Average Error: 29.1 → 18.0
Time: 2.5s
Precision: 64
Internal Precision: 128
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.7204358706938567 \cdot 10^{+149}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -6.947967300568486 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -2.4458756963213023 \cdot 10^{-235}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.3311994961664106 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 4.651686048724964 \cdot 10^{-195}:\\ \;\;\;\;re\\ \mathbf{elif}\;re \le 9.552953101375515 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -1.7204358706938567e+149 or -6.947967300568486e-177 < re < -2.4458756963213023e-235

    1. Initial program 49.5

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

    2. Taylor expanded around -inf 20.6

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

    3. Simplified20.6

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

    if -1.7204358706938567e+149 < re < -6.947967300568486e-177 or -2.4458756963213023e-235 < re < 2.3311994961664106e-215 or 4.651686048724964e-195 < re < 9.552953101375515e+122

    1. Initial program 18.7

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

    if 2.3311994961664106e-215 < re < 4.651686048724964e-195 or 9.552953101375515e+122 < re

    1. Initial program 48.8

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

    2. Taylor expanded around inf 12.2

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.7204358706938567 \cdot 10^{+149}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -6.947967300568486 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -2.4458756963213023 \cdot 10^{-235}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.3311994961664106 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 4.651686048724964 \cdot 10^{-195}:\\ \;\;\;\;re\\ \mathbf{elif}\;re \le 9.552953101375515 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))