Average Error: 29.7 → 17.1
Time: 32.4s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.702493838381082 \cdot 10^{+139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -5.063359694018908 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 4.6011664423220654 \cdot 10^{-185}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \le 1.777647936367165 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 < -7.702493838381082e+139

    1. Initial program 55.8

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

    2. Taylor expanded around -inf 7.9

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

    3. Simplified7.9

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

    if -7.702493838381082e+139 < re < -5.063359694018908e-273 or 4.6011664423220654e-185 < re < 1.777647936367165e+126

    1. Initial program 17.9

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

    if -5.063359694018908e-273 < re < 4.6011664423220654e-185

    1. Initial program 30.0

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

    3. Applied add-cube-cbrt30.5

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}}\]

    4. Applied sqrt-prod30.5

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\]

    5. Simplified30.5

      \[\leadsto \color{blue}{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\]

    6. Taylor expanded around -inf 33.5

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

    7. Simplified33.5

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

    if 1.777647936367165e+126 < re

    1. Initial program 52.4

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

    2. Taylor expanded around inf 8.3

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.702493838381082 \cdot 10^{+139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -5.063359694018908 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 4.6011664423220654 \cdot 10^{-185}:\\ \;\;\;\;-im\\ \mathbf{elif}\;re \le 1.777647936367165 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 32.4s)Debug logProfile

herbie shell --seed 2018214 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))