Average Error: 0.2 → 0.2
Time: 1.8m
Precision: 64
Internal Precision: 320
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[(\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left(\sqrt{b^2 + a^2}^* \cdot \sqrt{b^2 + a^2}^*\right) + \left((\left((a \cdot \left((a \cdot a + a)_*\right) + \left(b \cdot b\right))_*\right) \cdot 4 + \left(-1\right))_*\right))_*\]

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-cbrt0.7

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

  4. Applied unpow-prod-down0.7

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

  5. Applied simplify0.5

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

  6. Taylor expanded around 0 0.5

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

  7. Applied simplify0.2

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

  9. Applied add-sqr-sqrt0.2

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

  10. Applied simplify0.2

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

  11. Applied simplify0.2

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018206 +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))