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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

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

4. Applied associate-+r+2.8

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

6. Applied add-cbrt-cube2.9

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

8. Applied flip-+3.0

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

9. Applied flip-+3.1

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

10. Applied flip-+3.2

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

11. Applied frac-times3.2

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

12. Applied frac-times3.2

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

13. Simplified0.9

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

14. Simplified0.9

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

15. Final simplification0.9

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018217 
(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))