Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B Percentage Accurate: 96.7% → 99.6%
Time: 19.8s
Alternatives: 22
Speedup: 315.0×
20.6% of points produce a very large (infinite) output. You may want to add a precondition. (more) Specification ? \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Local Percentage Accuracy vs ?
The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples. Accuracy vs Speed? The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs. Alternative 1: 99.6% accurate, 0.8× speedup? \[x \cdot e^{\mathsf{fma}\left(a, \mathsf{log1p}\left(-z\right) - b, y \cdot \left(\log z - t\right)\right)}
\]
Derivation Initial program 95.6%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Step-by-step derivation +-commutative95.6%
\[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right) + y \cdot \left(\log z - t\right)}}
\]
fma-def96.4%
\[\leadsto x \cdot e^{\color{blue}{\mathsf{fma}\left(a, \log \left(1 - z\right) - b, y \cdot \left(\log z - t\right)\right)}}
\]
sub-neg96.4%
\[\leadsto x \cdot e^{\mathsf{fma}\left(a, \log \color{blue}{\left(1 + \left(-z\right)\right)} - b, y \cdot \left(\log z - t\right)\right)}
\]
log1p-def99.6%
\[\leadsto x \cdot e^{\mathsf{fma}\left(a, \color{blue}{\mathsf{log1p}\left(-z\right)} - b, y \cdot \left(\log z - t\right)\right)}
\]
Simplified99.6%
\[\leadsto \color{blue}{x \cdot e^{\mathsf{fma}\left(a, \mathsf{log1p}\left(-z\right) - b, y \cdot \left(\log z - t\right)\right)}}
\]
Final simplification99.6%
\[\leadsto x \cdot e^{\mathsf{fma}\left(a, \mathsf{log1p}\left(-z\right) - b, y \cdot \left(\log z - t\right)\right)}
\]
Alternative 2: 95.8% accurate, 1.0× speedup? \[\begin{array}{l}
\mathbf{if}\;a \leq 8.5 \cdot 10^{+134}:\\
\;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\
\mathbf{else}:\\
\;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\
\end{array}
\]
Derivation Split input into 2 regimes if a < 8.50000000000000024e134 Initial program 97.4%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
if 8.50000000000000024e134 < a Initial program 83.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in y around 0 83.2%
\[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}}
\]
Step-by-step derivation sub-neg83.2%
\[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(-z\right)\right)} - b\right) \cdot a}
\]
mul-1-neg83.2%
\[\leadsto x \cdot e^{\left(\log \left(1 + \color{blue}{-1 \cdot z}\right) - b\right) \cdot a}
\]
log1p-def96.8%
\[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot z\right)} - b\right) \cdot a}
\]
mul-1-neg96.8%
\[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a}
\]
Simplified96.8%
\[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}}
\]
Taylor expanded in z around 0 96.8%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot z\right) + -1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*96.8%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot z} + -1 \cdot \left(a \cdot b\right)}
\]
associate-*r*96.8%
\[\leadsto x \cdot e^{\left(-1 \cdot a\right) \cdot z + \color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
distribute-lft-out96.8%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot \left(z + b\right)}}
\]
mul-1-neg96.8%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot \left(z + b\right)}
\]
Simplified96.8%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot \left(z + b\right)}}
\]
Recombined 2 regimes into one program. Final simplification97.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;a \leq 8.5 \cdot 10^{+134}:\\
\;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\
\mathbf{else}:\\
\;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\
\end{array}
\]
Alternative 3: 87.0% accurate, 1.5× speedup? \[\begin{array}{l}
\mathbf{if}\;y \leq -3.8 \cdot 10^{-5} \lor \neg \left(y \leq 2.8 \cdot 10^{-41}\right):\\
\;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\
\mathbf{else}:\\
\;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\
\end{array}
\]
Derivation Split input into 2 regimes if y < -3.8000000000000002e-5 or 2.8000000000000002e-41 < y Initial program 95.9%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in y around inf 91.1%
\[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}}
\]
if -3.8000000000000002e-5 < y < 2.8000000000000002e-41 Initial program 95.4%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in y around 0 80.7%
\[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}}
\]
Step-by-step derivation sub-neg80.7%
\[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(-z\right)\right)} - b\right) \cdot a}
\]
mul-1-neg80.7%
\[\leadsto x \cdot e^{\left(\log \left(1 + \color{blue}{-1 \cdot z}\right) - b\right) \cdot a}
\]
log1p-def85.3%
\[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot z\right)} - b\right) \cdot a}
\]
mul-1-neg85.3%
\[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a}
\]
Simplified85.3%
\[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}}
\]
Taylor expanded in z around 0 85.3%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot z\right) + -1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*85.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot z} + -1 \cdot \left(a \cdot b\right)}
\]
associate-*r*85.3%
\[\leadsto x \cdot e^{\left(-1 \cdot a\right) \cdot z + \color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
distribute-lft-out85.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot \left(z + b\right)}}
\]
mul-1-neg85.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot \left(z + b\right)}
\]
Simplified85.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot \left(z + b\right)}}
\]
Recombined 2 regimes into one program. Final simplification88.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;y \leq -3.8 \cdot 10^{-5} \lor \neg \left(y \leq 2.8 \cdot 10^{-41}\right):\\
\;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\
\mathbf{else}:\\
\;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\
\end{array}
\]
Alternative 4: 70.8% accurate, 2.8× speedup? \[\begin{array}{l}
t_1 := x \cdot e^{b \cdot \left(-a\right)}\\
t_2 := x \cdot e^{y \cdot \left(-t\right)}\\
\mathbf{if}\;t \leq -1.6 \cdot 10^{+132}:\\
\;\;\;\;t_2\\
\mathbf{elif}\;t \leq 1.35 \cdot 10^{-303}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;t \leq 1.26 \cdot 10^{-101}:\\
\;\;\;\;x \cdot {z}^{y}\\
\mathbf{elif}\;t \leq 7.2 \cdot 10^{+69}:\\
\;\;\;\;t_1\\
\mathbf{else}:\\
\;\;\;\;t_2\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -1.5999999999999999e132 or 7.2000000000000005e69 < t Initial program 94.5%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 85.0%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg85.0%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out85.0%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified85.0%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
if -1.5999999999999999e132 < t < 1.34999999999999993e-303 or 1.26e-101 < t < 7.2000000000000005e69 Initial program 96.2%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 76.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*76.5%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-176.5%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative76.5%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified76.5%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
if 1.34999999999999993e-303 < t < 1.26e-101 Initial program 96.2%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in y around inf 72.2%
\[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}}
\]
Taylor expanded in t around 0 72.2%
\[\leadsto \color{blue}{{z}^{y} \cdot x}
\]
Recombined 3 regimes into one program. Final simplification78.7%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -1.6 \cdot 10^{+132}:\\
\;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\
\mathbf{elif}\;t \leq 1.35 \cdot 10^{-303}:\\
\;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\
\mathbf{elif}\;t \leq 1.26 \cdot 10^{-101}:\\
\;\;\;\;x \cdot {z}^{y}\\
\mathbf{elif}\;t \leq 7.2 \cdot 10^{+69}:\\
\;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\
\mathbf{else}:\\
\;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\
\end{array}
\]
Alternative 5: 73.3% accurate, 2.8× speedup? \[\begin{array}{l}
\mathbf{if}\;y \leq -2.9 \cdot 10^{+23}:\\
\;\;\;\;x \cdot {z}^{y}\\
\mathbf{elif}\;y \leq 5.8 \cdot 10^{-41}:\\
\;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\
\mathbf{else}:\\
\;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\
\end{array}
\]
Derivation Split input into 3 regimes if y < -2.90000000000000013e23 Initial program 93.2%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in y around inf 91.7%
\[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}}
\]
Taylor expanded in t around 0 76.6%
\[\leadsto \color{blue}{{z}^{y} \cdot x}
\]
if -2.90000000000000013e23 < y < 5.79999999999999955e-41 Initial program 95.5%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in y around 0 79.3%
\[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}}
\]
Step-by-step derivation sub-neg79.3%
\[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(-z\right)\right)} - b\right) \cdot a}
\]
mul-1-neg79.3%
\[\leadsto x \cdot e^{\left(\log \left(1 + \color{blue}{-1 \cdot z}\right) - b\right) \cdot a}
\]
log1p-def83.8%
\[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot z\right)} - b\right) \cdot a}
\]
mul-1-neg83.8%
\[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a}
\]
Simplified83.8%
\[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}}
\]
Taylor expanded in z around 0 83.8%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot z\right) + -1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*83.8%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot z} + -1 \cdot \left(a \cdot b\right)}
\]
associate-*r*83.8%
\[\leadsto x \cdot e^{\left(-1 \cdot a\right) \cdot z + \color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
distribute-lft-out83.8%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot \left(z + b\right)}}
\]
mul-1-neg83.8%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot \left(z + b\right)}
\]
Simplified83.8%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right) \cdot \left(z + b\right)}}
\]
if 5.79999999999999955e-41 < y Initial program 98.3%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 67.6%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg67.6%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out67.6%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified67.6%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Recombined 3 regimes into one program. Final simplification78.5%
\[\leadsto \begin{array}{l}
\mathbf{if}\;y \leq -2.9 \cdot 10^{+23}:\\
\;\;\;\;x \cdot {z}^{y}\\
\mathbf{elif}\;y \leq 5.8 \cdot 10^{-41}:\\
\;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\
\mathbf{else}:\\
\;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\
\end{array}
\]
Alternative 6: 73.6% accurate, 2.9× speedup? \[\begin{array}{l}
\mathbf{if}\;y \leq -2.25 \cdot 10^{+23} \lor \neg \left(y \leq 1.12\right):\\
\;\;\;\;x \cdot {z}^{y}\\
\mathbf{else}:\\
\;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\
\end{array}
\]
Derivation Split input into 2 regimes if y < -2.2499999999999999e23 or 1.1200000000000001 < y Initial program 95.4%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in y around inf 91.0%
\[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}}
\]
Taylor expanded in t around 0 73.8%
