NMSE Section 6.1 mentioned, A Percentage Accurate: 73.0% → 99.8%
Time: 19.1s
Alternatives: 18
Speedup: 227.0×
38.5% of points produce a very large (infinite) output. You may want to add a precondition. (more) could not determine a ground truth (more) Specification ? \[\begin{array}{l}
t_0 := \frac{1}{\varepsilon}\\
\frac{\left(1 + t_0\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(t_0 - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\end{array}
\]
Enter valid numbers for all inputs
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: 99.8% accurate, 1.0× speedup? \[\begin{array}{l}
t_0 := \left(1 + x\right) \cdot e^{-x}\\
t_1 := e^{\varepsilon \cdot \left(-x\right)}\\
\mathbf{if}\;\varepsilon \leq -1:\\
\;\;\;\;\frac{t_1 + e^{\varepsilon \cdot x}}{2}\\
\mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-29}:\\
\;\;\;\;\frac{t_0 + t_0}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + t_1}{2}\\
\end{array}
\]
Derivation Split input into 3 regimes if eps < -1 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation cancel-sign-sub-inv100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
mul-1-neg100.0%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
metadata-eval100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{1} \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
mul-1-neg100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{-\varepsilon \cdot x}}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{-\color{blue}{x \cdot \varepsilon}}}{2}
\]
distribute-lft-neg-out100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{\left(-x\right) \cdot \varepsilon}}}{2}
\]
*-lft-identity100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{e^{\left(-x\right) \cdot \varepsilon}}}{2}
\]
+-commutative100.0%
\[\leadsto \frac{\color{blue}{e^{\left(-x\right) \cdot \varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot \left(-x\right)}} + e^{-x \cdot \left(1 - \varepsilon\right)}}{2}
\]
distribute-lft-neg-in100.0%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot \left(-x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\varepsilon \cdot x}}}{2}
\]
if -1 < eps < 4.99999999999999986e-29 Initial program 38.8%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub38.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity38.8%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub38.8%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified38.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around 0 98.9%
\[\leadsto \frac{\color{blue}{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}}{2}
\]
Step-by-step derivation *-commutative98.9%
\[\leadsto \frac{\left(\color{blue}{x \cdot e^{-1 \cdot x}} + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2}
\]
distribute-lft1-in98.9%
\[\leadsto \frac{\color{blue}{\left(x + 1\right) \cdot e^{-1 \cdot x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2}
\]
neg-mul-198.9%
\[\leadsto \frac{\left(x + 1\right) \cdot e^{\color{blue}{-x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2}
\]
distribute-lft-out98.9%
\[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{-1 \cdot \left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)}}{2}
\]
mul-1-neg98.9%
\[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{\left(-\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)\right)}}{2}
\]
*-commutative98.9%
\[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(\color{blue}{x \cdot e^{-1 \cdot x}} + e^{-1 \cdot x}\right)\right)}{2}
\]
distribute-lft1-in100.0%
\[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\color{blue}{\left(x + 1\right) \cdot e^{-1 \cdot x}}\right)}{2}
\]
neg-mul-1100.0%
\[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)}{2}
\]
Simplified100.0%
\[\leadsto \frac{\color{blue}{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}}{2}
\]
if 4.99999999999999986e-29 < eps Initial program 97.5%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub97.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity97.5%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub97.5%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified97.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation cancel-sign-sub-inv100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
mul-1-neg100.0%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
metadata-eval100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{1} \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
mul-1-neg100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{-\varepsilon \cdot x}}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{-\color{blue}{x \cdot \varepsilon}}}{2}
\]
distribute-lft-neg-out100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{\left(-x\right) \cdot \varepsilon}}}{2}
\]
*-lft-identity100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{e^{\left(-x\right) \cdot \varepsilon}}}{2}
\]
+-commutative100.0%
\[\leadsto \frac{\color{blue}{e^{\left(-x\right) \cdot \varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot \left(-x\right)}} + e^{-x \cdot \left(1 - \varepsilon\right)}}{2}
\]
distribute-lft-neg-in100.0%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot \left(-x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Recombined 3 regimes into one program. Final simplification100.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\varepsilon \leq -1:\\
\;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\varepsilon \cdot x}}{2}\\
\mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-29}:\\
\;\;\;\;\frac{\left(1 + x\right) \cdot e^{-x} + \left(1 + x\right) \cdot e^{-x}}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\
\end{array}
\]
Alternative 2: 98.8% accurate, 1.1× speedup? \[\begin{array}{l}
t_0 := e^{x \cdot \left(-1 + \varepsilon\right)}\\
t_1 := e^{\varepsilon \cdot \left(-x\right)}\\
\mathbf{if}\;\varepsilon \leq -1:\\
\;\;\;\;\frac{t_1 + e^{\varepsilon \cdot x}}{2}\\
\mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-29}:\\
\;\;\;\;\frac{t_0 + e^{-x}}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{t_0 + t_1}{2}\\
\end{array}
\]
Derivation Split input into 3 regimes if eps < -1 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation cancel-sign-sub-inv100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
mul-1-neg100.0%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
