Average Error: 32.0 → 18.1
Time: 5.9s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.199115232773121436978557182122313209889 \cdot 10^{121}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -9.538446928549385159760190961856882540607 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.446990464932721506135775225301175117932 \cdot 10^{-197}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.937572631944601419338780453027731481475 \cdot 10^{124}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.199115232773121436978557182122313209889 \cdot 10^{121}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -9.538446928549385159760190961856882540607 \cdot 10^{-200}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 2.446990464932721506135775225301175117932 \cdot 10^{-197}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 3.937572631944601419338780453027731481475 \cdot 10^{124}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r45106 = re;
        double r45107 = r45106 * r45106;
        double r45108 = im;
        double r45109 = r45108 * r45108;
        double r45110 = r45107 + r45109;
        double r45111 = sqrt(r45110);
        return r45111;
}
↓
double f(double re, double im) {
        double r45112 = re;
        double r45113 = -1.1991152327731214e+121;
        bool r45114 = r45112 <= r45113;
        double r45115 = -1.0;
        double r45116 = r45115 * r45112;
        double r45117 = -9.538446928549385e-200;
        bool r45118 = r45112 <= r45117;
        double r45119 = r45112 * r45112;
        double r45120 = im;
        double r45121 = r45120 * r45120;
        double r45122 = r45119 + r45121;
        double r45123 = sqrt(r45122);
        double r45124 = 2.4469904649327215e-197;
        bool r45125 = r45112 <= r45124;
        double r45126 = 3.9375726319446014e+124;
        bool r45127 = r45112 <= r45126;
        double r45128 = r45127 ? r45123 : r45112;
        double r45129 = r45125 ? r45120 : r45128;
        double r45130 = r45118 ? r45123 : r45129;
        double r45131 = r45114 ? r45116 : r45130;
        return r45131;
}

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 < -1.1991152327731214e+121

    1. Initial program 55.2

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

    2. Taylor expanded around -inf 8.9

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

    if -1.1991152327731214e+121 < re < -9.538446928549385e-200 or 2.4469904649327215e-197 < re < 3.9375726319446014e+124

    1. Initial program 18.3

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

    if -9.538446928549385e-200 < re < 2.4469904649327215e-197

    1. Initial program 31.0

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

    2. Taylor expanded around 0 33.6

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

    if 3.9375726319446014e+124 < re

    1. Initial program 57.1

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

    2. Taylor expanded around inf 8.7

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

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.199115232773121436978557182122313209889 \cdot 10^{121}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -9.538446928549385159760190961856882540607 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.446990464932721506135775225301175117932 \cdot 10^{-197}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.937572631944601419338780453027731481475 \cdot 10^{124}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))