Average Error: 29.5 → 17.5
Time: 14.9s
Precision: 64
Internal Precision: 384
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -1.8881957746629162 \cdot 10^{+157}:\\ \;\;\;\;re\\ \mathbf{if}\;-re \le 6.843876812000925 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;-re \le 4.41847917226255 \cdot 10^{-179}:\\ \;\;\;\;-re\\ \mathbf{if}\;-re \le 2.490683560060314 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if (- re) < -1.8881957746629162e+157

    1. Initial program 59.5

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

    2. Taylor expanded around inf 6.3

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

    if -1.8881957746629162e+157 < (- re) < 6.843876812000925e-244 or 4.41847917226255e-179 < (- re) < 2.490683560060314e+130

    1. Initial program 18.5

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

    if 6.843876812000925e-244 < (- re) < 4.41847917226255e-179 or 2.490683560060314e+130 < (- re)

    1. Initial program 48.0

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

    2. Taylor expanded around -inf 21.1

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

    3. Applied simplify21.1

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

Runtime

Time bar (total: 14.9s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))