Average Error: 30.6 → 16.8
Time: 5.8s
Precision: 64
Internal Precision: 320
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -1.1811150131582225 \cdot 10^{+66}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le -8.601886711245417 \cdot 10^{-273}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le 7.61426085424912 \cdot 10^{-201}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 2.2851485703139845 \cdot 10^{+142}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

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) < -1.1811150131582225e+66

    1. Initial program 44.8

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

    2. Taylor expanded around inf 9.8

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

    if -1.1811150131582225e+66 < (- re) < -8.601886711245417e-273 or 7.61426085424912e-201 < (- re) < 2.2851485703139845e+142

    1. Initial program 18.7

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

    if -8.601886711245417e-273 < (- re) < 7.61426085424912e-201

    1. Initial program 29.9

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

    2. Taylor expanded around 0 31.9

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

    if 2.2851485703139845e+142 < (- re)

    1. Initial program 58.7

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

    2. Taylor expanded around -inf 6.5

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

    3. Applied simplify6.5

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

Runtime

Time bar (total: 5.8s)Debug logProfile

herbie shell --seed 2018201 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))