Average Error: 31.7 → 18.0
Time: 3.4s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.176665368798762034403420210926666921837 \cdot 10^{117}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 1.048565751089651352045129138343128122191 \cdot 10^{96}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.176665368798762034403420210926666921837 \cdot 10^{117}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 1.048565751089651352045129138343128122191 \cdot 10^{96}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r28565 = re;
        double r28566 = r28565 * r28565;
        double r28567 = im;
        double r28568 = r28567 * r28567;
        double r28569 = r28566 + r28568;
        double r28570 = sqrt(r28569);
        return r28570;
}
↓
double f(double re, double im) {
        double r28571 = re;
        double r28572 = -2.176665368798762e+117;
        bool r28573 = r28571 <= r28572;
        double r28574 = -1.0;
        double r28575 = r28574 * r28571;
        double r28576 = 1.0485657510896514e+96;
        bool r28577 = r28571 <= r28576;
        double r28578 = r28571 * r28571;
        double r28579 = im;
        double r28580 = r28579 * r28579;
        double r28581 = r28578 + r28580;
        double r28582 = sqrt(r28581);
        double r28583 = r28577 ? r28582 : r28571;
        double r28584 = r28573 ? r28575 : r28583;
        return r28584;
}

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 < -2.176665368798762e+117

    1. Initial program 54.5

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

    2. Taylor expanded around -inf 10.2

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

    if -2.176665368798762e+117 < re < 1.0485657510896514e+96

    1. Initial program 21.6

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

    if 1.0485657510896514e+96 < re

    1. Initial program 51.5

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

    2. Taylor expanded around inf 10.6

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.176665368798762034403420210926666921837 \cdot 10^{117}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 1.048565751089651352045129138343128122191 \cdot 10^{96}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))