Average Error: 32.1 → 17.7
Time: 1.6s
Precision: binary64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.117569134945259 \cdot 10^{108}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.31910859188192068 \cdot 10^{109}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -8.117569134945259e108

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 8.4

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

    3. Simplified8.4

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

    if -8.117569134945259e108 < re < 1.31910859188192068e109

    1. Initial program 22.0

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

    if 1.31910859188192068e109 < re

    1. Initial program 53.1

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

    2. Taylor expanded around inf 9.4

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.117569134945259 \cdot 10^{108}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.31910859188192068 \cdot 10^{109}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))