Average Error: 29.4 → 17.8
Time: 9.7s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -1.1118641726789477 \cdot 10^{+157}:\\ \;\;\;\;re\\ \mathbf{if}\;-re \le 3.6121788646421074 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;-re \le 1.1934306545466922 \cdot 10^{-134}:\\ \;\;\;\;im\\ \mathbf{if}\;-re \le 4.567139333627045 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -1.1118641726789477e+157

    1. Initial program 59.4

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

    2. Taylor expanded around inf 8.0

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

    if -1.1118641726789477e+157 < (- re) < 3.6121788646421074e-250 or 1.1934306545466922e-134 < (- re) < 4.567139333627045e+124

    1. Initial program 18.7

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

    if 3.6121788646421074e-250 < (- re) < 1.1934306545466922e-134

    1. Initial program 26.3

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

    2. Taylor expanded around 0 35.9

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

    if 4.567139333627045e+124 < (- re)

    1. Initial program 52.6

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

    2. Taylor expanded around -inf 8.9

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

    3. Applied simplify8.9

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

Runtime

Time bar (total: 9.7s)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))