\[\leadsto \color{blue}{{z}^{y} \cdot x}
\]
if -2.2499999999999999e23 < y < 1.1200000000000001 Initial program 95.7%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 75.7%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*75.7%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-175.7%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative75.7%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified75.7%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Recombined 2 regimes into one program. Final simplification74.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;y \leq -2.25 \cdot 10^{+23} \lor \neg \left(y \leq 1.12\right):\\
\;\;\;\;x \cdot {z}^{y}\\
\mathbf{else}:\\
\;\;\;\;x \cdot e^{b \cdot \left(-a\right)}\\
\end{array}
\]
Alternative 7: 57.1% accurate, 2.9× speedup? \[\begin{array}{l}
\mathbf{if}\;a \leq -4 \cdot 10^{+161}:\\
\;\;\;\;x \cdot \left(1 + b \cdot \left(b \cdot \left(a \cdot \left(a \cdot 0.5\right)\right) - a\right)\right)\\
\mathbf{elif}\;a \leq 2.25 \cdot 10^{+116}:\\
\;\;\;\;x \cdot {z}^{y}\\
\mathbf{else}:\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if a < -4.0000000000000002e161 Initial program 87.2%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 77.8%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*77.8%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-177.8%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative77.8%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified77.8%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 55.0%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg55.0%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg55.0%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*55.0%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow255.0%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*71.2%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--74.6%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative74.6%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow274.6%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*74.6%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified74.6%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
if -4.0000000000000002e161 < a < 2.25000000000000008e116 Initial program 98.9%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in y around inf 80.6%
\[\leadsto x \cdot e^{\color{blue}{\left(\log z - t\right) \cdot y}}
\]
Taylor expanded in t around 0 55.8%
\[\leadsto \color{blue}{{z}^{y} \cdot x}
\]
if 2.25000000000000008e116 < a Initial program 86.5%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 76.6%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*76.6%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-176.6%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative76.6%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified76.6%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 33.9%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg33.9%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg33.9%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*33.9%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow233.9%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*34.3%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--34.5%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative34.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow234.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*34.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified34.5%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 33.9%
\[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)}
\]
Step-by-step derivation associate-*r*33.9%
\[\leadsto \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)}
\]
*-commutative33.9%
\[\leadsto \color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot \left({b}^{2} \cdot x\right)
\]
unpow233.9%
\[\leadsto \left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*33.9%
\[\leadsto \color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*54.9%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)\right)}
\]
unpow254.9%
\[\leadsto a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot x\right)\right)
\]
Simplified54.9%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\left(b \cdot b\right) \cdot x\right)\right)}
\]
Recombined 3 regimes into one program. Final simplification57.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;a \leq -4 \cdot 10^{+161}:\\
\;\;\;\;x \cdot \left(1 + b \cdot \left(b \cdot \left(a \cdot \left(a \cdot 0.5\right)\right) - a\right)\right)\\
\mathbf{elif}\;a \leq 2.25 \cdot 10^{+116}:\\
\;\;\;\;x \cdot {z}^{y}\\
\mathbf{else}:\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\end{array}
\]
Alternative 8: 37.1% accurate, 13.5× speedup? \[\begin{array}{l}
t_1 := a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\mathbf{if}\;t \leq -1.9 \cdot 10^{+137}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\mathbf{elif}\;t \leq -0.00065:\\
\;\;\;\;t_1\\
\mathbf{elif}\;t \leq 9.5 \cdot 10^{+15}:\\
\;\;\;\;x \cdot \left(1 + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(b \cdot b\right)\right)\right)\\
\mathbf{elif}\;t \leq 7.4 \cdot 10^{+245} \lor \neg \left(t \leq 6.8 \cdot 10^{+276}\right):\\
\;\;\;\;t_1\\
\mathbf{else}:\\
\;\;\;\;x + 0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\\
\end{array}
\]
Derivation Split input into 4 regimes if t < -1.89999999999999981e137 Initial program 97.1%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 88.6%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg88.6%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out88.6%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified88.6%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 37.1%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative37.1%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg37.1%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg37.1%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified37.1%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in x around 0 39.8%
\[\leadsto \color{blue}{\left(1 - y \cdot t\right) \cdot x}
\]
Step-by-step derivation *-commutative39.8%
\[\leadsto \color{blue}{x \cdot \left(1 - y \cdot t\right)}
\]
distribute-rgt-out--39.8%
\[\leadsto \color{blue}{1 \cdot x - \left(y \cdot t\right) \cdot x}
\]
associate-*r*37.1%
\[\leadsto 1 \cdot x - \color{blue}{y \cdot \left(t \cdot x\right)}
\]
*-commutative37.1%
\[\leadsto 1 \cdot x - y \cdot \color{blue}{\left(x \cdot t\right)}
\]
*-lft-identity37.1%
\[\leadsto \color{blue}{x} - y \cdot \left(x \cdot t\right)
\]
*-commutative37.1%
\[\leadsto x - \color{blue}{\left(x \cdot t\right) \cdot y}
\]
*-commutative37.1%
\[\leadsto x - \color{blue}{\left(t \cdot x\right)} \cdot y
\]
associate-*l*39.8%
\[\leadsto x - \color{blue}{t \cdot \left(x \cdot y\right)}
\]
Simplified39.8%
\[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)}
\]
if -1.89999999999999981e137 < t < -6.4999999999999997e-4 or 9.5e15 < t < 7.4000000000000002e245 or 6.79999999999999967e276 < t Initial program 92.9%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 46.3%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*46.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-146.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative46.3%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified46.3%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 25.9%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg25.9%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg25.9%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*25.9%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow225.9%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*27.5%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--27.6%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative27.6%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow227.6%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*27.6%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified27.6%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 37.4%
\[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)}
\]
Step-by-step derivation associate-*r*37.4%
\[\leadsto \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)}
\]
*-commutative37.4%
\[\leadsto \color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot \left({b}^{2} \cdot x\right)
\]
unpow237.4%
\[\leadsto \left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*37.4%
\[\leadsto \color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*47.2%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)\right)}
\]
unpow247.2%
\[\leadsto a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot x\right)\right)
\]
Simplified47.2%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\left(b \cdot b\right) \cdot x\right)\right)}
\]
if -6.4999999999999997e-4 < t < 9.5e15 Initial program 98.2%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 72.3%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*72.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-172.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative72.3%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified72.3%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 44.0%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg44.0%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg44.0%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*44.0%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow244.0%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*44.3%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--44.4%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative44.4%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow244.4%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*44.4%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified44.4%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 43.9%
\[\leadsto x \cdot \left(1 + \color{blue}{0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right)}\right)
\]
Step-by-step derivation associate-*r*43.9%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}}\right)
\]
*-commutative43.9%
\[\leadsto x \cdot \left(1 + \color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot {b}^{2}\right)
\]
unpow243.9%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot {b}^{2}\right)
\]
associate-*l*43.9%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot {b}^{2}\right)
\]
associate-*l*45.9%
\[\leadsto x \cdot \left(1 + \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot {b}^{2}\right)}\right)
\]
unpow245.9%
\[\leadsto x \cdot \left(1 + a \cdot \left(\left(a \cdot 0.5\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)
\]
Simplified45.9%
\[\leadsto x \cdot \left(1 + \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(b \cdot b\right)\right)}\right)
\]
if 7.4000000000000002e245 < t < 6.79999999999999967e276 Initial program 75.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 100.0%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg100.0%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out100.0%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified100.0%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 100.0%
\[\leadsto \color{blue}{0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + \left(-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x\right)}
\]
Step-by-step derivation associate-+r+100.0%
