Average Error: 31.6 → 17.7
Time: 3.2s
Precision: 64
Internal Precision: 128
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.4938863573813225 \cdot 10^{+120}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.6903051380317462 \cdot 10^{+57}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -1.4938863573813225e+120

    1. Initial program 53.9

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

    2. Taylor expanded around -inf 8.2

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

    3. Simplified8.2

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

    if -1.4938863573813225e+120 < re < 1.6903051380317462e+57

    1. Initial program 22.0

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

    if 1.6903051380317462e+57 < re

    1. Initial program 44.7

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

    2. Taylor expanded around inf 11.2

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.4938863573813225 \cdot 10^{+120}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.6903051380317462 \cdot 10^{+57}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))