Average Error: 31.5 → 17.4
Time: 946.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.5872918038759956 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.5752696805163673 \cdot 10^{135}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.5872918038759956 \cdot 10^{153}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 4.5752696805163673 \cdot 10^{135}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r41415 = re;
        double r41416 = r41415 * r41415;
        double r41417 = im;
        double r41418 = r41417 * r41417;
        double r41419 = r41416 + r41418;
        double r41420 = sqrt(r41419);
        return r41420;
}
↓
double f(double re, double im) {
        double r41421 = re;
        double r41422 = -2.5872918038759956e+153;
        bool r41423 = r41421 <= r41422;
        double r41424 = -1.0;
        double r41425 = r41424 * r41421;
        double r41426 = 4.575269680516367e+135;
        bool r41427 = r41421 <= r41426;
        double r41428 = r41421 * r41421;
        double r41429 = im;
        double r41430 = r41429 * r41429;
        double r41431 = r41428 + r41430;
        double r41432 = sqrt(r41431);
        double r41433 = r41427 ? r41432 : r41421;
        double r41434 = r41423 ? r41425 : r41433;
        return r41434;
}

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 < -2.5872918038759956e+153

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 7.5

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

    if -2.5872918038759956e+153 < re < 4.575269680516367e+135

    1. Initial program 20.7

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

    if 4.575269680516367e+135 < re

    1. Initial program 59.0

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

    2. Taylor expanded around inf 9.0

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.5872918038759956 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.5752696805163673 \cdot 10^{135}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))