Average Error: 30.9 → 17.0
Time: 5.1s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -2.1800769502294067 \cdot 10^{+121}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le -7.12795805645718 \cdot 10^{-265}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le 1.884002029149176 \cdot 10^{-286}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 9.992510366287537 \cdot 10^{+86}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

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) < -2.1800769502294067e+121

    1. Initial program 52.7

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

    2. Taylor expanded around inf 6.9

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

    if -2.1800769502294067e+121 < (- re) < -7.12795805645718e-265 or 1.884002029149176e-286 < (- re) < 9.992510366287537e+86

    1. Initial program 20.3

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

    if -7.12795805645718e-265 < (- re) < 1.884002029149176e-286

    1. Initial program 30.9

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

    2. Taylor expanded around 0 33.3

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

    if 9.992510366287537e+86 < (- re)

    1. Initial program 47.9

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

    2. Taylor expanded around -inf 9.5

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

    3. Applied simplify9.5

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

Runtime

Time bar (total: 5.1s)Debug logProfile

herbie shell --seed 2018167 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))