Average Error: 29.1 → 16.7
Time: 2.5s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -3.1161956129599666 \cdot 10^{+136}:\\ \;\;\;\;re\\ \mathbf{elif}\;-re \le -3.745115008526117 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;-re \le -7.688320651497413 \cdot 10^{-286}:\\ \;\;\;\;im\\ \mathbf{elif}\;-re \le 2.47360553936312 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;-re \le 8.263368924409109 \cdot 10^{-182}:\\ \;\;\;\;im\\ \mathbf{elif}\;-re \le 3.251636425894959 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

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) < -3.1161956129599666e+136

    1. Initial program 54.4

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

    2. Taylor expanded around inf 7.5

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

    if -3.1161956129599666e+136 < (- re) < -3.745115008526117e-236 or -7.688320651497413e-286 < (- re) < 2.47360553936312e-268 or 8.263368924409109e-182 < (- re) < 3.251636425894959e+153

    1. Initial program 17.3

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

    if -3.745115008526117e-236 < (- re) < -7.688320651497413e-286 or 2.47360553936312e-268 < (- re) < 8.263368924409109e-182

    1. Initial program 30.6

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

    2. Taylor expanded around 0 36.0

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

    if 3.251636425894959e+153 < (- re)

    1. Initial program 59.1

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

    2. Taylor expanded around -inf 6.5

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

    3. Simplified6.5

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

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;-re \le -3.1161956129599666 \cdot 10^{+136}:\\ \;\;\;\;re\\ \mathbf{elif}\;-re \le -3.745115008526117 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;-re \le -7.688320651497413 \cdot 10^{-286}:\\ \;\;\;\;im\\ \mathbf{elif}\;-re \le 2.47360553936312 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;-re \le 8.263368924409109 \cdot 10^{-182}:\\ \;\;\;\;im\\ \mathbf{elif}\;-re \le 3.251636425894959 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

Runtime

Time bar (total: 2.5s)Debug logProfile

herbie shell --seed 2018216 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))