Average Error: 30.0 → 17.5
Time: 5.3s
Precision: 64
Internal Precision: 320
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -1.0534237471986007 \cdot 10^{+144}:\\ \;\;\;\;re\\ \mathbf{if}\;-re \le -7.855272437351815 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;-re \le 1.8009350273539524 \cdot 10^{-255}:\\ \;\;\;\;im\\ \mathbf{if}\;-re \le 6.158089157670478 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;-re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -1.0534237471986007e+144

    1. Initial program 57.1

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

    2. Taylor expanded around inf 8.6

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

    if -1.0534237471986007e+144 < (- re) < -7.855272437351815e-200 or 1.8009350273539524e-255 < (- re) < 6.158089157670478e+121

    1. Initial program 18.0

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

    if -7.855272437351815e-200 < (- re) < 1.8009350273539524e-255

    1. Initial program 29.6

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

    2. Taylor expanded around 0 32.8

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

    if 6.158089157670478e+121 < (- re)

    1. Initial program 51.7

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

    2. Taylor expanded around -inf 9.6

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

    3. Applied simplify9.6

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

Runtime

Time bar (total: 5.3s)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))