Average Error: 30.3 → 16.9
Time: 4.8s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -3.0365432001457886 \cdot 10^{+112}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le -4.970880198459218 \cdot 10^{-198}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le 9.073699737263314 \cdot 10^{-265}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 8.409728143308716 \cdot 10^{+104}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

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) < -3.0365432001457886e+112

    1. Initial program 52.9

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

    2. Taylor expanded around inf 7.2

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

    if -3.0365432001457886e+112 < (- re) < -4.970880198459218e-198 or 9.073699737263314e-265 < (- re) < 8.409728143308716e+104

    1. Initial program 18.4

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

    if -4.970880198459218e-198 < (- re) < 9.073699737263314e-265

    1. Initial program 30.0

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

    2. Taylor expanded around 0 31.8

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

    if 8.409728143308716e+104 < (- re)

    1. Initial program 50.0

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

    2. Taylor expanded around -inf 9.4

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

    3. Applied simplify9.4

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

Runtime

Time bar (total: 4.8s)Debug logProfile

herbie shell --seed 2018170 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))