Average Error: 29.5 → 17.3
Time: 3.5s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -8.569875015320511 \cdot 10^{+160}:\\ \;\;\;\;re\\ \mathbf{if}\;-re \le -6.776184305828227 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;-re \le 9.93053661753333 \cdot 10^{-286}:\\ \;\;\;\;im\\ \mathbf{if}\;-re \le 5.962744466236316 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

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) < -8.569875015320511e+160

    1. Initial program 59.3

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

    2. Taylor expanded around inf 7.4

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

    if -8.569875015320511e+160 < (- re) < -6.776184305828227e-289 or 9.93053661753333e-286 < (- re) < 5.962744466236316e+114

    1. Initial program 20.0

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

    if -6.776184305828227e-289 < (- re) < 9.93053661753333e-286

    1. Initial program 27.9

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

    2. Taylor expanded around 0 34.7

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

    if 5.962744466236316e+114 < (- re)

    1. Initial program 50.4

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

    2. Taylor expanded around -inf 8.2

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

    3. Applied simplify8.2

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

Runtime

Time bar (total: 3.5s)Debug logProfile

herbie shell --seed 2018166 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))