Average Error: 31.0 → 17.6
Time: 6.4s
Precision: 64
Internal Precision: 320
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -3.6077188296375152 \cdot 10^{+31}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le -3.713042994084451 \cdot 10^{-251}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le 5.643018672793251 \cdot 10^{-301}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 1.5954214813937854 \cdot 10^{+49}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

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 4 regimes

  2. if (- re) < -3.6077188296375152e+31

    1. Initial program 42.2

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

    2. Taylor expanded around inf 12.1

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

    if -3.6077188296375152e+31 < (- re) < -3.713042994084451e-251 or 5.643018672793251e-301 < (- re) < 1.5954214813937854e+49

    1. Initial program 21.0

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

    if -3.713042994084451e-251 < (- re) < 5.643018672793251e-301

    1. Initial program 32.0

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

    2. Taylor expanded around 0 31.6

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

    if 1.5954214813937854e+49 < (- re)

    1. Initial program 43.7

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

    2. Taylor expanded around -inf 11.4

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

    3. Applied simplify11.4

      \[\leadsto \color{blue}{\log \left(-re\right)}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 6.4s)Debug logProfile

herbie shell --seed 2019053 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))