math.log/1 on complex, real part

Average Error: 31.2 → 17.4

Time: 3.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.8015950926867568 \cdot 10^{144}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.603232334857776 \cdot 10^{-212}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -5.74125170767144492 \cdot 10^{-228}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 4.4853367152010175 \cdot 10^{105}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r85154 = re;
        double r85155 = r85154 * r85154;
        double r85156 = im;
        double r85157 = r85156 * r85156;
        double r85158 = r85155 + r85157;
        double r85159 = sqrt(r85158);
        double r85160 = log(r85159);
        return r85160;
}

↓

double f(double re, double im) {
        double r85161 = re;
        double r85162 = -2.8015950926867568e+144;
        bool r85163 = r85161 <= r85162;
        double r85164 = -r85161;
        double r85165 = log(r85164);
        double r85166 = -2.6032323348577763e-212;
        bool r85167 = r85161 <= r85166;
        double r85168 = r85161 * r85161;
        double r85169 = im;
        double r85170 = r85169 * r85169;
        double r85171 = r85168 + r85170;
        double r85172 = sqrt(r85171);
        double r85173 = log(r85172);
        double r85174 = -5.741251707671445e-228;
        bool r85175 = r85161 <= r85174;
        double r85176 = 4.4853367152010175e+105;
        bool r85177 = r85161 <= r85176;
        double r85178 = log(r85161);
        double r85179 = r85177 ? r85173 : r85178;
        double r85180 = r85175 ? r85165 : r85179;
        double r85181 = r85167 ? r85173 : r85180;
        double r85182 = r85163 ? r85165 : r85181;
        return r85182;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -2.8015950926867568e+144 or -2.6032323348577763e-212 < re < -5.741251707671445e-228

   1. Initial program 57.8

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

   2. Taylor expanded around -inf 11.0

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

   3. Simplified 11.0

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

   if -2.8015950926867568e+144 < re < -2.6032323348577763e-212 or -5.741251707671445e-228 < re < 4.4853367152010175e+105

   1. Initial program 20.9

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

   if 4.4853367152010175e+105 < re

   1. Initial program 51.2

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

   2. Taylor expanded around inf 7.9

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

4. Final simplification 17.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.8015950926867568 \cdot 10^{144}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.603232334857776 \cdot 10^{-212}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -5.74125170767144492 \cdot 10^{-228}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 4.4853367152010175 \cdot 10^{105}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))