Average Error: 30.2 → 17.6
Time: 2.8s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -6.20102674233657 \cdot 10^{+128}:\\ \;\;\;\;re\\ \mathbf{elif}\;-re \le -2.978005804291019 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;-re \le 1.8210497627404317 \cdot 10^{-280}:\\ \;\;\;\;im\\ \mathbf{elif}\;-re \le 1.2236301320960694 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

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) < -6.20102674233657e+128

    1. Initial program 53.2

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

    2. Taylor expanded around inf 8.2

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

    if -6.20102674233657e+128 < (- re) < -2.978005804291019e-145 or 1.8210497627404317e-280 < (- re) < 1.2236301320960694e+137

    1. Initial program 17.8

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

    if -2.978005804291019e-145 < (- re) < 1.8210497627404317e-280

    1. Initial program 28.5

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

    2. Taylor expanded around 0 34.5

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

    if 1.2236301320960694e+137 < (- re)

    1. Initial program 55.2

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

    2. Taylor expanded around -inf 8.1

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

    3. Simplified8.1

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;-re \le -6.20102674233657 \cdot 10^{+128}:\\ \;\;\;\;re\\ \mathbf{elif}\;-re \le -2.978005804291019 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;-re \le 1.8210497627404317 \cdot 10^{-280}:\\ \;\;\;\;im\\ \mathbf{elif}\;-re \le 1.2236301320960694 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

Runtime

Time bar (total: 2.8s)Debug logProfile

herbie shell --seed 2018217 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))