math.log/1 on complex, real part

Average Error: 32.3 → 17.8

Time: 4.4s

Precision: 64

Math

\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.219332295965777137041720193068407814529 \cdot 10^{82}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -3.743447547042940916879606925039648794356 \cdot 10^{-217}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -1.6117218241464792617406634329935791946 \cdot 10^{-282}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 124645931887550053482496:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r24156 = re;
        double r24157 = r24156 * r24156;
        double r24158 = im;
        double r24159 = r24158 * r24158;
        double r24160 = r24157 + r24159;
        double r24161 = sqrt(r24160);
        double r24162 = log(r24161);
        return r24162;
}

↓

double f(double re, double im) {
        double r24163 = re;
        double r24164 = -4.219332295965777e+82;
        bool r24165 = r24163 <= r24164;
        double r24166 = -r24163;
        double r24167 = log(r24166);
        double r24168 = -3.743447547042941e-217;
        bool r24169 = r24163 <= r24168;
        double r24170 = r24163 * r24163;
        double r24171 = im;
        double r24172 = r24171 * r24171;
        double r24173 = r24170 + r24172;
        double r24174 = sqrt(r24173);
        double r24175 = log(r24174);
        double r24176 = -1.6117218241464793e-282;
        bool r24177 = r24163 <= r24176;
        double r24178 = log(r24171);
        double r24179 = 1.2464593188755005e+23;
        bool r24180 = r24163 <= r24179;
        double r24181 = log(r24163);
        double r24182 = r24180 ? r24175 : r24181;
        double r24183 = r24177 ? r24178 : r24182;
        double r24184 = r24169 ? r24175 : r24183;
        double r24185 = r24165 ? r24167 : r24184;
        return r24185;
}

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

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around -inf 10.0

      \[\leadsto \log \color{blue}{\left(-1 \cdot re\right)}\]

   3. Simplified 10.0

      \[\leadsto \log \color{blue}{\left(-re\right)}\]

   if -4.219332295965777e+82 < re < -3.743447547042941e-217 or -1.6117218241464793e-282 < re < 1.2464593188755005e+23

   1. Initial program 21.7

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   if -3.743447547042941e-217 < re < -1.6117218241464793e-282

   1. Initial program 34.5

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around 0 34.1

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

   if 1.2464593188755005e+23 < re

   1. Initial program 41.7

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around inf 11.8

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.219332295965777137041720193068407814529 \cdot 10^{82}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -3.743447547042940916879606925039648794356 \cdot 10^{-217}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -1.6117218241464792617406634329935791946 \cdot 10^{-282}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 124645931887550053482496:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))