Average Error: 32.3 → 18.5
Time: 974.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.150193189996774185649058855667814599131 \cdot 10^{145}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -8.463201405765729115322317591242469230422 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -3.068368918555544922720895417815131992438 \cdot 10^{-192}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.162005679260343971879420197094042200479 \cdot 10^{70}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.150193189996774185649058855667814599131 \cdot 10^{145}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -8.463201405765729115322317591242469230422 \cdot 10^{-144}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le -3.068368918555544922720895417815131992438 \cdot 10^{-192}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 3.162005679260343971879420197094042200479 \cdot 10^{70}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r35407 = re;
        double r35408 = r35407 * r35407;
        double r35409 = im;
        double r35410 = r35409 * r35409;
        double r35411 = r35408 + r35410;
        double r35412 = sqrt(r35411);
        return r35412;
}
↓
double f(double re, double im) {
        double r35413 = re;
        double r35414 = -2.1501931899967742e+145;
        bool r35415 = r35413 <= r35414;
        double r35416 = -1.0;
        double r35417 = r35416 * r35413;
        double r35418 = -8.463201405765729e-144;
        bool r35419 = r35413 <= r35418;
        double r35420 = r35413 * r35413;
        double r35421 = im;
        double r35422 = r35421 * r35421;
        double r35423 = r35420 + r35422;
        double r35424 = sqrt(r35423);
        double r35425 = -3.068368918555545e-192;
        bool r35426 = r35413 <= r35425;
        double r35427 = 3.162005679260344e+70;
        bool r35428 = r35413 <= r35427;
        double r35429 = r35428 ? r35424 : r35413;
        double r35430 = r35426 ? r35421 : r35429;
        double r35431 = r35419 ? r35424 : r35430;
        double r35432 = r35415 ? r35417 : r35431;
        return r35432;
}

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 < -2.1501931899967742e+145

    1. Initial program 61.4

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

    2. Taylor expanded around -inf 8.3

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

    if -2.1501931899967742e+145 < re < -8.463201405765729e-144 or -3.068368918555545e-192 < re < 3.162005679260344e+70

    1. Initial program 21.3

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

    if -8.463201405765729e-144 < re < -3.068368918555545e-192

    1. Initial program 26.8

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

    2. Taylor expanded around 0 39.2

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

    if 3.162005679260344e+70 < re

    1. Initial program 48.1

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

    2. Taylor expanded around inf 12.1

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

  4. Final simplification18.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.150193189996774185649058855667814599131 \cdot 10^{145}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -8.463201405765729115322317591242469230422 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -3.068368918555544922720895417815131992438 \cdot 10^{-192}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.162005679260343971879420197094042200479 \cdot 10^{70}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))