\[\leadsto \color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)\right) + x}
\]
+-commutative100.0%
\[\leadsto \color{blue}{x + \left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)\right)}
\]
mul-1-neg100.0%
\[\leadsto x + \left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}\right)
\]
unsub-neg100.0%
\[\leadsto x + \color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) - y \cdot \left(t \cdot x\right)\right)}
\]
associate-*r*100.0%
\[\leadsto x + \left(0.5 \cdot \color{blue}{\left(\left({y}^{2} \cdot {t}^{2}\right) \cdot x\right)} - y \cdot \left(t \cdot x\right)\right)
\]
associate-*r*100.0%
\[\leadsto x + \left(\color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right)\right) \cdot x} - y \cdot \left(t \cdot x\right)\right)
\]
associate-*r*100.0%
\[\leadsto x + \left(\left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right)\right) \cdot x - \color{blue}{\left(y \cdot t\right) \cdot x}\right)
\]
distribute-rgt-out--100.0%
\[\leadsto x + \color{blue}{x \cdot \left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right) - y \cdot t\right)}
\]
unpow2100.0%
\[\leadsto x + x \cdot \left(0.5 \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot {t}^{2}\right) - y \cdot t\right)
\]
unpow2100.0%
\[\leadsto x + x \cdot \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) - y \cdot t\right)
\]
unswap-sqr88.6%
\[\leadsto x + x \cdot \left(0.5 \cdot \color{blue}{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} - y \cdot t\right)
\]
Simplified88.6%
\[\leadsto \color{blue}{x + x \cdot \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - y \cdot t\right)}
\]
Taylor expanded in y around inf 100.0%
\[\leadsto x + \color{blue}{0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right)}
\]
Step-by-step derivation unpow2100.0%
\[\leadsto x + 0.5 \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left({t}^{2} \cdot x\right)\right)
\]
*-commutative100.0%
\[\leadsto x + 0.5 \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(x \cdot {t}^{2}\right)}\right)
\]
unpow2100.0%
\[\leadsto x + 0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)
\]
Simplified100.0%
\[\leadsto x + \color{blue}{0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)}
\]
Recombined 4 regimes into one program. Final simplification47.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -1.9 \cdot 10^{+137}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\mathbf{elif}\;t \leq -0.00065:\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\mathbf{elif}\;t \leq 9.5 \cdot 10^{+15}:\\
\;\;\;\;x \cdot \left(1 + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(b \cdot b\right)\right)\right)\\
\mathbf{elif}\;t \leq 7.4 \cdot 10^{+245} \lor \neg \left(t \leq 6.8 \cdot 10^{+276}\right):\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;x + 0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\\
\end{array}
\]
Alternative 9: 42.1% accurate, 13.5× speedup? \[\begin{array}{l}
\mathbf{if}\;a \leq -1.25 \cdot 10^{+48}:\\
\;\;\;\;x \cdot \left(1 + b \cdot \left(b \cdot \left(a \cdot \left(a \cdot 0.5\right)\right) - a\right)\right)\\
\mathbf{elif}\;a \leq -8 \cdot 10^{-133} \lor \neg \left(a \leq -3.4 \cdot 10^{-295}\right) \land a \leq 1.05 \cdot 10^{+116}:\\
\;\;\;\;x + x \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot \left(t \cdot 0.5\right) + -1\right)\right)\\
\mathbf{else}:\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if a < -1.24999999999999993e48 Initial program 89.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 72.7%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*72.7%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-172.7%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative72.7%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified72.7%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 45.4%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg45.4%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg45.4%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*45.4%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow245.4%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*54.7%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--58.5%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative58.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow258.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*58.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified58.5%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
if -1.24999999999999993e48 < a < -8.0000000000000005e-133 or -3.40000000000000007e-295 < a < 1.0500000000000001e116 Initial program 100.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 68.0%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg68.0%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out68.0%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified68.0%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 36.1%
\[\leadsto \color{blue}{0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + \left(-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x\right)}
\]
Step-by-step derivation associate-+r+36.1%
\[\leadsto \color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)\right) + x}
\]
+-commutative36.1%
\[\leadsto \color{blue}{x + \left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)\right)}
\]
mul-1-neg36.1%
\[\leadsto x + \left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}\right)
\]
unsub-neg36.1%
\[\leadsto x + \color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) - y \cdot \left(t \cdot x\right)\right)}
\]
associate-*r*38.9%
\[\leadsto x + \left(0.5 \cdot \color{blue}{\left(\left({y}^{2} \cdot {t}^{2}\right) \cdot x\right)} - y \cdot \left(t \cdot x\right)\right)
\]
associate-*r*38.9%
\[\leadsto x + \left(\color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right)\right) \cdot x} - y \cdot \left(t \cdot x\right)\right)
\]
associate-*r*39.6%
\[\leadsto x + \left(\left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right)\right) \cdot x - \color{blue}{\left(y \cdot t\right) \cdot x}\right)
\]
distribute-rgt-out--39.6%
\[\leadsto x + \color{blue}{x \cdot \left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right) - y \cdot t\right)}
\]
unpow239.6%
\[\leadsto x + x \cdot \left(0.5 \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot {t}^{2}\right) - y \cdot t\right)
\]
unpow239.6%
\[\leadsto x + x \cdot \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) - y \cdot t\right)
\]
unswap-sqr41.7%
\[\leadsto x + x \cdot \left(0.5 \cdot \color{blue}{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} - y \cdot t\right)
\]
Simplified41.7%
\[\leadsto \color{blue}{x + x \cdot \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - y \cdot t\right)}
\]
Taylor expanded in x around 0 39.6%
\[\leadsto x + \color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right) - y \cdot t\right) \cdot x}
\]
Step-by-step derivation *-commutative39.6%
\[\leadsto x + \color{blue}{x \cdot \left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right) - y \cdot t\right)}
\]
sub-neg39.6%
\[\leadsto x + x \cdot \color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right) + \left(-y \cdot t\right)\right)}
\]
fma-def39.6%
\[\leadsto x + x \cdot \color{blue}{\mathsf{fma}\left(0.5, {y}^{2} \cdot {t}^{2}, -y \cdot t\right)}
\]
unpow239.6%
\[\leadsto x + x \cdot \mathsf{fma}\left(0.5, \color{blue}{\left(y \cdot y\right)} \cdot {t}^{2}, -y \cdot t\right)
\]
unpow239.6%
\[\leadsto x + x \cdot \mathsf{fma}\left(0.5, \left(y \cdot y\right) \cdot \color{blue}{\left(t \cdot t\right)}, -y \cdot t\right)
\]
swap-sqr41.7%
\[\leadsto x + x \cdot \mathsf{fma}\left(0.5, \color{blue}{\left(y \cdot t\right) \cdot \left(y \cdot t\right)}, -y \cdot t\right)
\]
unpow241.7%
\[\leadsto x + x \cdot \mathsf{fma}\left(0.5, \color{blue}{{\left(y \cdot t\right)}^{2}}, -y \cdot t\right)
\]
fma-def41.7%
\[\leadsto x + x \cdot \color{blue}{\left(0.5 \cdot {\left(y \cdot t\right)}^{2} + \left(-y \cdot t\right)\right)}
\]
unpow241.7%
\[\leadsto x + x \cdot \left(0.5 \cdot \color{blue}{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} + \left(-y \cdot t\right)\right)
\]
associate-*r*41.7%
\[\leadsto x + x \cdot \left(\color{blue}{\left(0.5 \cdot \left(y \cdot t\right)\right) \cdot \left(y \cdot t\right)} + \left(-y \cdot t\right)\right)
\]
neg-mul-141.7%
\[\leadsto x + x \cdot \left(\left(0.5 \cdot \left(y \cdot t\right)\right) \cdot \left(y \cdot t\right) + \color{blue}{-1 \cdot \left(y \cdot t\right)}\right)
\]
distribute-rgt-out41.8%
\[\leadsto x + x \cdot \color{blue}{\left(\left(y \cdot t\right) \cdot \left(0.5 \cdot \left(y \cdot t\right) + -1\right)\right)}
\]
*-commutative41.8%
\[\leadsto x + x \cdot \left(\left(y \cdot t\right) \cdot \left(\color{blue}{\left(y \cdot t\right) \cdot 0.5} + -1\right)\right)
\]
associate-*l*41.8%
\[\leadsto x + x \cdot \left(\left(y \cdot t\right) \cdot \left(\color{blue}{y \cdot \left(t \cdot 0.5\right)} + -1\right)\right)
\]
Simplified41.8%
\[\leadsto x + \color{blue}{x \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot \left(t \cdot 0.5\right) + -1\right)\right)}
\]
if -8.0000000000000005e-133 < a < -3.40000000000000007e-295 or 1.0500000000000001e116 < a Initial program 91.9%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 57.7%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*57.7%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-157.7%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative57.7%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified57.7%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 30.3%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg30.3%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg30.3%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*30.3%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow230.3%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*27.9%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--28.0%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative28.0%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow228.0%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*28.0%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified28.0%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 37.7%
\[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)}
\]
Step-by-step derivation associate-*r*37.7%
\[\leadsto \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)}
\]
*-commutative37.7%
\[\leadsto \color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot \left({b}^{2} \cdot x\right)
\]
unpow237.7%
\[\leadsto \left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*37.7%
\[\leadsto \color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*53.7%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)\right)}
\]
unpow253.7%
\[\leadsto a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot x\right)\right)
\]
Simplified53.7%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\left(b \cdot b\right) \cdot x\right)\right)}
\]
Recombined 3 regimes into one program. Final simplification48.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;a \leq -1.25 \cdot 10^{+48}:\\
\;\;\;\;x \cdot \left(1 + b \cdot \left(b \cdot \left(a \cdot \left(a \cdot 0.5\right)\right) - a\right)\right)\\
\mathbf{elif}\;a \leq -8 \cdot 10^{-133} \lor \neg \left(a \leq -3.4 \cdot 10^{-295}\right) \land a \leq 1.05 \cdot 10^{+116}:\\
\;\;\;\;x + x \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot \left(t \cdot 0.5\right) + -1\right)\right)\\
