268892598073539 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.505752205836537605611230467447200313868 \cdot 10^{136}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -3.200563398436491693418328268892598073539 \cdot 10^{-257}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r40505 = re;
        double r40506 = r40505 * r40505;
        double r40507 = im;
        double r40508 = r40507 * r40507;
        double r40509 = r40506 + r40508;
        double r40510 = sqrt(r40509);
        return r40510;
}
↓
double f(double re, double im) {
        double r40511 = re;
        double r40512 = -1.5057522058365376e+136;
        bool r40513 = r40511 <= r40512;
        double r40514 = -1.0;
        double r40515 = r40514 * r40511;
        double r40516 = -3.2005633984364917e-257;
        bool r40517 = r40511 <= r40516;
        double r40518 = r40511 * r40511;
        double r40519 = im;
        double r40520 = r40519 * r40519;
        double r40521 = r40518 + r40520;
        double r40522 = sqrt(r40521);
        double r40523 = 3.8197786805557845e-227;
        bool r40524 = r40511 <= r40523;
        double r40525 = 8.439330033545885e+67;
        bool r40526 = r40511 <= r40525;
        double r40527 = r40526 ? r40522 : r40511;
        double r40528 = r40524 ? r40519 : r40527;
        double r40529 = r40517 ? r40522 : r40528;
        double r40530 = r40513 ? r40515 : r40529;
        return r40530;
}

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 < -1.5057522058365376e+136

    1. Initial program 58.9

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

    2. Taylor expanded around -inf 9.2

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

    if -1.5057522058365376e+136 < re < -3.2005633984364917e-257 or 3.8197786805557845e-227 < re < 8.439330033545885e+67

    1. Initial program 18.7

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

    if -3.2005633984364917e-257 < re < 3.8197786805557845e-227

    1. Initial program 30.2

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

    2. Taylor expanded around 0 32.1

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

    if 8.439330033545885e+67 < re

    1. Initial program 46.7

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

    2. Taylor expanded around inf 12.0

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.505752205836537605611230467447200313868 \cdot 10^{136}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.200563398436491693418328268892598073539 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))