Average Error: 45.2 → 21.1
Time: 48.3s
Precision: 64
Internal Precision: 2432
\[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -327746434365658.8:\\ \;\;\;\;\left((x \cdot y + z)_* - z\right) - \left(\sqrt[3]{1 + y \cdot x} \cdot \left(\sqrt[3]{\sqrt[3]{1 + y \cdot x} \cdot \sqrt[3]{1 + y \cdot x}} \cdot \sqrt[3]{\sqrt[3]{1 + y \cdot x}}\right)\right) \cdot \sqrt[3]{1 + y \cdot x}\\ \mathbf{if}\;z \le 3.6869688938574516 \cdot 10^{-06}:\\ \;\;\;\;\log \left(e^{\left((x \cdot y + z)_* - y \cdot x\right) - \left(z + 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left((x \cdot y + z)_* - z\right) - \frac{1}{\frac{\frac{-1}{y}}{\frac{x}{-1}}}\right) - \frac{\frac{1}{x}}{\frac{-1}{y}} \cdot \frac{\frac{1}{y}}{\frac{-1}{x}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original45.2
Target0
Herbie21.1
\[-1\]

Derivation

  1. Split input into 3 regimes

  2. if z < -327746434365658.8

    1. Initial program 62.1

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

    3. Applied add-cube-cbrt62.1

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

    4. Taylor expanded around 0 62.1

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

    5. Applied simplify32.3

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

    7. Applied add-cube-cbrt32.4

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

    9. Applied add-cube-cbrt32.4

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

    10. Applied cbrt-prod32.4

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

    if -327746434365658.8 < z < 3.6869688938574516e-06

    1. Initial program 29.5

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

    3. Applied add-cube-cbrt29.5

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

    4. Taylor expanded around 0 29.5

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

    5. Applied simplify29.5

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

    7. Applied add-cube-cbrt30.1

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

    9. Applied add-log-exp31.3

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

    10. Applied add-log-exp31.6

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

    11. Applied diff-log31.6

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

    12. Applied simplify8.7

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

    if 3.6869688938574516e-06 < z

    1. Initial program 59.8

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

    3. Applied add-cube-cbrt59.8

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

    4. Taylor expanded around 0 59.8

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

    5. Applied simplify32.5

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

    7. Applied add-cube-cbrt32.5

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

    8. Taylor expanded around -inf 62.8

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

    9. Applied simplify34.9

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

Runtime

Time bar (total: 48.3s)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1

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