\[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -72609.22878773515:\\ \;\;\;\;\log \left(e^{\left(\sqrt[3]{\left(\left((x \cdot y + z)_* - z\right) - 1\right) - y \cdot x} \cdot \sqrt[3]{\left(\left((x \cdot y + z)_* - z\right) - 1\right) - y \cdot x}\right) \cdot \sqrt[3]{\left(\left((x \cdot y + z)_* - z\right) - 1\right) - y \cdot x}}\right)\\ \mathbf{if}\;z \le 681227749634197.2:\\ \;\;\;\;\left((x \cdot y + z)_* - y \cdot x\right) - \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\sqrt[3]{\left(\left((x \cdot y + z)_* - z\right) - 1\right) - y \cdot x} \cdot \sqrt[3]{\left(\left((x \cdot y + z)_* - z\right) - 1\right) - y \cdot x}\right) \cdot \sqrt[3]{\left(\left((x \cdot y + z)_* - z\right) - 1\right) - y \cdot x}}\right)\\ \end{array}\]

Original	45.1
Target	0
Herbie	20.1

Derivation

Split input into 2 regimes
if z < -72609.22878773515 or 681227749634197.2 < z
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-log-exp62.9
  \[\leadsto (x \cdot y + z)_* - \color{blue}{\log \left(e^{1 + \left(x \cdot y + z\right)}\right)}\]
4. Applied add-log-exp63.6
  \[\leadsto \color{blue}{\log \left(e^{(x \cdot y + z)_*}\right)} - \log \left(e^{1 + \left(x \cdot y + z\right)}\right)\]
5. Applied diff-log63.6
  \[\leadsto \color{blue}{\log \left(\frac{e^{(x \cdot y + z)_*}}{e^{1 + \left(x \cdot y + z\right)}}\right)}\]
6. Applied simplify61.5
  \[\leadsto \log \color{blue}{\left(e^{\left((x \cdot y + z)_* - \left(z + 1\right)\right) - y \cdot x}\right)}\]
7. Using strategy rm
8. Applied associate--r+31.4
  \[\leadsto \log \left(e^{\color{blue}{\left(\left((x \cdot y + z)_* - z\right) - 1\right)} - y \cdot x}\right)\]
9. Using strategy rm
10. Applied add-cube-cbrt31.4
  \[\leadsto \log \left(e^{\color{blue}{\left(\sqrt[3]{\left(\left((x \cdot y + z)_* - z\right) - 1\right) - y \cdot x} \cdot \sqrt[3]{\left(\left((x \cdot y + z)_* - z\right) - 1\right) - y \cdot x}\right) \cdot \sqrt[3]{\left(\left((x \cdot y + z)_* - z\right) - 1\right) - y \cdot x}}}\right)\]
if -72609.22878773515 < z < 681227749634197.2
1. Initial program 29.7
  \[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
2. Using strategy rm
3. Applied add-log-exp31.9
  \[\leadsto (x \cdot y + z)_* - \color{blue}{\log \left(e^{1 + \left(x \cdot y + z\right)}\right)}\]
4. Applied add-log-exp32.2
  \[\leadsto \color{blue}{\log \left(e^{(x \cdot y + z)_*}\right)} - \log \left(e^{1 + \left(x \cdot y + z\right)}\right)\]
5. Applied diff-log32.2
  \[\leadsto \color{blue}{\log \left(\frac{e^{(x \cdot y + z)_*}}{e^{1 + \left(x \cdot y + z\right)}}\right)}\]
6. Applied simplify29.7
  \[\leadsto \log \color{blue}{\left(e^{\left((x \cdot y + z)_* - \left(z + 1\right)\right) - y \cdot x}\right)}\]
7. Taylor expanded around 0 29.7
  \[\leadsto \log \left(e^{\color{blue}{(x \cdot y + z)_* - \left(z + \left(1 + y \cdot x\right)\right)}}\right)\]
8. Applied simplify9.5
  \[\leadsto \color{blue}{\left((x \cdot y + z)_* - y \cdot x\right) - \left(z + 1\right)}\]
Recombined 2 regimes into one program.

Error

Target

Derivation

`if z < -72609.22878773515 or 681227749634197.2 < z`

`if -72609.22878773515 < z < 681227749634197.2`

Runtime