metadata-eval100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{1} \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
mul-1-neg100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{-\varepsilon \cdot x}}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{-\color{blue}{x \cdot \varepsilon}}}{2}
\]
distribute-lft-neg-out100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{\left(-x\right) \cdot \varepsilon}}}{2}
\]
*-lft-identity100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{e^{\left(-x\right) \cdot \varepsilon}}}{2}
\]
+-commutative100.0%
\[\leadsto \frac{\color{blue}{e^{\left(-x\right) \cdot \varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot \left(-x\right)}} + e^{-x \cdot \left(1 - \varepsilon\right)}}{2}
\]
distribute-lft-neg-in100.0%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot \left(-x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\varepsilon \cdot x}}}{2}
\]
if -1 < eps < 4.99999999999999986e-29 Initial program 38.8%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub38.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity38.8%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub38.8%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified38.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 95.7%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around 0 95.7%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{x}}}{2}
\]
if 4.99999999999999986e-29 < eps Initial program 97.5%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub97.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity97.5%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub97.5%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified97.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation cancel-sign-sub-inv100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
mul-1-neg100.0%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
metadata-eval100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{1} \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
mul-1-neg100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{-\varepsilon \cdot x}}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{-\color{blue}{x \cdot \varepsilon}}}{2}
\]
distribute-lft-neg-out100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{\left(-x\right) \cdot \varepsilon}}}{2}
\]
*-lft-identity100.0%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{e^{\left(-x\right) \cdot \varepsilon}}}{2}
\]
+-commutative100.0%
\[\leadsto \frac{\color{blue}{e^{\left(-x\right) \cdot \varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
*-commutative100.0%
\[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot \left(-x\right)}} + e^{-x \cdot \left(1 - \varepsilon\right)}}{2}
\]
distribute-lft-neg-in100.0%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot \left(-x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Recombined 3 regimes into one program. Final simplification98.4%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\varepsilon \leq -1:\\
\;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\varepsilon \cdot x}}{2}\\
\mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-29}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\
\end{array}
\]
Alternative 3: 98.8% accurate, 1.1× speedup? \[\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}
\]
Derivation Initial program 76.5%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub76.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity76.5%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub76.5%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified76.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 98.4%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Final simplification98.4%
\[\leadsto \frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}
\]
Alternative 4: 92.1% accurate, 1.1× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq 135:\\
\;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\varepsilon \cdot x}}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\
\end{array}
\]
Derivation Split input into 2 regimes if x < 135 Initial program 67.1%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub67.1%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity67.1%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub67.1%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified67.1%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 98.2%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 98.2%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative98.2%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified98.2%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 98.2%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation cancel-sign-sub-inv98.2%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
mul-1-neg98.2%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
*-commutative98.2%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
metadata-eval98.2%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{1} \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
mul-1-neg98.2%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{-\varepsilon \cdot x}}}{2}
\]
*-commutative98.2%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{-\color{blue}{x \cdot \varepsilon}}}{2}
\]
distribute-lft-neg-out98.2%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{\left(-x\right) \cdot \varepsilon}}}{2}
\]
*-lft-identity98.2%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{e^{\left(-x\right) \cdot \varepsilon}}}{2}
\]
+-commutative98.2%
\[\leadsto \frac{\color{blue}{e^{\left(-x\right) \cdot \varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
*-commutative98.2%
\[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot \left(-x\right)}} + e^{-x \cdot \left(1 - \varepsilon\right)}}{2}
\]
distribute-lft-neg-in98.2%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified98.2%
\[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot \left(-x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 98.4%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\varepsilon \cdot x}}}{2}
\]
if 135 < x Initial program 98.7%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub98.7%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity98.7%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub98.7%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified98.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 34.1%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around 0 31.2%
