Average Error: 31.0 → 17.2
Time: 5.8s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -9.198771123190312 \cdot 10^{+144}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le -7.889187209979888 \cdot 10^{-287}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le 2.1944387002171865 \cdot 10^{-217}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 1.0223090613384444 \cdot 10^{+106}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -9.198771123190312e+144

    1. Initial program 59.7

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

    2. Taylor expanded around inf 6.8

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

    if -9.198771123190312e+144 < (- re) < -7.889187209979888e-287 or 2.1944387002171865e-217 < (- re) < 1.0223090613384444e+106

    1. Initial program 19.4

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

    if -7.889187209979888e-287 < (- re) < 2.1944387002171865e-217

    1. Initial program 29.8

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

    2. Taylor expanded around 0 33.5

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

    if 1.0223090613384444e+106 < (- re)

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 8.0

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

    3. Applied simplify8.0

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

Runtime

Time bar (total: 5.8s)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))