\mathbf{else}:\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\end{array}
\]
Alternative 10: 35.6% accurate, 16.3× speedup? \[\begin{array}{l}
\mathbf{if}\;a \leq -2.05 \cdot 10^{+49} \lor \neg \left(a \leq -1.15 \cdot 10^{-77} \lor \neg \left(a \leq -3.2 \cdot 10^{-295}\right) \land a \leq 1.2 \cdot 10^{+116}\right):\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if a < -2.05e49 or -1.14999999999999999e-77 < a < -3.2e-295 or 1.2e116 < a Initial program 91.2%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 64.3%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*64.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-164.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative64.3%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified64.3%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 37.1%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg37.1%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg37.1%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*37.1%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow237.1%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*38.3%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--40.0%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative40.0%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow240.0%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*40.0%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified40.0%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 39.6%
\[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)}
\]
Step-by-step derivation associate-*r*39.6%
\[\leadsto \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)}
\]
*-commutative39.6%
\[\leadsto \color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot \left({b}^{2} \cdot x\right)
\]
unpow239.6%
\[\leadsto \left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*39.6%
\[\leadsto \color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*45.6%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)\right)}
\]
unpow245.6%
\[\leadsto a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot x\right)\right)
\]
Simplified45.6%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\left(b \cdot b\right) \cdot x\right)\right)}
\]
if -2.05e49 < a < -1.14999999999999999e-77 or -3.2e-295 < a < 1.2e116 Initial program 100.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 69.0%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg69.0%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out69.0%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified69.0%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 36.0%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative36.0%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg36.0%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg36.0%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified36.0%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in x around 0 38.9%
\[\leadsto \color{blue}{\left(1 - y \cdot t\right) \cdot x}
\]
Step-by-step derivation *-commutative38.9%
\[\leadsto \color{blue}{x \cdot \left(1 - y \cdot t\right)}
\]
distribute-rgt-out--38.9%
\[\leadsto \color{blue}{1 \cdot x - \left(y \cdot t\right) \cdot x}
\]
associate-*r*36.0%
\[\leadsto 1 \cdot x - \color{blue}{y \cdot \left(t \cdot x\right)}
\]
*-commutative36.0%
\[\leadsto 1 \cdot x - y \cdot \color{blue}{\left(x \cdot t\right)}
\]
*-lft-identity36.0%
\[\leadsto \color{blue}{x} - y \cdot \left(x \cdot t\right)
\]
*-commutative36.0%
\[\leadsto x - \color{blue}{\left(x \cdot t\right) \cdot y}
\]
*-commutative36.0%
\[\leadsto x - \color{blue}{\left(t \cdot x\right)} \cdot y
\]
associate-*l*39.6%
\[\leadsto x - \color{blue}{t \cdot \left(x \cdot y\right)}
\]
Simplified39.6%
\[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)}
\]
Recombined 2 regimes into one program. Final simplification42.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;a \leq -2.05 \cdot 10^{+49} \lor \neg \left(a \leq -1.15 \cdot 10^{-77} \lor \neg \left(a \leq -3.2 \cdot 10^{-295}\right) \land a \leq 1.2 \cdot 10^{+116}\right):\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\end{array}
\]
Alternative 11: 35.0% accurate, 16.4× speedup? \[\begin{array}{l}
\mathbf{if}\;a \leq -8 \cdot 10^{+48}:\\
\;\;\;\;x \cdot \left(\left(a \cdot b\right) \cdot \left(0.5 \cdot \left(a \cdot b\right)\right)\right)\\
\mathbf{elif}\;a \leq -1.42 \cdot 10^{-77} \lor \neg \left(a \leq -1.75 \cdot 10^{-297}\right) \land a \leq 1.15 \cdot 10^{+116}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\mathbf{else}:\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if a < -8.00000000000000035e48 Initial program 89.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 72.7%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*72.7%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-172.7%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative72.7%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified72.7%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 45.4%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg45.4%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg45.4%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*45.4%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow245.4%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*54.7%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--58.5%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative58.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow258.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*58.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified58.5%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 43.8%
\[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)}
\]
Step-by-step derivation associate-*r*43.8%
\[\leadsto \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)}
\]
unpow243.8%
\[\leadsto \left(0.5 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left({b}^{2} \cdot x\right)
\]
unpow243.8%
\[\leadsto \left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot x\right)
\]
Simplified43.8%
\[\leadsto \color{blue}{\left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(\left(b \cdot b\right) \cdot x\right)}
\]
Taylor expanded in a around 0 43.8%
\[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)}
\]
Step-by-step derivation unpow243.8%
\[\leadsto 0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left({b}^{2} \cdot x\right)\right)
\]
unpow243.8%
\[\leadsto 0.5 \cdot \left(\left(a \cdot a\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot x\right)\right)
\]
associate-*r*49.2%
\[\leadsto 0.5 \cdot \color{blue}{\left(\left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right) \cdot x\right)}
\]
associate-*r*49.2%
\[\leadsto \color{blue}{\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right) \cdot x}
\]
*-commutative49.2%
\[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right)}
\]
*-commutative49.2%
\[\leadsto x \cdot \color{blue}{\left(\left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right) \cdot 0.5\right)}
\]
*-commutative49.2%
\[\leadsto x \cdot \left(\color{blue}{\left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)} \cdot 0.5\right)
\]
swap-sqr46.0%
\[\leadsto x \cdot \left(\color{blue}{\left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right)} \cdot 0.5\right)
\]
associate-*l*46.0%
\[\leadsto x \cdot \color{blue}{\left(\left(b \cdot a\right) \cdot \left(\left(b \cdot a\right) \cdot 0.5\right)\right)}
\]
*-commutative46.0%
\[\leadsto x \cdot \left(\color{blue}{\left(a \cdot b\right)} \cdot \left(\left(b \cdot a\right) \cdot 0.5\right)\right)
\]
*-commutative46.0%
\[\leadsto x \cdot \left(\left(a \cdot b\right) \cdot \left(\color{blue}{\left(a \cdot b\right)} \cdot 0.5\right)\right)
\]
Simplified46.0%
\[\leadsto \color{blue}{x \cdot \left(\left(a \cdot b\right) \cdot \left(\left(a \cdot b\right) \cdot 0.5\right)\right)}
\]
if -8.00000000000000035e48 < a < -1.42e-77 or -1.7499999999999999e-297 < a < 1.14999999999999997e116 Initial program 100.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 69.0%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg69.0%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out69.0%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified69.0%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 36.0%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative36.0%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg36.0%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg36.0%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified36.0%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in x around 0 38.9%
\[\leadsto \color{blue}{\left(1 - y \cdot t\right) \cdot x}
\]
Step-by-step derivation *-commutative38.9%
\[\leadsto \color{blue}{x \cdot \left(1 - y \cdot t\right)}
\]
distribute-rgt-out--38.9%
\[\leadsto \color{blue}{1 \cdot x - \left(y \cdot t\right) \cdot x}
\]
associate-*r*36.0%
\[\leadsto 1 \cdot x - \color{blue}{y \cdot \left(t \cdot x\right)}
\]
*-commutative36.0%
\[\leadsto 1 \cdot x - y \cdot \color{blue}{\left(x \cdot t\right)}
\]
*-lft-identity36.0%
\[\leadsto \color{blue}{x} - y \cdot \left(x \cdot t\right)
\]
*-commutative36.0%
\[\leadsto x - \color{blue}{\left(x \cdot t\right) \cdot y}
\]
*-commutative36.0%
\[\leadsto x - \color{blue}{\left(t \cdot x\right)} \cdot y
\]
associate-*l*39.6%
\[\leadsto x - \color{blue}{t \cdot \left(x \cdot y\right)}
\]
Simplified39.6%
\[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)}
\]
if -1.42e-77 < a < -1.7499999999999999e-297 or 1.14999999999999997e116 < a Initial program 92.8%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 58.1%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*58.1%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-158.1%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative58.1%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified58.1%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 31.0%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg31.0%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg31.0%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*31.0%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow231.0%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*26.3%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--26.4%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative26.4%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow226.4%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*26.4%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified26.4%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 36.5%
\[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)}
\]
Step-by-step derivation associate-*r*36.5%
\[\leadsto \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)}
\]
*-commutative36.5%
\[\leadsto \color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot \left({b}^{2} \cdot x\right)
\]
unpow236.5%
\[\leadsto \left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*36.5%
\[\leadsto \color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*50.7%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)\right)}
\]
unpow250.7%
\[\leadsto a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot x\right)\right)
\]
Simplified50.7%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\left(b \cdot b\right) \cdot x\right)\right)}
\]