\[\leadsto \frac{\color{blue}{\frac{1 + -1 \cdot x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Step-by-step derivation mul-1-neg31.2%
\[\leadsto \frac{\frac{1 + \color{blue}{\left(-x\right)}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
unsub-neg31.2%
\[\leadsto \frac{\frac{\color{blue}{1 - x}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Simplified31.2%
\[\leadsto \frac{\color{blue}{\frac{1 - x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around inf 75.6%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Step-by-step derivation exp-prod75.6%
\[\leadsto \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
+-commutative75.6%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}}{2}
\]
*-lft-identity75.6%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}}{2}
\]
metadata-eval75.6%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}}{2}
\]
cancel-sign-sub-inv75.6%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}}{2}
\]
exp-prod75.6%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}}{2}
\]
mul-1-neg75.6%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}}{2}
\]
*-commutative75.6%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}}{2}
\]
sub-neg75.6%
\[\leadsto \frac{e^{-x \cdot \color{blue}{\left(1 + \left(--1 \cdot \varepsilon\right)\right)}}}{2}
\]
mul-1-neg75.6%
\[\leadsto \frac{e^{-x \cdot \left(1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)}}{2}
\]
remove-double-neg75.6%
\[\leadsto \frac{e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}}{2}
\]
Simplified75.6%
\[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 + \varepsilon\right)}}}{2}
\]
Recombined 2 regimes into one program. Final simplification91.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq 135:\\
\;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\varepsilon \cdot x}}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\
\end{array}
\]
Alternative 5: 82.1% accurate, 1.7× speedup? \[\begin{array}{l}
t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\
t_1 := \frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 6.5 \cdot 10^{-99}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;x \leq 3.3 \cdot 10^{-57}:\\
\;\;\;\;\frac{\left(1 - \left(1 - \varepsilon\right) \cdot x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + t_0 \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\
\mathbf{elif}\;x \leq 1.3 \cdot 10^{-12}:\\
\;\;\;\;t_1\\
\mathbf{else}:\\
\;\;\;\;\frac{t_0}{2}\\
\end{array}
\]
Derivation Split input into 4 regimes if x < -4900 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around 0 100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot x}}}{2}
\]
Step-by-step derivation neg-mul-1100.0%
\[\leadsto \frac{1 + e^{\color{blue}{-x}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-x}}}{2}
\]
if -4900 < x < 6.50000000000000033e-99 or 3.2999999999999998e-57 < x < 1.29999999999999991e-12 Initial program 51.5%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub51.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity51.5%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub51.5%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified51.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 97.4%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 97.4%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative97.4%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified97.4%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 97.4%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation cancel-sign-sub-inv97.4%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
mul-1-neg97.4%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
*-commutative97.4%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
metadata-eval97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{1} \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
mul-1-neg97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{-\varepsilon \cdot x}}}{2}
\]
*-commutative97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{-\color{blue}{x \cdot \varepsilon}}}{2}
\]
distribute-lft-neg-out97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{\left(-x\right) \cdot \varepsilon}}}{2}
\]
*-lft-identity97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{e^{\left(-x\right) \cdot \varepsilon}}}{2}
\]
+-commutative97.4%
\[\leadsto \frac{\color{blue}{e^{\left(-x\right) \cdot \varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
*-commutative97.4%
\[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot \left(-x\right)}} + e^{-x \cdot \left(1 - \varepsilon\right)}}{2}
\]
distribute-lft-neg-in97.4%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified97.4%
\[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot \left(-x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around 0 82.2%
\[\leadsto \frac{\color{blue}{\left(-1 \cdot \left(\varepsilon \cdot x\right) + 1\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
Step-by-step derivation +-commutative82.2%
\[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(\varepsilon \cdot x\right)\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
mul-1-neg82.2%
\[\leadsto \frac{\left(1 + \color{blue}{\left(-\varepsilon \cdot x\right)}\right) + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
unsub-neg82.2%
\[\leadsto \frac{\color{blue}{\left(1 - \varepsilon \cdot x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
Simplified82.2%
\[\leadsto \frac{\color{blue}{\left(1 - \varepsilon \cdot x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
if 6.50000000000000033e-99 < x < 3.2999999999999998e-57 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified78.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 92.6%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
if 1.29999999999999991e-12 < x Initial program 98.8%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub98.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity98.8%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub98.8%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified98.9%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 35.4%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around 0 32.6%
