Average Error: 30.1 → 18.0
Time: 12.4s
Precision: 64
Internal Precision: 384
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -1.1502505807603953 \cdot 10^{+91}:\\ \;\;\;\;re\\ \mathbf{if}\;-re \le -1.2199555271435537 \cdot 10^{-214}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;-re \le 1.1673529293673539 \cdot 10^{-240}:\\ \;\;\;\;im\\ \mathbf{if}\;-re \le 2.1648414854408086 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -1.1502505807603953e+91

    1. Initial program 45.8

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

    2. Taylor expanded around inf 11.1

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

    if -1.1502505807603953e+91 < (- re) < -1.2199555271435537e-214 or 1.1673529293673539e-240 < (- re) < 2.1648414854408086e+147

    1. Initial program 18.7

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

    if -1.2199555271435537e-214 < (- re) < 1.1673529293673539e-240

    1. Initial program 29.2

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

    2. Taylor expanded around 0 34.6

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

    if 2.1648414854408086e+147 < (- re)

    1. Initial program 57.9

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

    2. Taylor expanded around -inf 7.8

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

    3. Applied simplify7.8

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

Runtime

Time bar (total: 12.4s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))