math.abs on complex

Average Error: 32.1 → 17.8

Time: 1.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.219332295965777137041720193068407814529 \cdot 10^{82}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.743447547042940916879606925039648794356 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 4.769025653725654548986941102749859144285 \cdot 10^{-305}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.419375064734749687649336536979338940651 \cdot 10^{132}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r129154 = re;
        double r129155 = r129154 * r129154;
        double r129156 = im;
        double r129157 = r129156 * r129156;
        double r129158 = r129155 + r129157;
        double r129159 = sqrt(r129158);
        return r129159;
}

↓

double f(double re, double im) {
        double r129160 = re;
        double r129161 = -4.219332295965777e+82;
        bool r129162 = r129160 <= r129161;
        double r129163 = -1.0;
        double r129164 = r129163 * r129160;
        double r129165 = -3.743447547042941e-217;
        bool r129166 = r129160 <= r129165;
        double r129167 = r129160 * r129160;
        double r129168 = im;
        double r129169 = r129168 * r129168;
        double r129170 = r129167 + r129169;
        double r129171 = sqrt(r129170);
        double r129172 = 4.769025653725655e-305;
        bool r129173 = r129160 <= r129172;
        double r129174 = 2.4193750647347497e+132;
        bool r129175 = r129160 <= r129174;
        double r129176 = r129175 ? r129171 : r129160;
        double r129177 = r129173 ? r129168 : r129176;
        double r129178 = r129166 ? r129171 : r129177;
        double r129179 = r129162 ? r129164 : r129178;
        return r129179;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -4.219332295965777e+82

   1. Initial program 49.1

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

   2. Taylor expanded around -inf 11.7

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

   if -4.219332295965777e+82 < re < -3.743447547042941e-217 or 4.769025653725655e-305 < re < 2.4193750647347497e+132

   1. Initial program 20.0

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

   if -3.743447547042941e-217 < re < 4.769025653725655e-305

   1. Initial program 33.9

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

   2. Taylor expanded around 0 33.5

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

   if 2.4193750647347497e+132 < re

   1. Initial program 58.5

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

   2. Taylor expanded around inf 8.4

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.219332295965777137041720193068407814529 \cdot 10^{82}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.743447547042940916879606925039648794356 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 4.769025653725654548986941102749859144285 \cdot 10^{-305}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.419375064734749687649336536979338940651 \cdot 10^{132}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))