\[\leadsto \frac{\color{blue}{\frac{1 + -1 \cdot x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Step-by-step derivation mul-1-neg32.6%
\[\leadsto \frac{\frac{1 + \color{blue}{\left(-x\right)}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
unsub-neg32.6%
\[\leadsto \frac{\frac{\color{blue}{1 - x}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Simplified32.6%
\[\leadsto \frac{\color{blue}{\frac{1 - x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around inf 75.2%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Step-by-step derivation exp-prod75.2%
\[\leadsto \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
+-commutative75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}}{2}
\]
*-lft-identity75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}}{2}
\]
metadata-eval75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}}{2}
\]
cancel-sign-sub-inv75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}}{2}
\]
exp-prod75.2%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}}{2}
\]
mul-1-neg75.2%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}}{2}
\]
*-commutative75.2%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}}{2}
\]
sub-neg75.2%
\[\leadsto \frac{e^{-x \cdot \color{blue}{\left(1 + \left(--1 \cdot \varepsilon\right)\right)}}}{2}
\]
mul-1-neg75.2%
\[\leadsto \frac{e^{-x \cdot \left(1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)}}{2}
\]
remove-double-neg75.2%
\[\leadsto \frac{e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}}{2}
\]
Simplified75.2%
\[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 + \varepsilon\right)}}}{2}
\]
Recombined 4 regimes into one program. Final simplification83.5%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 6.5 \cdot 10^{-99}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\
\mathbf{elif}\;x \leq 3.3 \cdot 10^{-57}:\\
\;\;\;\;\frac{\left(1 - \left(1 - \varepsilon\right) \cdot x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\
\mathbf{elif}\;x \leq 1.3 \cdot 10^{-12}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\
\end{array}
\]
Alternative 6: 82.0% accurate, 1.8× speedup? \[\begin{array}{l}
t_0 := \frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\
t_1 := e^{x \cdot \left(-1 - \varepsilon\right)}\\
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 6.5 \cdot 10^{-99}:\\
\;\;\;\;t_0\\
\mathbf{elif}\;x \leq 3.7 \cdot 10^{-57}:\\
\;\;\;\;\frac{\left(1 + \varepsilon \cdot x\right) + t_1 \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\
\mathbf{elif}\;x \leq 1.3 \cdot 10^{-12}:\\
\;\;\;\;t_0\\
\mathbf{else}:\\
\;\;\;\;\frac{t_1}{2}\\
\end{array}
\]
Derivation Split input into 4 regimes if x < -4900 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around 0 100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot x}}}{2}
\]
Step-by-step derivation neg-mul-1100.0%
\[\leadsto \frac{1 + e^{\color{blue}{-x}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-x}}}{2}
\]
if -4900 < x < 6.50000000000000033e-99 or 3.7e-57 < x < 1.29999999999999991e-12 Initial program 51.5%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub51.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity51.5%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub51.5%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified51.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 97.4%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 97.4%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative97.4%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified97.4%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 97.4%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation cancel-sign-sub-inv97.4%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
mul-1-neg97.4%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
*-commutative97.4%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
metadata-eval97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{1} \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
mul-1-neg97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{-\varepsilon \cdot x}}}{2}
\]
*-commutative97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{-\color{blue}{x \cdot \varepsilon}}}{2}
\]
distribute-lft-neg-out97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{\left(-x\right) \cdot \varepsilon}}}{2}
\]
*-lft-identity97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{e^{\left(-x\right) \cdot \varepsilon}}}{2}
\]
+-commutative97.4%
\[\leadsto \frac{\color{blue}{e^{\left(-x\right) \cdot \varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
*-commutative97.4%
\[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot \left(-x\right)}} + e^{-x \cdot \left(1 - \varepsilon\right)}}{2}
\]
distribute-lft-neg-in97.4%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified97.4%
\[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot \left(-x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around 0 82.2%
\[\leadsto \frac{\color{blue}{\left(-1 \cdot \left(\varepsilon \cdot x\right) + 1\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
Step-by-step derivation +-commutative82.2%
\[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(\varepsilon \cdot x\right)\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
mul-1-neg82.2%
\[\leadsto \frac{\left(1 + \color{blue}{\left(-\varepsilon \cdot x\right)}\right) + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
unsub-neg82.2%
\[\leadsto \frac{\color{blue}{\left(1 - \varepsilon \cdot x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
Simplified82.2%
\[\leadsto \frac{\color{blue}{\left(1 - \varepsilon \cdot x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
if 6.50000000000000033e-99 < x < 3.7e-57 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified78.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 92.6%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around inf 89.1%
\[\leadsto \frac{\color{blue}{\left(1 + \left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Step-by-step derivation +-commutative89.1%
\[\leadsto \frac{\color{blue}{\left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
distribute-lft1-in89.1%
\[\leadsto \frac{\left(\left(\varepsilon \cdot x + \color{blue}{\left(-1 + 1\right) \cdot x}\right) + 1\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
metadata-eval89.1%
