\[\log \left(1 + e^{x}\right) - x \cdot y\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.040718665343158 \cdot 10^{-06}:\\ \;\;\;\;\log \left(\sqrt{1 + e^{x}}\right) + \left(\left(\log \left(\left|\sqrt[3]{1 + e^{x}}\right|\right) + \sqrt[3]{{\left(\log \left(\sqrt{\sqrt[3]{1 + e^{x}}}\right)\right)}^{3}}\right) - x \cdot y\right)\\ \mathbf{if}\;x \le 2.7308670765476417 \cdot 10^{-09}:\\ \;\;\;\;\log \left(\frac{1}{2} \cdot {x}^{2} + \left(2 + x\right)\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + e^{x}}\right) + \left(\left(\log \left(\left|\sqrt[3]{1 + e^{x}}\right|\right) + \sqrt[3]{{\left(\log \left(\sqrt{\sqrt[3]{1 + e^{x}}}\right)\right)}^{3}}\right) - x \cdot y\right)\\ \end{array}\]

Original	0.5
Target	0.1
Herbie	0.5

Derivation

Split input into 2 regimes
if x < -9.040718665343158e-06 or 2.7308670765476417e-09 < x
1. Initial program 1.7
  \[\log \left(1 + e^{x}\right) - x \cdot y\]
2. Using strategy rm
3. Applied add-sqr-sqrt1.7
  \[\leadsto \log \color{blue}{\left(\sqrt{1 + e^{x}} \cdot \sqrt{1 + e^{x}}\right)} - x \cdot y\]
4. Applied log-prod1.7
  \[\leadsto \color{blue}{\left(\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{1 + e^{x}}\right)\right)} - x \cdot y\]
5. Applied associate--l+1.7
  \[\leadsto \color{blue}{\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{1 + e^{x}}\right) - x \cdot y\right)}\]
6. Using strategy rm
7. Applied add-cube-cbrt1.7
  \[\leadsto \log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{\color{blue}{\left(\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}\right) \cdot \sqrt[3]{1 + e^{x}}}}\right) - x \cdot y\right)\]
8. Applied sqrt-prod1.7
  \[\leadsto \log \left(\sqrt{1 + e^{x}}\right) + \left(\log \color{blue}{\left(\sqrt{\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}} \cdot \sqrt{\sqrt[3]{1 + e^{x}}}\right)} - x \cdot y\right)\]
9. Applied log-prod1.7
  \[\leadsto \log \left(\sqrt{1 + e^{x}}\right) + \left(\color{blue}{\left(\log \left(\sqrt{\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}}\right) + \log \left(\sqrt{\sqrt[3]{1 + e^{x}}}\right)\right)} - x \cdot y\right)\]
10. Applied simplify1.7
  \[\leadsto \log \left(\sqrt{1 + e^{x}}\right) + \left(\left(\color{blue}{\log \left(\left|\sqrt[3]{1 + e^{x}}\right|\right)} + \log \left(\sqrt{\sqrt[3]{1 + e^{x}}}\right)\right) - x \cdot y\right)\]
11. Using strategy rm
12. Applied add-cbrt-cube1.7
  \[\leadsto \log \left(\sqrt{1 + e^{x}}\right) + \left(\left(\log \left(\left|\sqrt[3]{1 + e^{x}}\right|\right) + \color{blue}{\sqrt[3]{\left(\log \left(\sqrt{\sqrt[3]{1 + e^{x}}}\right) \cdot \log \left(\sqrt{\sqrt[3]{1 + e^{x}}}\right)\right) \cdot \log \left(\sqrt{\sqrt[3]{1 + e^{x}}}\right)}}\right) - x \cdot y\right)\]
13. Applied simplify1.7
  \[\leadsto \log \left(\sqrt{1 + e^{x}}\right) + \left(\left(\log \left(\left|\sqrt[3]{1 + e^{x}}\right|\right) + \sqrt[3]{\color{blue}{{\left(\log \left(\sqrt{\sqrt[3]{1 + e^{x}}}\right)\right)}^{3}}}\right) - x \cdot y\right)\]
if -9.040718665343158e-06 < x < 2.7308670765476417e-09
1. Initial program 0.0
  \[\log \left(1 + e^{x}\right) - x \cdot y\]
2. Taylor expanded around 0 0.0
  \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + \left(2 + x\right)\right)} - x \cdot y\]
Recombined 2 regimes into one program.

Error

Target

Derivation

`if x < -9.040718665343158e-06 or 2.7308670765476417e-09 < x`

`if -9.040718665343158e-06 < x < 2.7308670765476417e-09`

Runtime