Average Error: 30.7 → 17.3
Time: 5.7s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -1.2461487395816108 \cdot 10^{+150}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le 6.843876812000925 \cdot 10^{-244}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le 4.41847917226255 \cdot 10^{-179}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;-re \le 5.2679063627160525 \cdot 10^{+78}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if (- re) < -1.2461487395816108e+150

    1. Initial program 60.7

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

    2. Taylor expanded around inf 5.5

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

    if -1.2461487395816108e+150 < (- re) < 6.843876812000925e-244 or 4.41847917226255e-179 < (- re) < 5.2679063627160525e+78

    1. Initial program 19.1

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

    if 6.843876812000925e-244 < (- re) < 4.41847917226255e-179 or 5.2679063627160525e+78 < (- re)

    1. Initial program 44.5

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

    2. Taylor expanded around -inf 19.0

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

    3. Applied simplify19.0

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

Runtime

Time bar (total: 5.7s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))