Average Error: 30.7 → 16.2
Time: 7.1s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.9227540988468336 \cdot 10^{+133}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le -1.7096067135741732 \cdot 10^{-274}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le -3.2042830049889953 \cdot 10^{-304}:\\ \;\;\;\;\log im\\ \mathbf{if}\;re \le 9.064697562001681 \cdot 10^{+141}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Try it out

  1. Inputs

  2. Original Output:

    Herbie Output:

Derivation

  1. Split input into 4 regimes

  2. if re < -1.9227540988468336e+133

    1. Initial program 56.4

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

    2. Taylor expanded around -inf 7.0

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

    3. Applied simplify7.0

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

    if -1.9227540988468336e+133 < re < -1.7096067135741732e-274 or -3.2042830049889953e-304 < re < 9.064697562001681e+141

    1. Initial program 19.6

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

    if -1.7096067135741732e-274 < re < -3.2042830049889953e-304

    1. Initial program 31.5

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

    2. Taylor expanded around 0 30.3

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

    if 9.064697562001681e+141 < re

    1. Initial program 58.9

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

    2. Taylor expanded around inf 6.0

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

Runtime

Time bar (total: 7.1s)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))