Average Error: 30.7 → 17.5
Time: 6.5s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.2313825745467258 \cdot 10^{+145}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.934855094064062 \cdot 10^{-140}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.4709787711092824 \cdot 10^{-185}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.5119874841553792 \cdot 10^{+117}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if re < -2.2313825745467258e+145

    1. Initial program 60.2

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

    2. Taylor expanded around -inf 6.8

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

    3. Applied simplify6.8

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

    if -2.2313825745467258e+145 < re < -2.934855094064062e-140 or 2.4709787711092824e-185 < re < 2.5119874841553792e+117

    1. Initial program 16.1

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

    if -2.934855094064062e-140 < re < 2.4709787711092824e-185

    1. Initial program 29.0

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

    2. Taylor expanded around 0 33.4

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

    if 2.5119874841553792e+117 < re

    1. Initial program 53.7

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

    2. Taylor expanded around inf 6.8

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

Runtime

Time bar (total: 6.5s)Debug logProfile

herbie shell --seed 2018208 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))