Average Error: 3.6 → 0
Time: 3.4s
Precision: 64
Internal Precision: 128
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[2 \cdot \left(\left(d + a\right) + \left(c + b\right)\right)\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original3.6
Target3.8
Herbie0
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Derivation

  1. Initial program 3.6

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

  3. Applied associate-+r+2.8

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

  5. Applied +-commutative2.8

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

  7. Applied associate-+l+0

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

  8. Final simplification0

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(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))