Average Error: 30.4 → 14.4
Time: 19.9s
Precision: 64
Internal Precision: 384
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.993615069801488 \cdot 10^{+125}:\\ \;\;\;\;-re\\ \mathbf{if}\;re \le 2.447855958190132 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re \le 2.240485188259365 \cdot 10^{-180}:\\ \;\;\;\;re\\ \mathbf{if}\;re \le 3.0139115645243634 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -3.993615069801488e+125

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 0

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

    3. Applied simplify0

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

    if -3.993615069801488e+125 < re < 2.447855958190132e-195 or 2.240485188259365e-180 < re < 3.0139115645243634e+153

    1. Initial program 20.2

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

    if 2.447855958190132e-195 < re < 2.240485188259365e-180 or 3.0139115645243634e+153 < re

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0

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

Runtime

Time bar (total: 19.9s)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))