Average Error: 0.2 → 0.4
Time: 22.6s
Precision: 64
Internal Precision: 576
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -0.963636567382646:\\ \;\;\;\;(\left((\left(b \cdot \left(a + 3\right)\right) \cdot b + \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right))_*\right) \cdot 4 + \left((2 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right))_*\right))_*\\ \mathbf{elif}\;a \le 1.491133146671688 \cdot 10^{-06}:\\ \;\;\;\;(\left((\left(b \cdot \left(a + 3\right)\right) \cdot b + \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right))_*\right) \cdot 4 + \left({b}^{4} + -1\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\left((b \cdot b + a)_* - a \cdot a\right) \cdot a\right) \cdot 4 + \left((\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left((b \cdot b + \left(a \cdot a\right))_*\right) + -1)_*\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Split input into 3 regimes

  2. if a < -0.963636567382646

    1. Initial program 0.5

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

    2. Initial simplification0.5

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

    3. Taylor expanded around inf 1.3

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

    4. Simplified1.3

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

    if -0.963636567382646 < a < 1.491133146671688e-06

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied expm1-log1p-u1.1

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

    5. Taylor expanded around -inf 31.4

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

    6. Simplified0.2

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

    if 1.491133146671688e-06 < a

    1. Initial program 0.5

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

    2. Initial simplification0.5

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

    3. Taylor expanded around inf 0.6

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

    4. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -0.963636567382646:\\ \;\;\;\;(\left((\left(b \cdot \left(a + 3\right)\right) \cdot b + \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right))_*\right) \cdot 4 + \left((2 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right))_*\right))_*\\ \mathbf{elif}\;a \le 1.491133146671688 \cdot 10^{-06}:\\ \;\;\;\;(\left((\left(b \cdot \left(a + 3\right)\right) \cdot b + \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right))_*\right) \cdot 4 + \left({b}^{4} + -1\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\left((b \cdot b + a)_* - a \cdot a\right) \cdot a\right) \cdot 4 + \left((\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left((b \cdot b + \left(a \cdot a\right))_*\right) + -1)_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 22.6s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))