Average Error: 45.5 → 30.8
Time: 1.0m
Precision: 64
Internal Precision: 2432
\[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\left(\sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)} \cdot \sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)}\right) \cdot \sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original45.5
Target0
Herbie30.8
\[-1\]

Derivation

  1. Initial program 45.5

    \[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt45.5

    \[\leadsto \color{blue}{\left(\sqrt[3]{(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)} \cdot \sqrt[3]{(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)}\right) \cdot \sqrt[3]{(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)}}\]

  4. Taylor expanded around 0 45.5

    \[\leadsto \left(\sqrt[3]{(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)} \cdot \sqrt[3]{\color{blue}{(x \cdot y + z)_* - \left(z + \left(1 + y \cdot x\right)\right)}}\right) \cdot \sqrt[3]{(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)}\]

  5. Applied simplify30.8

    \[\leadsto \color{blue}{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt30.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)} \cdot \sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)}\right) \cdot \sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)}}\]
  8. Removed slow pow expressions.

Runtime

Time bar (total: 1.0m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1

  (- (fma x y z) (+ 1 (+ (* x y) z))))