Average Error: 29.3 → 17.6
Time: 11.1s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -1.8434839462812739 \cdot 10^{+140}:\\ \;\;\;\;re\\ \mathbf{if}\;-re \le -4.158026975480099 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;-re \le -8.601110130196432 \cdot 10^{-247}:\\ \;\;\;\;im\\ \mathbf{if}\;-re \le 7.498172526912422 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

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) < -1.8434839462812739e+140

    1. Initial program 56.4

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

    2. Taylor expanded around inf 8.7

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

    if -1.8434839462812739e+140 < (- re) < -4.158026975480099e-184 or -8.601110130196432e-247 < (- re) < 7.498172526912422e+177

    1. Initial program 19.6

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

    if -4.158026975480099e-184 < (- re) < -8.601110130196432e-247

    1. Initial program 30.3

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

    2. Taylor expanded around 0 35.5

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

    if 7.498172526912422e+177 < (- re)

    1. Initial program 59.2

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

    2. Taylor expanded around -inf 6.8

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

    3. Applied simplify6.8

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

Runtime

Time bar (total: 11.1s)Debug logProfile

herbie shell --seed 2018206 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))