Average Error: 30.0 → 17.2
Time: 13.2s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -8.626721421935331 \cdot 10^{+122}:\\ \;\;\;\;re\\ \mathbf{if}\;-re \le -2.746672438769142 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;-re \le -5.438961245257397 \cdot 10^{-277}:\\ \;\;\;\;im\\ \mathbf{if}\;-re \le 6.907662975370443 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

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) < -8.626721421935331e+122

    1. Initial program 51.3

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

    2. Taylor expanded around inf 8.5

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

    if -8.626721421935331e+122 < (- re) < -2.746672438769142e-253 or -5.438961245257397e-277 < (- re) < 6.907662975370443e+142

    1. Initial program 20.1

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

    if -2.746672438769142e-253 < (- re) < -5.438961245257397e-277

    1. Initial program 34.1

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

    2. Taylor expanded around 0 32.8

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

    if 6.907662975370443e+142 < (- re)

    1. Initial program 56.1

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

    2. Taylor expanded around -inf 9.3

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

    3. Applied simplify9.3

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

Runtime

Time bar (total: 13.2s)Debug logProfile

herbie shell --seed 2018199 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))