Average Error: 29.6 → 17.8
Time: 9.1s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.0300476806324841 \cdot 10^{+128}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.1623795656049374 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 5.321773148611292 \cdot 10^{-160}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.5761557902101205 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

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 < -1.0300476806324841e+128

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 8.5

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

    3. Simplified8.5

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

    if -1.0300476806324841e+128 < re < -2.1623795656049374e-236 or 5.321773148611292e-160 < re < 2.5761557902101205e+86

    1. Initial program 17.1

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

    if -2.1623795656049374e-236 < re < 5.321773148611292e-160

    1. Initial program 29.5

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

    2. Taylor expanded around 0 33.9

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

    if 2.5761557902101205e+86 < re

    1. Initial program 44.3

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

    2. Taylor expanded around inf 10.7

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.0300476806324841 \cdot 10^{+128}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.1623795656049374 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 5.321773148611292 \cdot 10^{-160}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.5761557902101205 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 9.1s)Debug logProfile

herbie shell --seed 2018219 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))