Recombined 3 regimes into one program. Final simplification44.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;a \leq -8 \cdot 10^{+48}:\\
\;\;\;\;x \cdot \left(\left(a \cdot b\right) \cdot \left(0.5 \cdot \left(a \cdot b\right)\right)\right)\\
\mathbf{elif}\;a \leq -1.42 \cdot 10^{-77} \lor \neg \left(a \leq -1.75 \cdot 10^{-297}\right) \land a \leq 1.15 \cdot 10^{+116}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\mathbf{else}:\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\end{array}
\]
Alternative 12: 35.4% accurate, 16.4× speedup? \[\begin{array}{l}
\mathbf{if}\;t \leq -1.85 \cdot 10^{+137}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\mathbf{elif}\;t \leq -2.05 \cdot 10^{-122} \lor \neg \left(t \leq 35\right):\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(1 + 0.5 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -1.8500000000000001e137 Initial program 97.1%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 88.6%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg88.6%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out88.6%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified88.6%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 37.1%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative37.1%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg37.1%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg37.1%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified37.1%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in x around 0 39.8%
\[\leadsto \color{blue}{\left(1 - y \cdot t\right) \cdot x}
\]
Step-by-step derivation *-commutative39.8%
\[\leadsto \color{blue}{x \cdot \left(1 - y \cdot t\right)}
\]
distribute-rgt-out--39.8%
\[\leadsto \color{blue}{1 \cdot x - \left(y \cdot t\right) \cdot x}
\]
associate-*r*37.1%
\[\leadsto 1 \cdot x - \color{blue}{y \cdot \left(t \cdot x\right)}
\]
*-commutative37.1%
\[\leadsto 1 \cdot x - y \cdot \color{blue}{\left(x \cdot t\right)}
\]
*-lft-identity37.1%
\[\leadsto \color{blue}{x} - y \cdot \left(x \cdot t\right)
\]
*-commutative37.1%
\[\leadsto x - \color{blue}{\left(x \cdot t\right) \cdot y}
\]
*-commutative37.1%
\[\leadsto x - \color{blue}{\left(t \cdot x\right)} \cdot y
\]
associate-*l*39.8%
\[\leadsto x - \color{blue}{t \cdot \left(x \cdot y\right)}
\]
Simplified39.8%
\[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)}
\]
if -1.8500000000000001e137 < t < -2.05e-122 or 35 < t Initial program 92.7%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 47.9%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*47.9%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-147.9%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative47.9%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified47.9%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 25.7%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg25.7%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg25.7%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*25.7%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow225.7%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*29.8%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--31.7%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative31.7%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow231.7%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*31.7%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified31.7%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 35.5%
\[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)}
\]
Step-by-step derivation associate-*r*35.5%
\[\leadsto \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)}
\]
*-commutative35.5%
\[\leadsto \color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot \left({b}^{2} \cdot x\right)
\]
unpow235.5%
\[\leadsto \left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*35.5%
\[\leadsto \color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*44.9%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)\right)}
\]
unpow244.9%
\[\leadsto a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot x\right)\right)
\]
Simplified44.9%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\left(b \cdot b\right) \cdot x\right)\right)}
\]
if -2.05e-122 < t < 35 Initial program 98.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 71.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*71.5%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-171.5%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative71.5%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified71.5%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 46.1%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg46.1%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg46.1%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*46.1%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow246.1%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*43.7%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--43.8%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative43.8%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow243.8%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*43.8%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified43.8%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 45.9%
\[\leadsto x \cdot \left(1 + \color{blue}{0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right)}\right)
\]
Step-by-step derivation unpow245.9%
\[\leadsto x \cdot \left(1 + 0.5 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right)\right)
\]
unpow245.9%
\[\leadsto x \cdot \left(1 + 0.5 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)
\]
Simplified45.9%
\[\leadsto x \cdot \left(1 + \color{blue}{0.5 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}\right)
\]
Recombined 3 regimes into one program. Final simplification44.7%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -1.85 \cdot 10^{+137}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\mathbf{elif}\;t \leq -2.05 \cdot 10^{-122} \lor \neg \left(t \leq 35\right):\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(1 + 0.5 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)\right)\\
\end{array}
\]
Alternative 13: 37.1% accurate, 16.4× speedup? \[\begin{array}{l}
\mathbf{if}\;t \leq -1.9 \cdot 10^{+137}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\mathbf{elif}\;t \leq -1.15 \cdot 10^{-5} \lor \neg \left(t \leq 1.1 \cdot 10^{+14}\right):\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(1 + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(b \cdot b\right)\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -1.89999999999999981e137 Initial program 97.1%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 88.6%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg88.6%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out88.6%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified88.6%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 37.1%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative37.1%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg37.1%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg37.1%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified37.1%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in x around 0 39.8%
\[\leadsto \color{blue}{\left(1 - y \cdot t\right) \cdot x}
\]
Step-by-step derivation *-commutative39.8%
\[\leadsto \color{blue}{x \cdot \left(1 - y \cdot t\right)}
\]
distribute-rgt-out--39.8%
\[\leadsto \color{blue}{1 \cdot x - \left(y \cdot t\right) \cdot x}
\]
associate-*r*37.1%
\[\leadsto 1 \cdot x - \color{blue}{y \cdot \left(t \cdot x\right)}
\]
*-commutative37.1%
\[\leadsto 1 \cdot x - y \cdot \color{blue}{\left(x \cdot t\right)}
\]
*-lft-identity37.1%
\[\leadsto \color{blue}{x} - y \cdot \left(x \cdot t\right)
\]
*-commutative37.1%
\[\leadsto x - \color{blue}{\left(x \cdot t\right) \cdot y}
\]
*-commutative37.1%
\[\leadsto x - \color{blue}{\left(t \cdot x\right)} \cdot y
\]
associate-*l*39.8%
\[\leadsto x - \color{blue}{t \cdot \left(x \cdot y\right)}
\]
Simplified39.8%
\[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)}
\]
if -1.89999999999999981e137 < t < -1.15e-5 or 1.1e14 < t Initial program 91.3%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 42.4%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*42.4%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-142.4%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative42.4%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified42.4%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 24.9%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg24.9%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg24.9%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*24.9%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow224.9%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*26.3%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--28.6%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative28.6%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow228.6%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*28.6%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified28.6%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 37.5%
\[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)}
\]
Step-by-step derivation associate-*r*37.5%
\[\leadsto \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)}
\]
*-commutative37.5%
\[\leadsto \color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot \left({b}^{2} \cdot x\right)
\]
unpow237.5%
\[\leadsto \left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*37.5%
\[\leadsto \color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*46.4%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)\right)}
\]
unpow246.4%
\[\leadsto a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot x\right)\right)
\]
Simplified46.4%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\left(b \cdot b\right) \cdot x\right)\right)}
\]
if -1.15e-5 < t < 1.1e14 Initial program 98.2%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 72.3%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*72.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-172.3%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative72.3%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified72.3%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 44.0%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg44.0%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg44.0%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*44.0%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow244.0%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*44.3%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--44.4%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative44.4%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow244.4%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*44.4%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified44.4%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 43.9%
\[\leadsto x \cdot \left(1 + \color{blue}{0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right)}\right)
\]
Step-by-step derivation associate-*r*43.9%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}}\right)
\]
*-commutative43.9%
\[\leadsto x \cdot \left(1 + \color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot {b}^{2}\right)
\]
unpow243.9%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot {b}^{2}\right)
