Average Error: 0.2 → 0.6
Time: 19.2s
Precision: 64
Internal Precision: 576
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[(\left((b \cdot b + \left(a \cdot a\right))_* \cdot \sqrt[3]{(b \cdot b + \left(a \cdot a\right))_*}\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{a \cdot a + b \cdot b}}\right)}^{2}\right) + \left(\left(b \cdot b\right) \cdot 4\right))_* - 1\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  3. Applied add-cube-cbrt0.7

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

  4. Applied unpow-prod-down0.7

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

  5. Applied fma-def0.7

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

  6. Simplified0.5

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

  8. Applied add-cube-cbrt0.5

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

  9. Applied cbrt-prod0.6

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

  10. Applied unpow-prod-down0.6

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

  11. Final simplification0.6

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

Runtime

Time bar (total: 19.2s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))