Average Error: 32.1 → 10.6
Time: 29.6s
Precision: 64
Internal Precision: 384
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.460645605766947 \cdot 10^{+149}:\\ \;\;\;\;-re\\ \mathbf{if}\;re \le 2.2531375296527453 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re \le 1.099940540275573 \cdot 10^{-139}:\\ \;\;\;\;im\\ \mathbf{if}\;re \le 9.614861284507057 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -5.460645605766947e+149

    1. Initial program 58.3

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around -inf 0

      \[\leadsto \color{blue}{-1 \cdot re}\]

    3. Applied simplify0

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

    if -5.460645605766947e+149 < re < 2.2531375296527453e-307 or 1.099940540275573e-139 < re < 9.614861284507057e+142

    1. Initial program 17.7

      \[\sqrt{re \cdot re + im \cdot im}\]

    if 2.2531375296527453e-307 < re < 1.099940540275573e-139

    1. Initial program 44.7

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around 0 0

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

    if 9.614861284507057e+142 < re

    1. Initial program 57.1

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around inf 0

      \[\leadsto \color{blue}{re}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 29.6s)Debug logProfile

herbie shell --seed '#(1062930989 876886121 3990119081 3032829768 3060892583 1929069376)' 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))