\]
associate-*l*43.9%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot {b}^{2}\right)
\]
associate-*l*45.9%
\[\leadsto x \cdot \left(1 + \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot {b}^{2}\right)}\right)
\]
unpow245.9%
\[\leadsto x \cdot \left(1 + a \cdot \left(\left(a \cdot 0.5\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)
\]
Simplified45.9%
\[\leadsto x \cdot \left(1 + \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(b \cdot b\right)\right)}\right)
\]
Recombined 3 regimes into one program. Final simplification45.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -1.9 \cdot 10^{+137}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\mathbf{elif}\;t \leq -1.15 \cdot 10^{-5} \lor \neg \left(t \leq 1.1 \cdot 10^{+14}\right):\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(1 + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(b \cdot b\right)\right)\right)\\
\end{array}
\]
Alternative 14: 43.8% accurate, 18.4× speedup? \[\begin{array}{l}
\mathbf{if}\;a \leq -7 \cdot 10^{+48}:\\
\;\;\;\;x \cdot \left(1 + b \cdot \left(b \cdot \left(a \cdot \left(a \cdot 0.5\right)\right) - a\right)\right)\\
\mathbf{elif}\;a \leq 5.1 \cdot 10^{+116}:\\
\;\;\;\;x + x \cdot \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(t \cdot t\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if a < -6.9999999999999995e48 Initial program 89.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 72.7%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*72.7%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-172.7%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative72.7%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified72.7%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 45.4%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg45.4%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg45.4%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*45.4%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow245.4%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*54.7%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--58.5%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative58.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow258.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*58.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified58.5%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
if -6.9999999999999995e48 < a < 5.09999999999999999e116 Initial program 100.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 69.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg69.5%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out69.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified69.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 36.0%
\[\leadsto \color{blue}{0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + \left(-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x\right)}
\]
Step-by-step derivation associate-+r+36.0%
\[\leadsto \color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)\right) + x}
\]
+-commutative36.0%
\[\leadsto \color{blue}{x + \left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)\right)}
\]
mul-1-neg36.0%
\[\leadsto x + \left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}\right)
\]
unsub-neg36.0%
\[\leadsto x + \color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot \left({t}^{2} \cdot x\right)\right) - y \cdot \left(t \cdot x\right)\right)}
\]
associate-*r*37.8%
\[\leadsto x + \left(0.5 \cdot \color{blue}{\left(\left({y}^{2} \cdot {t}^{2}\right) \cdot x\right)} - y \cdot \left(t \cdot x\right)\right)
\]
associate-*r*37.8%
\[\leadsto x + \left(\color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right)\right) \cdot x} - y \cdot \left(t \cdot x\right)\right)
\]
associate-*r*38.4%
\[\leadsto x + \left(\left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right)\right) \cdot x - \color{blue}{\left(y \cdot t\right) \cdot x}\right)
\]
distribute-rgt-out--38.5%
\[\leadsto x + \color{blue}{x \cdot \left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right) - y \cdot t\right)}
\]
unpow238.5%
\[\leadsto x + x \cdot \left(0.5 \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot {t}^{2}\right) - y \cdot t\right)
\]
unpow238.5%
\[\leadsto x + x \cdot \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) - y \cdot t\right)
\]
unswap-sqr40.2%
\[\leadsto x + x \cdot \left(0.5 \cdot \color{blue}{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right)} - y \cdot t\right)
\]
Simplified40.2%
\[\leadsto \color{blue}{x + x \cdot \left(0.5 \cdot \left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) - y \cdot t\right)}
\]
Taylor expanded in y around inf 38.6%
\[\leadsto x + x \cdot \color{blue}{\left(0.5 \cdot \left({y}^{2} \cdot {t}^{2}\right)\right)}
\]
Step-by-step derivation unpow238.6%
\[\leadsto x + x \cdot \left(0.5 \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot {t}^{2}\right)\right)
\]
unpow238.6%
\[\leadsto x + x \cdot \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)
\]
Simplified38.6%
\[\leadsto x + x \cdot \color{blue}{\left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(t \cdot t\right)\right)\right)}
\]
if 5.09999999999999999e116 < a Initial program 86.5%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 76.6%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*76.6%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-176.6%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative76.6%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified76.6%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 33.9%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + -1 \cdot \left(a \cdot b\right)\right)\right)}
\]
Step-by-step derivation mul-1-neg33.9%
\[\leadsto x \cdot \left(1 + \left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left(-a \cdot b\right)}\right)\right)
\]
unsub-neg33.9%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(0.5 \cdot \left({a}^{2} \cdot {b}^{2}\right) - a \cdot b\right)}\right)
\]
associate-*r*33.9%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot {b}^{2}} - a \cdot b\right)\right)
\]
unpow233.9%
\[\leadsto x \cdot \left(1 + \left(\left(0.5 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} - a \cdot b\right)\right)
\]
associate-*r*34.3%
\[\leadsto x \cdot \left(1 + \left(\color{blue}{\left(\left(0.5 \cdot {a}^{2}\right) \cdot b\right) \cdot b} - a \cdot b\right)\right)
\]
distribute-rgt-out--34.5%
\[\leadsto x \cdot \left(1 + \color{blue}{b \cdot \left(\left(0.5 \cdot {a}^{2}\right) \cdot b - a\right)}\right)
\]
*-commutative34.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot b - a\right)\right)
\]
unpow234.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot b - a\right)\right)
\]
associate-*l*34.5%
\[\leadsto x \cdot \left(1 + b \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot b - a\right)\right)
\]
Simplified34.5%
\[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(\left(a \cdot \left(a \cdot 0.5\right)\right) \cdot b - a\right)\right)}
\]
Taylor expanded in b around inf 33.9%
\[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)}
\]
Step-by-step derivation associate-*r*33.9%
\[\leadsto \color{blue}{\left(0.5 \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)}
\]
*-commutative33.9%
\[\leadsto \color{blue}{\left({a}^{2} \cdot 0.5\right)} \cdot \left({b}^{2} \cdot x\right)
\]
unpow233.9%
\[\leadsto \left(\color{blue}{\left(a \cdot a\right)} \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*33.9%
\[\leadsto \color{blue}{\left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot \left({b}^{2} \cdot x\right)
\]
associate-*l*54.9%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left({b}^{2} \cdot x\right)\right)}
\]
unpow254.9%
\[\leadsto a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot x\right)\right)
\]
Simplified54.9%
\[\leadsto \color{blue}{a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(\left(b \cdot b\right) \cdot x\right)\right)}
\]
Recombined 3 regimes into one program. Final simplification45.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;a \leq -7 \cdot 10^{+48}:\\
\;\;\;\;x \cdot \left(1 + b \cdot \left(b \cdot \left(a \cdot \left(a \cdot 0.5\right)\right) - a\right)\right)\\
\mathbf{elif}\;a \leq 5.1 \cdot 10^{+116}:\\
\;\;\;\;x + x \cdot \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(t \cdot t\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot \left(b \cdot b\right)\right)\right)\\
\end{array}
\]
Alternative 15: 23.3% accurate, 25.8× speedup? \[\begin{array}{l}
t_1 := x \cdot \left(y \cdot \left(-t\right)\right)\\
\mathbf{if}\;t \leq -5 \cdot 10^{+66}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;t \leq 1.15 \cdot 10^{+57}:\\
\;\;\;\;x\\
\mathbf{elif}\;t \leq 3.9 \cdot 10^{+246}:\\
\;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\
\mathbf{else}:\\
\;\;\;\;t_1\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -4.99999999999999991e66 or 3.9e246 < t Initial program 93.5%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 85.8%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg85.8%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out85.8%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified85.8%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 33.3%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative33.3%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg33.3%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg33.3%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified33.3%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in y around inf 32.0%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
Step-by-step derivation mul-1-neg32.0%
\[\leadsto \color{blue}{-y \cdot \left(t \cdot x\right)}
\]
associate-*r*35.0%
\[\leadsto -\color{blue}{\left(y \cdot t\right) \cdot x}
\]
*-commutative35.0%
\[\leadsto -\color{blue}{x \cdot \left(y \cdot t\right)}
\]
distribute-rgt-neg-in35.0%
\[\leadsto \color{blue}{x \cdot \left(-y \cdot t\right)}
\]
distribute-rgt-neg-in35.0%
\[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-t\right)\right)}
\]
Simplified35.0%
\[\leadsto \color{blue}{x \cdot \left(y \cdot \left(-t\right)\right)}
\]
if -4.99999999999999991e66 < t < 1.1499999999999999e57 Initial program 96.5%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 39.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg39.5%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out39.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified39.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 25.8%
\[\leadsto \color{blue}{x}
\]
if 1.1499999999999999e57 < t < 3.9e246 Initial program 95.4%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 41.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*41.5%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-141.5%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative41.5%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified41.5%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 21.2%
\[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}
\]
Step-by-step derivation mul-1-neg21.2%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right)
\]
unsub-neg21.2%
\[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)}
\]
Simplified21.2%
\[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)}
\]
Taylor expanded in a around inf 25.4%
\[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)}
\]
