Average Error: 0.2 → 0.0
Time: 1.7m
Precision: 64
Internal Precision: 576
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left((\left((a \cdot 4 + 4)_* \cdot a\right) \cdot a + \left({\left(\sqrt{a^2 + b^2}^*\right)}^{\left(1 + 3\right)}\right))_* + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

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-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. Using strategy rm

  7. Applied fma-udef0.2

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

  8. Applied simplify0.0

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

  10. Applied distribute-lft-in0.0

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

  11. Applied associate-+r+0.0

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

  12. Applied simplify0.0

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2020178 +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))