Average Error: 0.2 → 0.3
Time: 19.9s
Precision: 64
Internal Precision: 576
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.0009869567736607:\\ \;\;\;\;(4 \cdot \left((\left(1 - 3 \cdot a\right) \cdot \left(b \cdot b\right) + \left((\left(a \cdot a\right) \cdot a + \left(a \cdot a\right))_*\right))_*\right) + \left((2 \cdot \left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) + \left({b}^{4} + {a}^{4}\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(4 \cdot \left((\left(1 - 3 \cdot a\right) \cdot \left(b \cdot b\right) + \left(e^{\log \left((\left(a \cdot a\right) \cdot a + \left(a \cdot a\right))_*\right)}\right))_*\right) + \left((\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left((b \cdot b + \left(a \cdot a\right))_*\right) + \left(-1\right))_*\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Split input into 2 regimes

  2. if a < -1.0009869567736607

    1. Initial program 0.5

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

    2. Initial simplification0.5

      \[\leadsto (4 \cdot \left((\left(1 - 3 \cdot a\right) \cdot \left(b \cdot b\right) + \left((\left(a \cdot a\right) \cdot a + \left(a \cdot a\right))_*\right))_*\right) + \left((\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left((b \cdot b + \left(a \cdot a\right))_*\right) + \left(-1\right))_*\right))_*\]

    3. Taylor expanded around -inf 1.4

      \[\leadsto (4 \cdot \left((\left(1 - 3 \cdot a\right) \cdot \left(b \cdot b\right) + \left((\left(a \cdot a\right) \cdot a + \left(a \cdot a\right))_*\right))_*\right) + \color{blue}{\left({b}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right)\right)})_*\]

    4. Simplified1.4

      \[\leadsto (4 \cdot \left((\left(1 - 3 \cdot a\right) \cdot \left(b \cdot b\right) + \left((\left(a \cdot a\right) \cdot a + \left(a \cdot a\right))_*\right))_*\right) + \color{blue}{\left((2 \cdot \left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) + \left({a}^{4} + {b}^{4}\right))_*\right)})_*\]

    if -1.0009869567736607 < a

    1. Initial program 0.1

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

    2. Initial simplification0.2

      \[\leadsto (4 \cdot \left((\left(1 - 3 \cdot a\right) \cdot \left(b \cdot b\right) + \left((\left(a \cdot a\right) \cdot a + \left(a \cdot a\right))_*\right))_*\right) + \left((\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left((b \cdot b + \left(a \cdot a\right))_*\right) + \left(-1\right))_*\right))_*\]
    3. Using strategy rm

    4. Applied add-exp-log0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.0009869567736607:\\ \;\;\;\;(4 \cdot \left((\left(1 - 3 \cdot a\right) \cdot \left(b \cdot b\right) + \left((\left(a \cdot a\right) \cdot a + \left(a \cdot a\right))_*\right))_*\right) + \left((2 \cdot \left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) + \left({b}^{4} + {a}^{4}\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(4 \cdot \left((\left(1 - 3 \cdot a\right) \cdot \left(b \cdot b\right) + \left(e^{\log \left((\left(a \cdot a\right) \cdot a + \left(a \cdot a\right))_*\right)}\right))_*\right) + \left((\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left((b \cdot b + \left(a \cdot a\right))_*\right) + \left(-1\right))_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 19.9s)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))