\[\leadsto \frac{\left(\left(\varepsilon \cdot x + \color{blue}{0} \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
mul0-lft89.1%
\[\leadsto \frac{\left(\left(\varepsilon \cdot x + \color{blue}{0}\right) + 1\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
associate-+l+89.1%
\[\leadsto \frac{\color{blue}{\left(\varepsilon \cdot x + \left(0 + 1\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
metadata-eval89.1%
\[\leadsto \frac{\left(\varepsilon \cdot x + \color{blue}{1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
*-commutative89.1%
\[\leadsto \frac{\left(\color{blue}{x \cdot \varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Simplified89.1%
\[\leadsto \frac{\color{blue}{\left(x \cdot \varepsilon + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
if 1.29999999999999991e-12 < x Initial program 98.8%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub98.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity98.8%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub98.8%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified98.9%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 35.4%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around 0 32.6%
\[\leadsto \frac{\color{blue}{\frac{1 + -1 \cdot x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Step-by-step derivation mul-1-neg32.6%
\[\leadsto \frac{\frac{1 + \color{blue}{\left(-x\right)}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
unsub-neg32.6%
\[\leadsto \frac{\frac{\color{blue}{1 - x}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Simplified32.6%
\[\leadsto \frac{\color{blue}{\frac{1 - x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around inf 75.2%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Step-by-step derivation exp-prod75.2%
\[\leadsto \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
+-commutative75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}}{2}
\]
*-lft-identity75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}}{2}
\]
metadata-eval75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}}{2}
\]
cancel-sign-sub-inv75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}}{2}
\]
exp-prod75.2%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}}{2}
\]
mul-1-neg75.2%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}}{2}
\]
*-commutative75.2%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}}{2}
\]
sub-neg75.2%
\[\leadsto \frac{e^{-x \cdot \color{blue}{\left(1 + \left(--1 \cdot \varepsilon\right)\right)}}}{2}
\]
mul-1-neg75.2%
\[\leadsto \frac{e^{-x \cdot \left(1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)}}{2}
\]
remove-double-neg75.2%
\[\leadsto \frac{e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}}{2}
\]
Simplified75.2%
\[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 + \varepsilon\right)}}}{2}
\]
Recombined 4 regimes into one program. Final simplification83.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 6.5 \cdot 10^{-99}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\
\mathbf{elif}\;x \leq 3.7 \cdot 10^{-57}:\\
\;\;\;\;\frac{\left(1 + \varepsilon \cdot x\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\
\mathbf{elif}\;x \leq 1.3 \cdot 10^{-12}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\
\end{array}
\]
Alternative 7: 82.1% accurate, 1.8× speedup? \[\begin{array}{l}
t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\
t_1 := \frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 5.5 \cdot 10^{-99}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;x \leq 3.7 \cdot 10^{-57}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right) \cdot t_0}{2}\\
\mathbf{elif}\;x \leq 5.8 \cdot 10^{-13}:\\
\;\;\;\;t_1\\
\mathbf{else}:\\
\;\;\;\;\frac{t_0}{2}\\
\end{array}
\]
Derivation Split input into 4 regimes if x < -4900 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around 0 100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot x}}}{2}
\]
Step-by-step derivation neg-mul-1100.0%
\[\leadsto \frac{1 + e^{\color{blue}{-x}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-x}}}{2}
\]
if -4900 < x < 5.49999999999999991e-99 or 3.7e-57 < x < 5.7999999999999995e-13 Initial program 51.5%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub51.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity51.5%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub51.5%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified51.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 97.4%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 97.4%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative97.4%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified97.4%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 97.4%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation cancel-sign-sub-inv97.4%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
mul-1-neg97.4%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
*-commutative97.4%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
metadata-eval97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{1} \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
mul-1-neg97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{-\varepsilon \cdot x}}}{2}
\]
*-commutative97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{-\color{blue}{x \cdot \varepsilon}}}{2}
\]
distribute-lft-neg-out97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{\left(-x\right) \cdot \varepsilon}}}{2}
\]
*-lft-identity97.4%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{e^{\left(-x\right) \cdot \varepsilon}}}{2}
\]
+-commutative97.4%
\[\leadsto \frac{\color{blue}{e^{\left(-x\right) \cdot \varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
*-commutative97.4%
\[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot \left(-x\right)}} + e^{-x \cdot \left(1 - \varepsilon\right)}}{2}
\]
distribute-lft-neg-in97.4%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified97.4%
\[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot \left(-x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around 0 82.2%
\[\leadsto \frac{\color{blue}{\left(-1 \cdot \left(\varepsilon \cdot x\right) + 1\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
Step-by-step derivation +-commutative82.2%
\[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(\varepsilon \cdot x\right)\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
