Average Error: 29.5 → 17.5
Time: 2.4s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.1619932690089387 \cdot 10^{+147}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.8413088542688936 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -7.565496371114485 \cdot 10^{-306}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 9.129554876024421 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if re < -1.1619932690089387e+147

    1. Initial program 57.2

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

    2. Taylor expanded around -inf 8.8

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

    3. Simplified8.8

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

    if -1.1619932690089387e+147 < re < -2.8413088542688936e-141 or -7.565496371114485e-306 < re < 9.129554876024421e+87

    1. Initial program 17.5

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

    if -2.8413088542688936e-141 < re < -7.565496371114485e-306

    1. Initial program 28.5

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

    2. Taylor expanded around 0 35.4

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

    if 9.129554876024421e+87 < re

    1. Initial program 46.6

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

    2. Taylor expanded around inf 10.5

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1619932690089387 \cdot 10^{+147}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.8413088542688936 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -7.565496371114485 \cdot 10^{-306}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 9.129554876024421 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 2.4s)Debug logProfile

herbie shell --seed 2018221 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))