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Derivation

1. Initial program 3.7

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]

2. Initial simplification3.1

   \[\leadsto 2 \cdot \left(\left(a + c\right) + \left(b + d\right)\right)\]
3. Using strategy rm

4. Applied associate-+r+3.1

   \[\leadsto 2 \cdot \color{blue}{\left(\left(\left(a + c\right) + b\right) + d\right)}\]

5. Taylor expanded around 0 2.8

   \[\leadsto 2 \cdot \left(\color{blue}{\left(a + \left(b + c\right)\right)} + d\right)\]
6. Using strategy rm

7. Applied add-cbrt-cube2.9

   \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{\left(\left(\left(a + \left(b + c\right)\right) + d\right) \cdot \left(\left(a + \left(b + c\right)\right) + d\right)\right) \cdot \left(\left(a + \left(b + c\right)\right) + d\right)}}\]
8. Using strategy rm

9. Applied flip-+3.2

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

10. Applied flip-+3.2

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

11. Applied flip-+3.4

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

12. Applied frac-times3.4

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

13. Applied frac-times3.4

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

14. Applied cbrt-div3.4

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

15. Simplified0.6

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

16. Final simplification0.6

    \[\leadsto 2 \cdot \frac{\left(\left(b + a\right) + \left(c - d\right)\right) \cdot \left(\left(b + c\right) + \left(d + a\right)\right)}{\sqrt[3]{\left(\left(\left(\left(b + c\right) + a\right) - d\right) \cdot \left(\left(\left(b + c\right) + a\right) - d\right)\right) \cdot \left(\left(\left(b + c\right) + a\right) - d\right)}}\]

Runtime

Time bar (total: 29.7s)Debug logProfile

herbie shell --seed 2018304 
(FPCore (a b c d)
  :name "Expression, p6"
  :pre (and (<= -14 a -13) (<= -3 b -2) (<= 3 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2) (* (+ c d) 2))

  (* (+ a (+ b (+ c d))) 2))