Average Error: 30.2 → 18.3
Time: 16.1s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.3382925664035833 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -7.220768591101605 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 5.283325772797005 \cdot 10^{-145}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.2122680560109031 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 < -1.3382925664035833e+154

    1. Initial program 59.4

      \[\sqrt{re \cdot re + im \cdot im}\]
    2. Using strategy rm

    3. Applied add-exp-log59.4

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]
    4. Using strategy rm

    5. Applied pow159.4

      \[\leadsto e^{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}\]

    6. Applied log-pow59.4

      \[\leadsto e^{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

    7. Applied exp-prod59.4

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}}\]

    8. Simplified59.4

      \[\leadsto {\color{blue}{e}}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}\]

    9. Taylor expanded around -inf 7.2

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

    10. Simplified7.2

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

    if -1.3382925664035833e+154 < re < -7.220768591101605e-173 or 5.283325772797005e-145 < re < 1.2122680560109031e+156

    1. Initial program 16.0

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

    if -7.220768591101605e-173 < re < 5.283325772797005e-145

    1. Initial program 29.1

      \[\sqrt{re \cdot re + im \cdot im}\]
    2. Using strategy rm

    3. Applied add-exp-log31.5

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

    4. Taylor expanded around 0 35.1

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

    if 1.2122680560109031e+156 < re

    1. Initial program 59.3

      \[\sqrt{re \cdot re + im \cdot im}\]
    2. Using strategy rm

    3. Applied add-exp-log59.3

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

    4. Taylor expanded around inf 13.6

      \[\leadsto \color{blue}{e^{-\log \left(\frac{1}{re}\right)}}\]

    5. Simplified7.8

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

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3382925664035833 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -7.220768591101605 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 5.283325772797005 \cdot 10^{-145}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.2122680560109031 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 16.1s)Debug logProfile

herbie shell --seed 2018235 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))