math.log/1 on complex, real part

Average Error: 31.2 → 17.4

Time: 2.6s

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 r42754 = re;
        double r42755 = r42754 * r42754;
        double r42756 = im;
        double r42757 = r42756 * r42756;
        double r42758 = r42755 + r42757;
        double r42759 = sqrt(r42758);
        double r42760 = log(r42759);
        return r42760;
}

↓

double f(double re, double im) {
        double r42761 = re;
        double r42762 = -2.8015950926867568e+144;
        bool r42763 = r42761 <= r42762;
        double r42764 = -r42761;
        double r42765 = log(r42764);
        double r42766 = -2.6032323348577763e-212;
        bool r42767 = r42761 <= r42766;
        double r42768 = r42761 * r42761;
        double r42769 = im;
        double r42770 = r42769 * r42769;
        double r42771 = r42768 + r42770;
        double r42772 = sqrt(r42771);
        double r42773 = log(r42772);
        double r42774 = -5.741251707671445e-228;
        bool r42775 = r42761 <= r42774;
        double r42776 = 4.4853367152010175e+105;
        bool r42777 = r42761 <= r42776;
        double r42778 = log(r42761);
        double r42779 = r42777 ? r42773 : r42778;
        double r42780 = r42775 ? r42765 : r42779;
        double r42781 = r42767 ? r42773 : r42780;
        double r42782 = r42763 ? r42765 : r42781;
        return r42782;
}

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)))))