Step-by-step derivation associate-*r*25.4%
\[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot x\right)}
\]
neg-mul-125.4%
\[\leadsto \color{blue}{\left(-a\right)} \cdot \left(b \cdot x\right)
\]
Simplified25.4%
\[\leadsto \color{blue}{\left(-a\right) \cdot \left(b \cdot x\right)}
\]
Recombined 3 regimes into one program. Final simplification27.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -5 \cdot 10^{+66}:\\
\;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\
\mathbf{elif}\;t \leq 1.15 \cdot 10^{+57}:\\
\;\;\;\;x\\
\mathbf{elif}\;t \leq 3.9 \cdot 10^{+246}:\\
\;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\
\end{array}
\]
Alternative 16: 23.4% accurate, 25.8× speedup? \[\begin{array}{l}
t_1 := x \cdot \left(y \cdot \left(-t\right)\right)\\
\mathbf{if}\;t \leq -6.6 \cdot 10^{+65}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;t \leq 2.2 \cdot 10^{+56}:\\
\;\;\;\;x\\
\mathbf{elif}\;t \leq 2.25 \cdot 10^{+244}:\\
\;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\
\mathbf{else}:\\
\;\;\;\;t_1\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -6.60000000000000046e65 or 2.2500000000000001e244 < t Initial program 93.5%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 85.8%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg85.8%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out85.8%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified85.8%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 33.3%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative33.3%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg33.3%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg33.3%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified33.3%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in y around inf 32.0%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
Step-by-step derivation mul-1-neg32.0%
\[\leadsto \color{blue}{-y \cdot \left(t \cdot x\right)}
\]
associate-*r*35.0%
\[\leadsto -\color{blue}{\left(y \cdot t\right) \cdot x}
\]
*-commutative35.0%
\[\leadsto -\color{blue}{x \cdot \left(y \cdot t\right)}
\]
distribute-rgt-neg-in35.0%
\[\leadsto \color{blue}{x \cdot \left(-y \cdot t\right)}
\]
distribute-rgt-neg-in35.0%
\[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-t\right)\right)}
\]
Simplified35.0%
\[\leadsto \color{blue}{x \cdot \left(y \cdot \left(-t\right)\right)}
\]
if -6.60000000000000046e65 < t < 2.20000000000000016e56 Initial program 96.5%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 39.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg39.5%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out39.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified39.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 25.8%
\[\leadsto \color{blue}{x}
\]
if 2.20000000000000016e56 < t < 2.2500000000000001e244 Initial program 95.4%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 41.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*41.5%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-141.5%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative41.5%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified41.5%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 21.2%
\[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}
\]
Step-by-step derivation mul-1-neg21.2%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right)
\]
unsub-neg21.2%
\[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)}
\]
Simplified21.2%
\[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)}
\]
Taylor expanded in a around inf 25.4%
\[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)}
\]
Step-by-step derivation mul-1-neg25.4%
\[\leadsto \color{blue}{-a \cdot \left(b \cdot x\right)}
\]
associate-*r*27.5%
\[\leadsto -\color{blue}{\left(a \cdot b\right) \cdot x}
\]
distribute-rgt-neg-in27.5%
\[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \left(-x\right)}
\]
Simplified27.5%
\[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \left(-x\right)}
\]
Recombined 3 regimes into one program. Final simplification28.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -6.6 \cdot 10^{+65}:\\
\;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\
\mathbf{elif}\;t \leq 2.2 \cdot 10^{+56}:\\
\;\;\;\;x\\
\mathbf{elif}\;t \leq 2.25 \cdot 10^{+244}:\\
\;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\
\end{array}
\]
Alternative 17: 30.7% accurate, 28.3× speedup? \[\begin{array}{l}
\mathbf{if}\;a \leq -7.6 \cdot 10^{+50}:\\
\;\;\;\;x - b \cdot \left(x \cdot a\right)\\
\mathbf{elif}\;a \leq 2.8 \cdot 10^{+127}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\mathbf{else}:\\
\;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if a < -7.59999999999999975e50 Initial program 88.8%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 74.1%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*74.1%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-174.1%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative74.1%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified74.1%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 26.7%
\[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}
\]
Step-by-step derivation mul-1-neg26.7%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right)
\]
unsub-neg26.7%
\[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)}
\]
Simplified26.7%
\[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)}
\]
Taylor expanded in x around 0 26.7%
\[\leadsto \color{blue}{\left(1 - a \cdot b\right) \cdot x}
\]
Step-by-step derivation *-commutative26.7%
\[\leadsto \color{blue}{x \cdot \left(1 - a \cdot b\right)}
\]
sub-neg26.7%
\[\leadsto x \cdot \color{blue}{\left(1 + \left(-a \cdot b\right)\right)}
\]
mul-1-neg26.7%
\[\leadsto x \cdot \left(1 + \color{blue}{-1 \cdot \left(a \cdot b\right)}\right)
\]
distribute-rgt-in26.7%
\[\leadsto \color{blue}{1 \cdot x + \left(-1 \cdot \left(a \cdot b\right)\right) \cdot x}
\]
*-lft-identity26.7%
\[\leadsto \color{blue}{x} + \left(-1 \cdot \left(a \cdot b\right)\right) \cdot x
\]
mul-1-neg26.7%
\[\leadsto x + \color{blue}{\left(-a \cdot b\right)} \cdot x
\]
distribute-lft-neg-in26.7%
\[\leadsto x + \color{blue}{\left(-\left(a \cdot b\right) \cdot x\right)}
\]
associate-*r*24.9%
\[\leadsto x + \left(-\color{blue}{a \cdot \left(b \cdot x\right)}\right)
\]
unsub-neg24.9%
\[\leadsto \color{blue}{x - a \cdot \left(b \cdot x\right)}
\]
associate-*r*26.7%
\[\leadsto x - \color{blue}{\left(a \cdot b\right) \cdot x}
\]
*-commutative26.7%
\[\leadsto x - \color{blue}{\left(b \cdot a\right)} \cdot x
\]
associate-*l*31.8%
\[\leadsto x - \color{blue}{b \cdot \left(a \cdot x\right)}
\]
Simplified31.8%
\[\leadsto \color{blue}{x - b \cdot \left(a \cdot x\right)}
\]
if -7.59999999999999975e50 < a < 2.8000000000000002e127 Initial program 100.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 68.9%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg68.9%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out68.9%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified68.9%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 32.7%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative32.7%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg32.7%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg32.7%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified32.7%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in x around 0 35.0%
\[\leadsto \color{blue}{\left(1 - y \cdot t\right) \cdot x}
\]
Step-by-step derivation *-commutative35.0%
\[\leadsto \color{blue}{x \cdot \left(1 - y \cdot t\right)}
\]
distribute-rgt-out--35.0%
\[\leadsto \color{blue}{1 \cdot x - \left(y \cdot t\right) \cdot x}
\]
associate-*r*32.7%
\[\leadsto 1 \cdot x - \color{blue}{y \cdot \left(t \cdot x\right)}
\]
*-commutative32.7%
\[\leadsto 1 \cdot x - y \cdot \color{blue}{\left(x \cdot t\right)}
\]
*-lft-identity32.7%
\[\leadsto \color{blue}{x} - y \cdot \left(x \cdot t\right)
\]
*-commutative32.7%
\[\leadsto x - \color{blue}{\left(x \cdot t\right) \cdot y}
\]
*-commutative32.7%
\[\leadsto x - \color{blue}{\left(t \cdot x\right)} \cdot y
\]
associate-*l*35.5%
\[\leadsto x - \color{blue}{t \cdot \left(x \cdot y\right)}
\]
Simplified35.5%
\[\leadsto \color{blue}{x - t \cdot \left(x \cdot y\right)}
\]
if 2.8000000000000002e127 < a Initial program 84.5%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 78.9%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*78.9%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-178.9%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative78.9%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified78.9%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 18.7%
\[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}
\]
Step-by-step derivation mul-1-neg18.7%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right)
\]
unsub-neg18.7%
\[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)}
\]
Simplified18.7%
\[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)}
\]
Taylor expanded in a around inf 23.0%
\[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)}
\]
Step-by-step derivation associate-*r*23.0%
\[\leadsto \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot x\right)}
\]
neg-mul-123.0%
\[\leadsto \color{blue}{\left(-a\right)} \cdot \left(b \cdot x\right)
\]
Simplified23.0%
\[\leadsto \color{blue}{\left(-a\right) \cdot \left(b \cdot x\right)}
\]
Recombined 3 regimes into one program. Final simplification33.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;a \leq -7.6 \cdot 10^{+50}:\\
\;\;\;\;x - b \cdot \left(x \cdot a\right)\\
\mathbf{elif}\;a \leq 2.8 \cdot 10^{+127}:\\
\;\;\;\;x - t \cdot \left(x \cdot y\right)\\
\mathbf{else}:\\
\;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\
\end{array}
\]
Alternative 18: 23.4% accurate, 31.1× speedup? \[\begin{array}{l}
\mathbf{if}\;t \leq -2.65 \cdot 10^{+66} \lor \neg \left(t \leq 2.1 \cdot 10^{-5}\right):\\
\;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\
\mathbf{else}:\\
\;\;\;\;x\\
\end{array}
\]
Derivation Split input into 2 regimes if t < -2.6499999999999998e66 or 2.09999999999999988e-5 < t Initial program 93.8%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 77.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg77.5%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out77.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified77.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 24.6%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative24.6%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg24.6%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg24.6%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified24.6%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in y around inf 24.7%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
Step-by-step derivation mul-1-neg24.7%
\[\leadsto \color{blue}{-y \cdot \left(t \cdot x\right)}
\]
associate-*r*24.7%
\[\leadsto -\color{blue}{\left(y \cdot t\right) \cdot x}
\]
*-commutative24.7%
\[\leadsto -\color{blue}{x \cdot \left(y \cdot t\right)}
\]
distribute-rgt-neg-in24.7%
\[\leadsto \color{blue}{x \cdot \left(-y \cdot t\right)}
\]
distribute-rgt-neg-in24.7%
\[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-t\right)\right)}
\]
Simplified24.7%
\[\leadsto \color{blue}{x \cdot \left(y \cdot \left(-t\right)\right)}
\]
