Average Error: 32.3 → 18.0
Time: 1.0s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.15024475259887937 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -9.52817244882649108 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.04745553524127593 \cdot 10^{-281}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.70835173311075 \cdot 10^{105}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.15024475259887937 \cdot 10^{153}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -9.52817244882649108 \cdot 10^{-265}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.04745553524127593 \cdot 10^{-281}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 2.70835173311075 \cdot 10^{105}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r48553 = re;
        double r48554 = r48553 * r48553;
        double r48555 = im;
        double r48556 = r48555 * r48555;
        double r48557 = r48554 + r48556;
        double r48558 = sqrt(r48557);
        return r48558;
}
↓
double f(double re, double im) {
        double r48559 = re;
        double r48560 = -8.15024475259888e+153;
        bool r48561 = r48559 <= r48560;
        double r48562 = -1.0;
        double r48563 = r48562 * r48559;
        double r48564 = -9.528172448826491e-265;
        bool r48565 = r48559 <= r48564;
        double r48566 = r48559 * r48559;
        double r48567 = im;
        double r48568 = r48567 * r48567;
        double r48569 = r48566 + r48568;
        double r48570 = sqrt(r48569);
        double r48571 = 1.047455535241276e-281;
        bool r48572 = r48559 <= r48571;
        double r48573 = 2.70835173311075e+105;
        bool r48574 = r48559 <= r48573;
        double r48575 = r48574 ? r48570 : r48559;
        double r48576 = r48572 ? r48567 : r48575;
        double r48577 = r48565 ? r48570 : r48576;
        double r48578 = r48561 ? r48563 : r48577;
        return r48578;
}

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 < -8.15024475259888e+153

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 7.8

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

    if -8.15024475259888e+153 < re < -9.528172448826491e-265 or 1.047455535241276e-281 < re < 2.70835173311075e+105

    1. Initial program 21.0

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

    if -9.528172448826491e-265 < re < 1.047455535241276e-281

    1. Initial program 30.8

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

    2. Taylor expanded around 0 32.9

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

    if 2.70835173311075e+105 < re

    1. Initial program 52.5

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.15024475259887937 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -9.52817244882649108 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.04745553524127593 \cdot 10^{-281}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.70835173311075 \cdot 10^{105}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))