Average Error: 29.3 → 16.3
Time: 3.3s
Precision: 64
Internal Precision: 128
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.475921366949707 \cdot 10^{+151}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 9.855353724210325 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -6.475921366949707e+151

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 8.4

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

    3. Simplified8.4

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

    if -6.475921366949707e+151 < re < 9.855353724210325e+145

    1. Initial program 19.3

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

    if 9.855353724210325e+145 < re

    1. Initial program 56.9

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

    2. Taylor expanded around inf 7.4

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.475921366949707 \cdot 10^{+151}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 9.855353724210325 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 3.3s)Debug logProfile

herbie shell --seed 2018273 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))