mul-1-neg82.2%
\[\leadsto \frac{\left(1 + \color{blue}{\left(-\varepsilon \cdot x\right)}\right) + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
unsub-neg82.2%
\[\leadsto \frac{\color{blue}{\left(1 - \varepsilon \cdot x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
Simplified82.2%
\[\leadsto \frac{\color{blue}{\left(1 - \varepsilon \cdot x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
if 5.49999999999999991e-99 < x < 3.7e-57 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 92.5%
\[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
if 5.7999999999999995e-13 < x Initial program 98.8%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub98.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity98.8%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub98.8%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified98.9%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 35.4%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around 0 32.6%
\[\leadsto \frac{\color{blue}{\frac{1 + -1 \cdot x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Step-by-step derivation mul-1-neg32.6%
\[\leadsto \frac{\frac{1 + \color{blue}{\left(-x\right)}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
unsub-neg32.6%
\[\leadsto \frac{\frac{\color{blue}{1 - x}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Simplified32.6%
\[\leadsto \frac{\color{blue}{\frac{1 - x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around inf 75.2%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Step-by-step derivation exp-prod75.2%
\[\leadsto \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
+-commutative75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}}{2}
\]
*-lft-identity75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}}{2}
\]
metadata-eval75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}}{2}
\]
cancel-sign-sub-inv75.2%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}}{2}
\]
exp-prod75.2%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}}{2}
\]
mul-1-neg75.2%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}}{2}
\]
*-commutative75.2%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}}{2}
\]
sub-neg75.2%
\[\leadsto \frac{e^{-x \cdot \color{blue}{\left(1 + \left(--1 \cdot \varepsilon\right)\right)}}}{2}
\]
mul-1-neg75.2%
\[\leadsto \frac{e^{-x \cdot \left(1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)}}{2}
\]
remove-double-neg75.2%
\[\leadsto \frac{e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}}{2}
\]
Simplified75.2%
\[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 + \varepsilon\right)}}}{2}
\]
Recombined 4 regimes into one program. Final simplification83.5%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 5.5 \cdot 10^{-99}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\
\mathbf{elif}\;x \leq 3.7 \cdot 10^{-57}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\
\mathbf{elif}\;x \leq 5.8 \cdot 10^{-13}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\
\end{array}
\]
Alternative 8: 83.1% accurate, 1.9× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 1.45 \cdot 10^{-77} \lor \neg \left(x \leq 3.5 \cdot 10^{-57}\right) \land x \leq 6.4 \cdot 10^{-12}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -4900 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around 0 100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot x}}}{2}
\]
Step-by-step derivation neg-mul-1100.0%
\[\leadsto \frac{1 + e^{\color{blue}{-x}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-x}}}{2}
\]
if -4900 < x < 1.4499999999999999e-77 or 3.49999999999999991e-57 < x < 6.4000000000000002e-12 Initial program 53.8%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub53.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity53.8%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub53.8%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified53.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 97.5%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 97.5%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative97.5%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified97.5%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 97.5%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation cancel-sign-sub-inv97.5%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2}
\]
mul-1-neg97.5%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
*-commutative97.5%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
metadata-eval97.5%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{1} \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2}
\]
mul-1-neg97.5%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{-\varepsilon \cdot x}}}{2}
\]
*-commutative97.5%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{-\color{blue}{x \cdot \varepsilon}}}{2}
\]
distribute-lft-neg-out97.5%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{\left(-x\right) \cdot \varepsilon}}}{2}
\]
*-lft-identity97.5%
\[\leadsto \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{e^{\left(-x\right) \cdot \varepsilon}}}{2}
\]
+-commutative97.5%
\[\leadsto \frac{\color{blue}{e^{\left(-x\right) \cdot \varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
*-commutative97.5%
\[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot \left(-x\right)}} + e^{-x \cdot \left(1 - \varepsilon\right)}}{2}
\]
distribute-lft-neg-in97.5%
\[\leadsto \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified97.5%
\[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot \left(-x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around 0 81.6%
\[\leadsto \frac{\color{blue}{\left(-1 \cdot \left(\varepsilon \cdot x\right) + 1\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
Step-by-step derivation +-commutative81.6%
\[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(\varepsilon \cdot x\right)\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
mul-1-neg81.6%
\[\leadsto \frac{\left(1 + \color{blue}{\left(-\varepsilon \cdot x\right)}\right) + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
unsub-neg81.6%
\[\leadsto \frac{\color{blue}{\left(1 - \varepsilon \cdot x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
Simplified81.6%
\[\leadsto \frac{\color{blue}{\left(1 - \varepsilon \cdot x\right)} + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}{2}
