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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot a \le 4.801699443824738 \cdot 10^{-27}:\\ \;\;\;\;(\left(4 \cdot b\right) \cdot b + \left({b}^{4} - 1\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left((\left(2 \cdot a\right) \cdot a + 4)_*\right) \cdot \left(b \cdot b\right) + \left({a}^{4} - 1\right))_*\\ \end{array}\]

Derivation

Split input into 2 regimes
if (* a a) < 4.801699443824738e-27
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 0.0
  \[\leadsto \left(\color{blue}{{b}^{4}} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
3. Applied simplify0.0
  \[\leadsto \color{blue}{(\left(4 \cdot b\right) \cdot b + \left({b}^{4} - 1\right))_*}\]
if 4.801699443824738e-27 < (* a a)
1. Initial program 0.4
  \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
2. Taylor expanded around 0 7.0
  \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left({a}^{4} + 2 \cdot \left({b}^{2} \cdot {a}^{2}\right)\right)\right)} - 1\]
3. Applied simplify7.0
  \[\leadsto \color{blue}{(\left((\left(2 \cdot a\right) \cdot a + 4)_*\right) \cdot \left(b \cdot b\right) + \left({a}^{4} - 1\right))_*}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))

Error

Derivation

`if (* a a) < 4.801699443824738e-27`

`if 4.801699443824738e-27 < (* a a)`

Runtime