Derivation

Split input into 4 regimes
if b < -9.153348824044421e-243
1. Initial program 11.4
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Using strategy rm
3. Applied sub-neg11.4
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(c \cdot t + \left(-i \cdot y\right)\right)}\]
4. Applied distribute-lft-in11.4
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)}\]
5. Taylor expanded around -inf 11.7
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{t \cdot \left(j \cdot c\right)} + j \cdot \left(-i \cdot y\right)\right)\]
6. Using strategy rm
7. Applied associate-*r*11.2
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(t \cdot j\right) \cdot c} + j \cdot \left(-i \cdot y\right)\right)\]
8. Using strategy rm
9. Applied add-cube-cbrt11.5
  \[\leadsto \left(\color{blue}{\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + j \cdot \left(-i \cdot y\right)\right)\]
if -9.153348824044421e-243 < b < 4.5584486085582877e-268
1. Initial program 17.5
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Taylor expanded around 0 16.2
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{0}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
if 4.5584486085582877e-268 < b < 1.0174711443687872e-190
1. Initial program 15.6
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Using strategy rm
3. Applied sub-neg15.6
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(c \cdot t + \left(-i \cdot y\right)\right)}\]
4. Applied distribute-lft-in15.6
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)}\]
5. Taylor expanded around -inf 15.4
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{t \cdot \left(j \cdot c\right)} + j \cdot \left(-i \cdot y\right)\right)\]
6. Using strategy rm
7. Applied associate-*r*16.3
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(t \cdot j\right) \cdot c} + j \cdot \left(-i \cdot y\right)\right)\]
8. Using strategy rm
9. Applied distribute-rgt-neg-in16.3
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + j \cdot \color{blue}{\left(i \cdot \left(-y\right)\right)}\right)\]
10. Applied associate-*r*15.6
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \color{blue}{\left(j \cdot i\right) \cdot \left(-y\right)}\right)\]
if 1.0174711443687872e-190 < b
1. Initial program 10.0
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Using strategy rm
3. Applied sub-neg10.0
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(c \cdot t + \left(-i \cdot y\right)\right)}\]
4. Applied distribute-lft-in10.0
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)}\]
5. Taylor expanded around -inf 10.0
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{t \cdot \left(j \cdot c\right)} + j \cdot \left(-i \cdot y\right)\right)\]
6. Using strategy rm
7. Applied associate-*r*10.4
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(t \cdot j\right) \cdot c} + j \cdot \left(-i \cdot y\right)\right)\]
8. Taylor expanded around -inf 10.5
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \color{blue}{-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)}\right)\]
9. Simplified10.5
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \color{blue}{\left(y \cdot j\right) \cdot \left(-i\right)}\right)\]
Recombined 4 regimes into one program.
Final simplification12.0
\[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.153348824044421 \cdot 10^{-243}:\\ \;\;\;\;\left(\left(\sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} \cdot \sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x}\right) \cdot \sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} - b \cdot \left(c \cdot z - a \cdot i\right)\right) + \left(\left(-j\right) \cdot \left(y \cdot i\right) + \left(j \cdot t\right) \cdot c\right)\\ \mathbf{elif}\;b \le 4.5584486085582877 \cdot 10^{-268}:\\ \;\;\;\;\left(y \cdot z - t \cdot a\right) \cdot x + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{elif}\;b \le 1.0174711443687872 \cdot 10^{-190}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - a \cdot i\right)\right) + \left(\left(-y\right) \cdot \left(i \cdot j\right) + \left(j \cdot t\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot j\right) \cdot \left(-i\right) + \left(j \cdot t\right) \cdot c\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - a \cdot i\right)\right)\\ \end{array}\]

Error

Try it out

Derivation

`if b < -9.153348824044421e-243`

`if -9.153348824044421e-243 < b < 4.5584486085582877e-268`

`if 4.5584486085582877e-268 < b < 1.0174711443687872e-190`

`if 1.0174711443687872e-190 < b`

Runtime

In
Out