Average Error: 0.2 → 0.2
Time: 1.2m
Precision: 64
Internal Precision: 320
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\sqrt[3]{\frac{1 - a \cdot a}{\frac{a + 1}{a \cdot a}}} \cdot \sqrt[3]{\frac{1 - a \cdot a}{\frac{a + 1}{a \cdot a}}}\right) \cdot \sqrt[3]{\frac{1 - a \cdot a}{\frac{a + 1}{a \cdot a}}} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  3. Applied flip--0.2

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

  4. Applied associate-*r/0.2

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

  6. Applied add-cube-cbrt0.2

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

  7. Applied simplify0.2

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

  8. Applied simplify0.2

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))