Average Error: 29.7 → 17.4
Time: 29.5s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -3.0346774368399296 \cdot 10^{+128}:\\ \;\;\;\;re\\ \mathbf{if}\;-re \le 3.885765596894332 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;-re \le 1.1734262626490164 \cdot 10^{-228}:\\ \;\;\;\;im\\ \mathbf{if}\;-re \le 8.778892043483209 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

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) < -3.0346774368399296e+128

    1. Initial program 53.5

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

    2. Taylor expanded around inf 8.9

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

    if -3.0346774368399296e+128 < (- re) < 3.885765596894332e-285 or 1.1734262626490164e-228 < (- re) < 8.778892043483209e+92

    1. Initial program 19.7

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

    if 3.885765596894332e-285 < (- re) < 1.1734262626490164e-228

    1. Initial program 29.2

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

    2. Taylor expanded around 0 35.9

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

    if 8.778892043483209e+92 < (- re)

    1. Initial program 46.1

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

    2. Taylor expanded around -inf 11.3

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

    3. Applied simplify11.3

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

Runtime

Time bar (total: 29.5s)Debug logProfile

herbie shell --seed 2018178 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))