Average Error: 29.4 → 18.1
Time: 11.0s
Precision: 64
Internal Precision: 384
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -4.641055197765252 \cdot 10^{+155}:\\ \;\;\;\;re\\ \mathbf{if}\;-re \le -7.812502410634447 \cdot 10^{-204}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;-re \le 1.4474198858501773 \cdot 10^{-233}:\\ \;\;\;\;im\\ \mathbf{if}\;-re \le 1.82619128701192 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -4.641055197765252e+155

    1. Initial program 59.4

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

    2. Taylor expanded around inf 8.2

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

    if -4.641055197765252e+155 < (- re) < -7.812502410634447e-204 or 1.4474198858501773e-233 < (- re) < 1.82619128701192e+84

    1. Initial program 17.7

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

    if -7.812502410634447e-204 < (- re) < 1.4474198858501773e-233

    1. Initial program 28.9

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

    2. Taylor expanded around 0 35.6

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

    if 1.82619128701192e+84 < (- re)

    1. Initial program 45.6

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

    2. Taylor expanded around -inf 11.9

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

    3. Applied simplify11.9

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

Runtime

Time bar (total: 11.0s)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))