math.abs on complex

Average Error: 31.4 → 17.8

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.273448994496035253530762983989520870583 \cdot 10^{87}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.763702591686904819827254628881572839528 \cdot 10^{111}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r1677531 = re;
        double r1677532 = r1677531 * r1677531;
        double r1677533 = im;
        double r1677534 = r1677533 * r1677533;
        double r1677535 = r1677532 + r1677534;
        double r1677536 = sqrt(r1677535);
        return r1677536;
}

↓

double f(double re, double im) {
        double r1677537 = re;
        double r1677538 = -2.2734489944960353e+87;
        bool r1677539 = r1677537 <= r1677538;
        double r1677540 = -r1677537;
        double r1677541 = 1.7637025916869048e+111;
        bool r1677542 = r1677537 <= r1677541;
        double r1677543 = im;
        double r1677544 = r1677543 * r1677543;
        double r1677545 = r1677537 * r1677537;
        double r1677546 = r1677544 + r1677545;
        double r1677547 = sqrt(r1677546);
        double r1677548 = r1677542 ? r1677547 : r1677537;
        double r1677549 = r1677539 ? r1677540 : r1677548;
        return r1677549;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -2.2734489944960353e+87

   1. Initial program 49.1

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

   2. Taylor expanded around -inf 11.1

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

   3. Simplified 11.1

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

   if -2.2734489944960353e+87 < re < 1.7637025916869048e+111

   1. Initial program 21.4

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

   if 1.7637025916869048e+111 < re

   1. Initial program 53.8

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

   2. Taylor expanded around inf 10.3

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.273448994496035253530762983989520870583 \cdot 10^{87}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.763702591686904819827254628881572839528 \cdot 10^{111}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))