Average Error: 0.2 → 1.4
Time: 1.0m
Precision: 64
Internal Precision: 576
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -145.74288793026716 \lor \neg \left(a \le 0.055311120503756486\right):\\ \;\;\;\;(\left((\left(2 \cdot a\right) \cdot a + 4)_*\right) \cdot \left(b \cdot b\right) + \left({a}^{4} - 1\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(4 \cdot b\right) \cdot b + \left({b}^{4} - 1\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Split input into 2 regimes

  2. if a < -145.74288793026716 or 0.055311120503756486 < a

    1. Initial program 0.5

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

    2. Taylor expanded around 0 6.1

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left({a}^{4} + 2 \cdot \left({b}^{2} \cdot {a}^{2}\right)\right)\right)} - 1\]

    3. Applied simplify6.1

      \[\leadsto \color{blue}{(\left((\left(2 \cdot a\right) \cdot a + 4)_*\right) \cdot \left(b \cdot b\right) + \left({a}^{4} - 1\right))_*}\]

    if -145.74288793026716 < a < 0.055311120503756486

    1. Initial program 0.1

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

    2. Taylor expanded around 0 0.3

      \[\leadsto \left(\color{blue}{{b}^{4}} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

    3. Applied simplify0.3

      \[\leadsto \color{blue}{(\left(4 \cdot b\right) \cdot b + \left({b}^{4} - 1\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify1.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;a \le -145.74288793026716 \lor \neg \left(a \le 0.055311120503756486\right):\\ \;\;\;\;(\left((\left(2 \cdot a\right) \cdot a + 4)_*\right) \cdot \left(b \cdot b\right) + \left({a}^{4} - 1\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(4 \cdot b\right) \cdot b + \left({b}^{4} - 1\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018207 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))