Average Error: 29.8 → 17.2
Time: 9.4s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -5.993473697805842 \cdot 10^{+159}:\\ \;\;\;\;re\\ \mathbf{if}\;-re \le -1.6128607142744344 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;-re \le 1.808438267145418 \cdot 10^{-286}:\\ \;\;\;\;im\\ \mathbf{if}\;-re \le 3.723451359700138 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

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) < -5.993473697805842e+159

    1. Initial program 59.3

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

    2. Taylor expanded around inf 7.2

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

    if -5.993473697805842e+159 < (- re) < -1.6128607142744344e-264 or 1.808438267145418e-286 < (- re) < 3.723451359700138e+108

    1. Initial program 19.4

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

    if -1.6128607142744344e-264 < (- re) < 1.808438267145418e-286

    1. Initial program 29.7

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

    2. Taylor expanded around 0 33.6

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

    if 3.723451359700138e+108 < (- re)

    1. Initial program 49.6

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

    2. Taylor expanded around -inf 10.2

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

    3. Applied simplify10.2

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

Runtime

Time bar (total: 9.4s)Debug logProfile

herbie shell --seed 2018167 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))