Average Error: 30.8 → 17.5
Time: 6.4s
Precision: 64
Internal Precision: 384
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.7302483150496375 \cdot 10^{+141}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 6.028235519277853 \cdot 10^{-295}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 2.1332931106726862 \cdot 10^{-159}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 7.97108900733419 \cdot 10^{+115}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -1.7302483150496375e+141

    1. Initial program 58.1

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

    2. Taylor expanded around -inf 7.1

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

    3. Applied simplify7.1

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

    if -1.7302483150496375e+141 < re < 6.028235519277853e-295 or 2.1332931106726862e-159 < re < 7.97108900733419e+115

    1. Initial program 19.1

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

    if 6.028235519277853e-295 < re < 2.1332931106726862e-159

    1. Initial program 29.5

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

    2. Taylor expanded around 0 35.1

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

    if 7.97108900733419e+115 < re

    1. Initial program 52.9

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

    2. Taylor expanded around inf 8.3

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

Runtime

Time bar (total: 6.4s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))