Average Error: 31.7 → 18.1
Time: 1.9s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -3.4362841072323272 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot im\\ \mathbf{elif}\;im \le -1.7874287404230692 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;im \le 5.8993776144081826 \cdot 10^{-308}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \le 2.4345784437110915 \cdot 10^{-199}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;im \le 1.7970794289179904 \cdot 10^{-166}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \le 1.0688990210562475 \cdot 10^{-162}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;im \le 2.23402097896517821 \cdot 10^{109}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;im \le -3.4362841072323272 \cdot 10^{150}:\\
\;\;\;\;-1 \cdot im\\

\mathbf{elif}\;im \le -1.7874287404230692 \cdot 10^{-225}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;im \le 5.8993776144081826 \cdot 10^{-308}:\\
\;\;\;\;re\\

\mathbf{elif}\;im \le 2.4345784437110915 \cdot 10^{-199}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;im \le 1.7970794289179904 \cdot 10^{-166}:\\
\;\;\;\;re\\

\mathbf{elif}\;im \le 1.0688990210562475 \cdot 10^{-162}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;im \le 2.23402097896517821 \cdot 10^{109}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;im\\

\end{array}
double f(double re, double im) {
        double r66206 = re;
        double r66207 = r66206 * r66206;
        double r66208 = im;
        double r66209 = r66208 * r66208;
        double r66210 = r66207 + r66209;
        double r66211 = sqrt(r66210);
        return r66211;
}
↓
double f(double re, double im) {
        double r66212 = im;
        double r66213 = -3.436284107232327e+150;
        bool r66214 = r66212 <= r66213;
        double r66215 = -1.0;
        double r66216 = r66215 * r66212;
        double r66217 = -1.7874287404230692e-225;
        bool r66218 = r66212 <= r66217;
        double r66219 = re;
        double r66220 = r66219 * r66219;
        double r66221 = r66212 * r66212;
        double r66222 = r66220 + r66221;
        double r66223 = sqrt(r66222);
        double r66224 = 5.899377614408183e-308;
        bool r66225 = r66212 <= r66224;
        double r66226 = 2.4345784437110915e-199;
        bool r66227 = r66212 <= r66226;
        double r66228 = r66215 * r66219;
        double r66229 = 1.7970794289179904e-166;
        bool r66230 = r66212 <= r66229;
        double r66231 = 1.0688990210562475e-162;
        bool r66232 = r66212 <= r66231;
        double r66233 = 2.2340209789651782e+109;
        bool r66234 = r66212 <= r66233;
        double r66235 = r66234 ? r66223 : r66212;
        double r66236 = r66232 ? r66228 : r66235;
        double r66237 = r66230 ? r66219 : r66236;
        double r66238 = r66227 ? r66228 : r66237;
        double r66239 = r66225 ? r66219 : r66238;
        double r66240 = r66218 ? r66223 : r66239;
        double r66241 = r66214 ? r66216 : r66240;
        return r66241;
}

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 5 regimes

  2. if im < -3.436284107232327e+150

    1. Initial program 62.6

      \[\sqrt{re \cdot re + im \cdot im}\]
    2. Using strategy rm

    3. Applied flip-+64.0

      \[\leadsto \sqrt{\color{blue}{\frac{\left(re \cdot re\right) \cdot \left(re \cdot re\right) - \left(im \cdot im\right) \cdot \left(im \cdot im\right)}{re \cdot re - im \cdot im}}}\]

    4. Simplified64.0

      \[\leadsto \sqrt{\frac{\color{blue}{\left(-{im}^{3}\right) \cdot im + {re}^{4}}}{re \cdot re - im \cdot im}}\]

    5. Taylor expanded around -inf 8.8

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

    if -3.436284107232327e+150 < im < -1.7874287404230692e-225 or 1.0688990210562475e-162 < im < 2.2340209789651782e+109

    1. Initial program 17.2

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

    if -1.7874287404230692e-225 < im < 5.899377614408183e-308 or 2.4345784437110915e-199 < im < 1.7970794289179904e-166

    1. Initial program 31.5

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

    2. Taylor expanded around inf 33.7

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

    if 5.899377614408183e-308 < im < 2.4345784437110915e-199 or 1.7970794289179904e-166 < im < 1.0688990210562475e-162

    1. Initial program 31.3

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

    2. Taylor expanded around -inf 35.2

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

    if 2.2340209789651782e+109 < im

    1. Initial program 53.4

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

    2. Taylor expanded around 0 9.8

      \[\leadsto \color{blue}{im}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -3.4362841072323272 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot im\\ \mathbf{elif}\;im \le -1.7874287404230692 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;im \le 5.8993776144081826 \cdot 10^{-308}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \le 2.4345784437110915 \cdot 10^{-199}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;im \le 1.7970794289179904 \cdot 10^{-166}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \le 1.0688990210562475 \cdot 10^{-162}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;im \le 2.23402097896517821 \cdot 10^{109}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))