Average Error: 0.2 → 1.8
Time: 55.3s
Precision: 64
Internal Precision: 384
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -264984.8036273524:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{if}\;b \le 6.423519984539504 \cdot 10^{-06}:\\ \;\;\;\;\left(4 \cdot {b}^{2} + \left({a}^{4} + 2 \cdot \left({b}^{2} \cdot {a}^{2}\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Split input into 2 regimes

  2. if b < -264984.8036273524 or 6.423519984539504e-06 < b

    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 6.8

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

    if -264984.8036273524 < b < 6.423519984539504e-06

    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 0.6

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

Runtime

Time bar (total: 55.3s)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))