Average Error: 0.2 → 0.0
Time: 47.0s
Precision: 64
Internal Precision: 384
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \le 8379299926568.48:\\ \;\;\;\;\left(\sqrt[3]{{\left((b \cdot b + \left(a \cdot a\right))_*\right)}^{3} \cdot {\left((b \cdot b + \left(a \cdot a\right))_*\right)}^{3}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;(4 \cdot \left((\left(a - a \cdot a\right) \cdot a + \left(\left(3 + a\right) \cdot \left(b \cdot b\right)\right))_*\right) + \left((\left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) \cdot 2 + \left({b}^{4} + {a}^{4}\right))_*\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Split input into 2 regimes

  2. if (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1) < 8379299926568.48

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube0.0

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

    4. Applied simplify0.0

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

    if 8379299926568.48 < (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1)

    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. Taylor expanded around 0 0.5

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

    3. Applied simplify0.5

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

    4. Taylor expanded around inf 0.1

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

    5. Applied simplify0.1

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

Runtime

Time bar (total: 47.0s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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))