Average Error: 30.9 → 16.7
Time: 6.3s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -3.4730439514362437 \cdot 10^{+92}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le 5.043600616547752 \cdot 10^{+103}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (- re) < -3.4730439514362437e+92

    1. Initial program 49.1

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

    2. Taylor expanded around inf 9.2

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

    if -3.4730439514362437e+92 < (- re) < 5.043600616547752e+103

    1. Initial program 20.9

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

    if 5.043600616547752e+103 < (- re)

    1. Initial program 51.0

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

    2. Taylor expanded around -inf 8.0

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

    3. Applied simplify8.0

      \[\leadsto \color{blue}{\log \left(-re\right)}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 6.3s)Debug logProfile

herbie shell --seed 2018195 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))