Average Error: 0.4 → 0.4
Time: 50.8s
Precision: 64
Internal Precision: 384
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[(\left(\sqrt[3]{a + b} \cdot \sqrt[3]{a + b}\right) \cdot \left(\sqrt[3]{a + b}\right) + \left(\frac{c + \left(d + e\right)}{1}\right))_*\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Target

Original0.4
Target0.2
Herbie0.4
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e\]

Derivation

  1. Initial program 0.4

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

  3. Applied flip-+0.5

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

  4. Taylor expanded around 0 0.5

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

  5. Applied simplify0.3

    \[\leadsto \color{blue}{\left(a + b\right) + \frac{c + \left(d + e\right)}{1}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt0.4

    \[\leadsto \color{blue}{\left(\sqrt[3]{a + b} \cdot \sqrt[3]{a + b}\right) \cdot \sqrt[3]{a + b}} + \frac{c + \left(d + e\right)}{1}\]

  8. Applied fma-def0.4

    \[\leadsto \color{blue}{(\left(\sqrt[3]{a + b} \cdot \sqrt[3]{a + b}\right) \cdot \left(\sqrt[3]{a + b}\right) + \left(\frac{c + \left(d + e\right)}{1}\right))_*}\]

Runtime

Time bar (total: 50.8s)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +o rules:numerics
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :pre (<= 1 a 2 b 4 c 8 d 16 e 32)

  :herbie-target
  (+ (+ d (+ c (+ a b))) e)

  (+ (+ (+ (+ e d) c) b) a))