Average Error: 35.7 → 21.1
Time: 43.6s
Precision: 64
Internal Precision: 384
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.2462366172629023 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{if}\;x \le 1.8809713604828146 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original35.7
Target25.5
Herbie21.1
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{if}\;z \lt 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.2462366172629023e+154

    1. Initial program 59.5

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

    2. Taylor expanded around -inf 0

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

    3. Applied simplify0

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

    if -1.2462366172629023e+154 < x < 1.8809713604828146e+158

    1. Initial program 27.9

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

    if 1.8809713604828146e+158 < x

    1. Initial program 59.5

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

    2. Taylor expanded around inf 0

      \[\leadsto \color{blue}{x}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 43.6s)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"

  :herbie-target
  (if (< z -6.396479394109776e+136) (- z) (if (< z 7.320293694404182e+117) (sqrt (+ (+ (* z z) (* x x)) (* y y))) z))

  (sqrt (+ (+ (* x x) (* y y)) (* z z))))