if -2.6499999999999998e66 < t < 2.09999999999999988e-5 Initial program 97.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 39.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg39.5%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out39.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified39.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 27.0%
\[\leadsto \color{blue}{x}
\]
Recombined 2 regimes into one program. Final simplification26.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -2.65 \cdot 10^{+66} \lor \neg \left(t \leq 2.1 \cdot 10^{-5}\right):\\
\;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\
\mathbf{else}:\\
\;\;\;\;x\\
\end{array}
\]
Alternative 19: 22.9% accurate, 31.1× speedup? \[\begin{array}{l}
\mathbf{if}\;t \leq -6.6 \cdot 10^{+65}:\\
\;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\
\mathbf{elif}\;t \leq 0.00032:\\
\;\;\;\;x\\
\mathbf{else}:\\
\;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -6.60000000000000046e65 Initial program 95.8%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 81.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg81.5%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out81.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified81.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 31.8%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative31.8%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg31.8%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg31.8%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified31.8%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in y around inf 30.0%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
Step-by-step derivation mul-1-neg30.0%
\[\leadsto \color{blue}{-y \cdot \left(t \cdot x\right)}
\]
associate-*r*30.1%
\[\leadsto -\color{blue}{\left(y \cdot t\right) \cdot x}
\]
*-commutative30.1%
\[\leadsto -\color{blue}{x \cdot \left(y \cdot t\right)}
\]
distribute-rgt-neg-in30.1%
\[\leadsto \color{blue}{x \cdot \left(-y \cdot t\right)}
\]
distribute-rgt-neg-in30.1%
\[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-t\right)\right)}
\]
Simplified30.1%
\[\leadsto \color{blue}{x \cdot \left(y \cdot \left(-t\right)\right)}
\]
if -6.60000000000000046e65 < t < 3.20000000000000026e-4 Initial program 97.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 39.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg39.5%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out39.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified39.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 27.0%
\[\leadsto \color{blue}{x}
\]
if 3.20000000000000026e-4 < t Initial program 92.4%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 74.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg74.5%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out74.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified74.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 19.4%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative19.4%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg19.4%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg19.4%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified19.4%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in y around inf 20.9%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
Step-by-step derivation mul-1-neg20.9%
\[\leadsto \color{blue}{-y \cdot \left(t \cdot x\right)}
\]
*-commutative20.9%
\[\leadsto -y \cdot \color{blue}{\left(x \cdot t\right)}
\]
distribute-rgt-neg-in20.9%
\[\leadsto \color{blue}{y \cdot \left(-x \cdot t\right)}
\]
distribute-lft-neg-in20.9%
\[\leadsto y \cdot \color{blue}{\left(\left(-x\right) \cdot t\right)}
\]
Simplified20.9%
\[\leadsto \color{blue}{y \cdot \left(\left(-x\right) \cdot t\right)}
\]
Recombined 3 regimes into one program. Final simplification26.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -6.6 \cdot 10^{+65}:\\
\;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\
\mathbf{elif}\;t \leq 0.00032:\\
\;\;\;\;x\\
\mathbf{else}:\\
\;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\
\end{array}
\]
Alternative 20: 21.4% accurate, 34.5× speedup? \[\begin{array}{l}
\mathbf{if}\;a \leq -7.3 \cdot 10^{+46} \lor \neg \left(a \leq 1.1 \cdot 10^{+125}\right):\\
\;\;\;\;y \cdot \left(x \cdot t\right)\\
\mathbf{else}:\\
\;\;\;\;x\\
\end{array}
\]
Derivation Split input into 2 regimes if a < -7.30000000000000028e46 or 1.09999999999999995e125 < a Initial program 87.7%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 31.1%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg31.1%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out31.1%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified31.1%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 10.8%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative10.8%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg10.8%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg10.8%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified10.8%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in y around inf 21.0%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
Step-by-step derivation mul-1-neg21.0%
\[\leadsto \color{blue}{-y \cdot \left(t \cdot x\right)}
\]
associate-*r*18.9%
\[\leadsto -\color{blue}{\left(y \cdot t\right) \cdot x}
\]
*-commutative18.9%
\[\leadsto -\color{blue}{x \cdot \left(y \cdot t\right)}
\]
distribute-rgt-neg-in18.9%
\[\leadsto \color{blue}{x \cdot \left(-y \cdot t\right)}
\]
distribute-rgt-neg-in18.9%
\[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-t\right)\right)}
\]
Simplified18.9%
\[\leadsto \color{blue}{x \cdot \left(y \cdot \left(-t\right)\right)}
\]
Step-by-step derivation expm1-log1p-u15.2%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(y \cdot \left(-t\right)\right)\right)\right)}
\]
expm1-udef29.5%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(y \cdot \left(-t\right)\right)\right)} - 1}
\]
*-commutative29.5%
\[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot \left(-t\right)\right) \cdot x}\right)} - 1
\]
associate-*l*29.5%
\[\leadsto e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left(\left(-t\right) \cdot x\right)}\right)} - 1
\]
add-sqr-sqrt17.9%
\[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(\color{blue}{\left(\sqrt{-t} \cdot \sqrt{-t}\right)} \cdot x\right)\right)} - 1
\]
sqrt-unprod30.5%
\[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(\color{blue}{\sqrt{\left(-t\right) \cdot \left(-t\right)}} \cdot x\right)\right)} - 1
\]
sqr-neg30.5%
\[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(\sqrt{\color{blue}{t \cdot t}} \cdot x\right)\right)} - 1
\]
sqrt-prod11.4%
\[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot x\right)\right)} - 1
\]
add-sqr-sqrt29.4%
\[\leadsto e^{\mathsf{log1p}\left(y \cdot \left(\color{blue}{t} \cdot x\right)\right)} - 1
\]
Applied egg-rr 29.4%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \left(t \cdot x\right)\right)} - 1}
\]
Step-by-step derivation expm1-def16.2%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(t \cdot x\right)\right)\right)}
\]
expm1-log1p18.8%
\[\leadsto \color{blue}{y \cdot \left(t \cdot x\right)}
\]
*-commutative18.8%
\[\leadsto y \cdot \color{blue}{\left(x \cdot t\right)}
\]
Simplified18.8%
\[\leadsto \color{blue}{y \cdot \left(x \cdot t\right)}
\]
if -7.30000000000000028e46 < a < 1.09999999999999995e125 Initial program 100.0%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 69.9%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg69.9%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out69.9%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified69.9%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 25.8%
\[\leadsto \color{blue}{x}
\]
Recombined 2 regimes into one program. Final simplification23.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;a \leq -7.3 \cdot 10^{+46} \lor \neg \left(a \leq 1.1 \cdot 10^{+125}\right):\\
\;\;\;\;y \cdot \left(x \cdot t\right)\\
\mathbf{else}:\\
\;\;\;\;x\\
\end{array}
\]
Alternative 21: 30.8% accurate, 34.8× speedup? \[\begin{array}{l}
\mathbf{if}\;y \leq 2.06 \cdot 10^{+21}:\\
\;\;\;\;x \cdot \left(1 - a \cdot b\right)\\
\mathbf{else}:\\
\;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if y < 2.06e21 Initial program 95.1%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in b around inf 64.2%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}}
\]
Step-by-step derivation associate-*r*64.2%
\[\leadsto x \cdot e^{\color{blue}{\left(-1 \cdot a\right) \cdot b}}
\]
neg-mul-164.2%
\[\leadsto x \cdot e^{\color{blue}{\left(-a\right)} \cdot b}
\]
*-commutative64.2%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Simplified64.2%
\[\leadsto x \cdot e^{\color{blue}{b \cdot \left(-a\right)}}
\]
Taylor expanded in b around 0 29.7%
\[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)}
\]
Step-by-step derivation mul-1-neg29.7%
\[\leadsto x \cdot \left(1 + \color{blue}{\left(-a \cdot b\right)}\right)
\]
unsub-neg29.7%
\[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)}
\]
Simplified29.7%
\[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)}
\]
if 2.06e21 < y Initial program 97.8%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 65.5%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg65.5%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out65.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified65.5%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 20.0%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right) + x}
\]
Step-by-step derivation +-commutative20.0%
\[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
mul-1-neg20.0%
\[\leadsto x + \color{blue}{\left(-y \cdot \left(t \cdot x\right)\right)}
\]
unsub-neg20.0%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Simplified20.0%
\[\leadsto \color{blue}{x - y \cdot \left(t \cdot x\right)}
\]
Taylor expanded in y around inf 28.4%
\[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(t \cdot x\right)\right)}
\]
Step-by-step derivation mul-1-neg28.4%
\[\leadsto \color{blue}{-y \cdot \left(t \cdot x\right)}
\]
*-commutative28.4%
\[\leadsto -y \cdot \color{blue}{\left(x \cdot t\right)}
\]
distribute-rgt-neg-in28.4%
\[\leadsto \color{blue}{y \cdot \left(-x \cdot t\right)}
\]
distribute-lft-neg-in28.4%
\[\leadsto y \cdot \color{blue}{\left(\left(-x\right) \cdot t\right)}
\]
Simplified28.4%
\[\leadsto \color{blue}{y \cdot \left(\left(-x\right) \cdot t\right)}
\]
Recombined 2 regimes into one program. Final simplification29.5%
\[\leadsto \begin{array}{l}
\mathbf{if}\;y \leq 2.06 \cdot 10^{+21}:\\
\;\;\;\;x \cdot \left(1 - a \cdot b\right)\\
\mathbf{else}:\\
\;\;\;\;y \cdot \left(t \cdot \left(-x\right)\right)\\
\end{array}
\]
Alternative 22: 19.3% accurate, 315.0× speedup? \[x
\]
Derivation Initial program 95.6%
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
\]
Taylor expanded in t around inf 56.1%
\[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}}
\]
Step-by-step derivation mul-1-neg56.1%
\[\leadsto x \cdot e^{\color{blue}{-y \cdot t}}
\]
distribute-rgt-neg-out56.1%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Simplified56.1%
\[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}}
\]
Taylor expanded in y around 0 18.5%
\[\leadsto \color{blue}{x}
\]
Final simplification18.5%
\[\leadsto x
\]
Reproduce ? herbie shell --seed 2023167
(FPCore (x y z t a b)
:name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
:precision binary64
(* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))
