Average Error: 30.3 → 17.5
Time: 5.1s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -1.0689382806433884 \cdot 10^{+136}:\\ \;\;\;\;\log re\\ \mathbf{elif}\;-re \le -3.944376596747617 \cdot 10^{-236}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;-re \le 8.603430461379705 \cdot 10^{-182}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;-re \le 1.0576382136883787 \cdot 10^{-07}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

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) < -1.0689382806433884e+136

    1. Initial program 56.6

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

    2. Taylor expanded around inf 6.3

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

    if -1.0689382806433884e+136 < (- re) < -3.944376596747617e-236 or 8.603430461379705e-182 < (- re) < 1.0576382136883787e-07

    1. Initial program 17.6

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

    if -3.944376596747617e-236 < (- re) < 8.603430461379705e-182

    1. Initial program 31.3

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

    2. Taylor expanded around 0 34.5

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

    if 1.0576382136883787e-07 < (- re)

    1. Initial program 37.5

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

    2. Taylor expanded around -inf 12.2

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

    3. Simplified12.2

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;-re \le -1.0689382806433884 \cdot 10^{+136}:\\ \;\;\;\;\log re\\ \mathbf{elif}\;-re \le -3.944376596747617 \cdot 10^{-236}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;-re \le 8.603430461379705 \cdot 10^{-182}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;-re \le 1.0576382136883787 \cdot 10^{-07}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Runtime

Time bar (total: 5.1s)Debug logProfile

herbie shell --seed 2018216 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))