Average Error: 0.2 → 0.2
Time: 3.5m
Precision: 64
Internal Precision: 384
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\sqrt[3]{\left(a \cdot a\right) \cdot \left(1 + a\right)} \cdot \sqrt[3]{\left(a \cdot a\right) \cdot \left(1 + a\right)}\right) \cdot \sqrt[3]{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  3. Applied add-cube-cbrt0.2

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

Runtime

Time bar (total: 3.5m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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))