Average Error: 30.9 → 17.2
Time: 11.3s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.389900545062201 \cdot 10^{+108}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -4.105762006508071 \cdot 10^{-213}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le -2.7427147486198195 \cdot 10^{-291}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 3.6413109223672616 \cdot 10^{+89}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

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 < -4.389900545062201e+108

    1. Initial program 50.7

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

    2. Taylor expanded around -inf 9.0

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

    3. Applied simplify9.0

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

    if -4.389900545062201e+108 < re < -4.105762006508071e-213 or -2.7427147486198195e-291 < re < 3.6413109223672616e+89

    1. Initial program 19.8

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

    if -4.105762006508071e-213 < re < -2.7427147486198195e-291

    1. Initial program 31.0

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

    2. Taylor expanded around 0 34.2

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

    if 3.6413109223672616e+89 < re

    1. Initial program 49.4

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

    2. Taylor expanded around inf 9.5

      \[\leadsto \log \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 11.3s)Debug logProfile

herbie shell --seed 2018178 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))