Average Error: 29.7 → 16.6
Time: 6.9s
Precision: 64
Internal Precision: 128
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.069661801608885 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.8034656734801727 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.2241568427137281 \cdot 10^{-244}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.0367130689854715 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 < -2.069661801608885e+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 7.5

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

    10. Simplified7.5

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

    if -2.069661801608885e+154 < re < -2.8034656734801727e-279 or 1.2241568427137281e-244 < re < 3.0367130689854715e+109

    1. Initial program 18.8

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

    if -2.8034656734801727e-279 < re < 1.2241568427137281e-244

    1. Initial program 28.2

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

    3. Applied add-exp-log30.7

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

    5. Applied pow130.7

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

    6. Applied log-pow30.7

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

    7. Applied exp-prod31.0

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

    8. Simplified31.0

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

    9. Taylor expanded around 0 31.6

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

    if 3.0367130689854715e+109 < re

    1. Initial program 49.0

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

    3. Applied add-exp-log50.1

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

    4. Taylor expanded around inf 8.7

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.069661801608885 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.8034656734801727 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.2241568427137281 \cdot 10^{-244}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.0367130689854715 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 6.9s)Debug logProfile

herbie shell --seed 2018346 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))