\]
if 1.4499999999999999e-77 < x < 3.49999999999999991e-57 or 6.4000000000000002e-12 < x Initial program 98.9%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub98.9%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity98.9%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub98.9%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified99.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 40.7%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around 0 38.1%
\[\leadsto \frac{\color{blue}{\frac{1 + -1 \cdot x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Step-by-step derivation mul-1-neg38.1%
\[\leadsto \frac{\frac{1 + \color{blue}{\left(-x\right)}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
unsub-neg38.1%
\[\leadsto \frac{\frac{\color{blue}{1 - x}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Simplified38.1%
\[\leadsto \frac{\color{blue}{\frac{1 - x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around inf 77.3%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Step-by-step derivation exp-prod77.3%
\[\leadsto \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
+-commutative77.3%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}}{2}
\]
*-lft-identity77.3%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}}{2}
\]
metadata-eval77.3%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}}{2}
\]
cancel-sign-sub-inv77.3%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}}{2}
\]
exp-prod77.3%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}}{2}
\]
mul-1-neg77.3%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}}{2}
\]
*-commutative77.3%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}}{2}
\]
sub-neg77.3%
\[\leadsto \frac{e^{-x \cdot \color{blue}{\left(1 + \left(--1 \cdot \varepsilon\right)\right)}}}{2}
\]
mul-1-neg77.3%
\[\leadsto \frac{e^{-x \cdot \left(1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)}}{2}
\]
remove-double-neg77.3%
\[\leadsto \frac{e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}}{2}
\]
Simplified77.3%
\[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 + \varepsilon\right)}}}{2}
\]
Recombined 3 regimes into one program. Final simplification83.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 1.45 \cdot 10^{-77} \lor \neg \left(x \leq 3.5 \cdot 10^{-57}\right) \land x \leq 6.4 \cdot 10^{-12}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\
\end{array}
\]
Alternative 9: 82.9% accurate, 2.0× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 1.45 \cdot 10^{-77} \lor \neg \left(x \leq 2 \cdot 10^{-56}\right) \land x \leq 6.4 \cdot 10^{-12}:\\
\;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -4900 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around 0 100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot x}}}{2}
\]
Step-by-step derivation neg-mul-1100.0%
\[\leadsto \frac{1 + e^{\color{blue}{-x}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-x}}}{2}
\]
if -4900 < x < 1.4499999999999999e-77 or 2.0000000000000001e-56 < x < 6.4000000000000002e-12 Initial program 53.8%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub53.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity53.8%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub53.8%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified53.8%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 38.1%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2}
\]
Taylor expanded in eps around inf 81.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + 1}}{2}
\]
Step-by-step derivation associate-*r*81.0%
\[\leadsto \frac{e^{\color{blue}{\left(-1 \cdot \left(1 - \varepsilon\right)\right) \cdot x}} + 1}{2}
\]
sub-neg81.0%
\[\leadsto \frac{e^{\left(-1 \cdot \color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right) \cdot x} + 1}{2}
\]
neg-mul-181.0%
\[\leadsto \frac{e^{\left(-1 \cdot \left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot x} + 1}{2}
\]
associate-*r*81.0%
\[\leadsto \frac{e^{\color{blue}{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2}
\]
+-commutative81.0%
\[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}}}{2}
\]
*-commutative81.0%
\[\leadsto \frac{1 + e^{-1 \cdot \color{blue}{\left(x \cdot \left(1 + -1 \cdot \varepsilon\right)\right)}}}{2}
\]
neg-mul-181.0%
\[\leadsto \frac{1 + e^{-1 \cdot \left(x \cdot \left(1 + \color{blue}{\left(-\varepsilon\right)}\right)\right)}}{2}
\]
sub-neg81.0%
\[\leadsto \frac{1 + e^{-1 \cdot \left(x \cdot \color{blue}{\left(1 - \varepsilon\right)}\right)}}{2}
\]
neg-mul-181.0%
\[\leadsto \frac{1 + e^{\color{blue}{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
distribute-lft-neg-in81.0%
\[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified81.0%
\[\leadsto \frac{\color{blue}{1 + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 81.2%
\[\leadsto \frac{1 + e^{\color{blue}{\varepsilon \cdot x}}}{2}
\]
if 1.4499999999999999e-77 < x < 2.0000000000000001e-56 or 6.4000000000000002e-12 < x Initial program 98.9%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub98.9%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity98.9%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub98.9%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified99.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 40.7%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around 0 38.1%
\[\leadsto \frac{\color{blue}{\frac{1 + -1 \cdot x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Step-by-step derivation mul-1-neg38.1%
\[\leadsto \frac{\frac{1 + \color{blue}{\left(-x\right)}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
unsub-neg38.1%
\[\leadsto \frac{\frac{\color{blue}{1 - x}}{\varepsilon} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Simplified38.1%
\[\leadsto \frac{\color{blue}{\frac{1 - x}{\varepsilon}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around inf 77.3%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Step-by-step derivation exp-prod77.3%
\[\leadsto \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
+-commutative77.3%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}}{2}
\]
*-lft-identity77.3%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{1 \cdot \varepsilon}\right) \cdot x\right)}}{2}
\]
metadata-eval77.3%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot x\right)}}{2}
\]
cancel-sign-sub-inv77.3%
