Average Error: 31.2 → 16.9
Time: 5.4s
Precision: 64
Internal Precision: 320
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.141969331767055 \cdot 10^{+99}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 9.557816170603889 \cdot 10^{+76}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -1.141969331767055e+99

    1. Initial program 49.7

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

    2. Taylor expanded around -inf 8.2

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

    3. Applied simplify8.2

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

    if -1.141969331767055e+99 < re < 9.557816170603889e+76

    1. Initial program 21.5

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

    if 9.557816170603889e+76 < re

    1. Initial program 46.1

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

    2. Taylor expanded around inf 10.0

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

Runtime

Time bar (total: 5.4s)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))