Average Error: 45.3 → 19.9
Time: 29.0s
Precision: 64
Internal Precision: 2432
\[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -757376874.9301252:\\ \;\;\;\;\left(\sqrt[3]{(x \cdot y + z)_* - z} \cdot \sqrt[3]{(x \cdot y + z)_* - z}\right) \cdot \sqrt[3]{(x \cdot y + z)_* - z} - \left(1 + y \cdot x\right)\\ \mathbf{if}\;z \le 4.290458252343054 \cdot 10^{+18}:\\ \;\;\;\;\left((x \cdot y + z)_* - y \cdot x\right) - \left(z + 1\right)\\ \mathbf{if}\;z \le +\infty:\\ \;\;\;\;\left(\sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)} \cdot \sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)}\right) \cdot \sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)} \cdot \sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)}\right) \cdot \sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original45.3
Target0
Herbie19.9
\[-1\]

Derivation

  1. Split input into 3 regimes

  2. if z < -757376874.9301252

    1. Initial program 61.4

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

    3. Applied add-cube-cbrt61.4

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

    4. Taylor expanded around 0 61.4

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

    5. Applied simplify31.0

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

    7. Applied add-cube-cbrt31.0

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

    if -757376874.9301252 < z < 4.290458252343054e+18

    1. Initial program 30.3

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

    3. Applied add-cube-cbrt30.3

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

    4. Taylor expanded around 0 30.3

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

    5. Applied simplify30.2

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

    7. Applied add-cube-cbrt30.2

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

    8. Taylor expanded around 0 30.3

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

    9. Applied simplify9.6

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

    if 4.290458252343054e+18 < z

    1. Initial program 62.2

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

    3. Applied add-cube-cbrt62.2

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

    4. Taylor expanded around 0 62.2

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

    5. Applied simplify31.2

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

    7. Applied add-cube-cbrt31.2

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)} \cdot \sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)}\right) \cdot \sqrt[3]{\left((x \cdot y + z)_* - z\right) - \left(1 + y \cdot x\right)}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 29.0s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1

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