Average Error: 30.6 → 17.2
Time: 5.8s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.734673720272803 \cdot 10^{+88}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -5.111710031589403 \cdot 10^{-285}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 7.788676175088279 \cdot 10^{-289}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 4.757030266529576 \cdot 10^{+71}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if re < -1.734673720272803e+88

    1. Initial program 47.4

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

    2. Taylor expanded around -inf 8.3

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

    3. Applied simplify8.3

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

    if -1.734673720272803e+88 < re < -5.111710031589403e-285 or 7.788676175088279e-289 < re < 4.757030266529576e+71

    1. Initial program 21.0

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

    if -5.111710031589403e-285 < re < 7.788676175088279e-289

    1. Initial program 28.6

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

    2. Taylor expanded around 0 34.4

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

    if 4.757030266529576e+71 < re

    1. Initial program 45.6

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

    2. Taylor expanded around inf 9.9

      \[\leadsto \log \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 5.8s)Debug logProfile

herbie shell --seed 2018166 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))