Average Error: 0.2 → 0.2
Time: 44.3s
Precision: 64
Internal Precision: 320
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[(\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}\right) + \left(\left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right) + \left(a + 1\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right))_* - 1\]

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-sqr-sqrt0.2

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

  4. Applied unpow-prod-down0.2

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

  5. Applied fma-def0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Runtime

Time bar (total: 44.3s)Debug logProfile

herbie shell --seed 2018221 +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))