\[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 - -1 \cdot \varepsilon\right)} \cdot x\right)}}{2}
\]
exp-prod77.3%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}}{2}
\]
mul-1-neg77.3%
\[\leadsto \frac{e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}}{2}
\]
*-commutative77.3%
\[\leadsto \frac{e^{-\color{blue}{x \cdot \left(1 - -1 \cdot \varepsilon\right)}}}{2}
\]
sub-neg77.3%
\[\leadsto \frac{e^{-x \cdot \color{blue}{\left(1 + \left(--1 \cdot \varepsilon\right)\right)}}}{2}
\]
mul-1-neg77.3%
\[\leadsto \frac{e^{-x \cdot \left(1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)}}{2}
\]
remove-double-neg77.3%
\[\leadsto \frac{e^{-x \cdot \left(1 + \color{blue}{\varepsilon}\right)}}{2}
\]
Simplified77.3%
\[\leadsto \frac{\color{blue}{e^{-x \cdot \left(1 + \varepsilon\right)}}}{2}
\]
Recombined 3 regimes into one program. Final simplification83.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 1.45 \cdot 10^{-77} \lor \neg \left(x \leq 2 \cdot 10^{-56}\right) \land x \leq 6.4 \cdot 10^{-12}:\\
\;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\
\end{array}
\]
Alternative 10: 77.1% accurate, 2.0× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 520000000000:\\
\;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -4900 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around 0 100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot x}}}{2}
\]
Step-by-step derivation neg-mul-1100.0%
\[\leadsto \frac{1 + e^{\color{blue}{-x}}}{2}
\]
Simplified100.0%
\[\leadsto \frac{\color{blue}{1 + e^{-x}}}{2}
\]
if -4900 < x < 5.2e11 Initial program 57.1%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub57.1%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity57.1%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub57.1%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified57.1%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 35.8%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2}
\]
Taylor expanded in eps around inf 75.1%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + 1}}{2}
\]
Step-by-step derivation associate-*r*75.1%
\[\leadsto \frac{e^{\color{blue}{\left(-1 \cdot \left(1 - \varepsilon\right)\right) \cdot x}} + 1}{2}
\]
sub-neg75.1%
\[\leadsto \frac{e^{\left(-1 \cdot \color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right) \cdot x} + 1}{2}
\]
neg-mul-175.1%
\[\leadsto \frac{e^{\left(-1 \cdot \left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot x} + 1}{2}
\]
associate-*r*75.1%
\[\leadsto \frac{e^{\color{blue}{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2}
\]
+-commutative75.1%
\[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}}}{2}
\]
*-commutative75.1%
\[\leadsto \frac{1 + e^{-1 \cdot \color{blue}{\left(x \cdot \left(1 + -1 \cdot \varepsilon\right)\right)}}}{2}
\]
neg-mul-175.1%
\[\leadsto \frac{1 + e^{-1 \cdot \left(x \cdot \left(1 + \color{blue}{\left(-\varepsilon\right)}\right)\right)}}{2}
\]
sub-neg75.1%
\[\leadsto \frac{1 + e^{-1 \cdot \left(x \cdot \color{blue}{\left(1 - \varepsilon\right)}\right)}}{2}
\]
neg-mul-175.1%
\[\leadsto \frac{1 + e^{\color{blue}{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
distribute-lft-neg-in75.1%
\[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified75.1%
\[\leadsto \frac{\color{blue}{1 + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around inf 75.2%
\[\leadsto \frac{1 + e^{\color{blue}{\varepsilon \cdot x}}}{2}
\]
if 5.2e11 < x Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}}
\]
Taylor expanded in eps around 0 46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
Step-by-step derivation div-sub46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
rec-exp46.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2}
\]
neg-mul-146.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2}
\]
+-inverses46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Simplified46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Recombined 3 regimes into one program. Final simplification71.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -4900:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{elif}\;x \leq 520000000000:\\
\;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Alternative 11: 70.3% accurate, 2.1× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq 13000:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Derivation Split input into 2 regimes if x < 13000 Initial program 67.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub67.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity67.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub67.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified67.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in eps around inf 97.8%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2}
\]
Taylor expanded in eps around inf 97.7%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}}}{2}
\]
Step-by-step derivation *-commutative97.7%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Simplified97.7%
\[\leadsto \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2}
\]
Taylor expanded in eps around 0 72.0%
\[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot x}}}{2}
\]
Step-by-step derivation neg-mul-172.0%
\[\leadsto \frac{1 + e^{\color{blue}{-x}}}{2}
\]
Simplified72.0%
\[\leadsto \frac{\color{blue}{1 + e^{-x}}}{2}
\]
if 13000 < x Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}}
\]
Taylor expanded in eps around 0 46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
Step-by-step derivation div-sub46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
rec-exp46.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2}
\]
neg-mul-146.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2}
\]
+-inverses46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Simplified46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Recombined 2 regimes into one program. Final simplification64.7%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq 13000:\\
\;\;\;\;\frac{1 + e^{-x}}{2}\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Alternative 12: 60.4% accurate, 8.4× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -1.4 \cdot 10^{+69}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \left(1 - \varepsilon\right) \cdot x\right) + \left(1 + \left(\left(x - x\right) - \varepsilon \cdot x\right)\right)}{2}\\
\mathbf{elif}\;x \leq 13000:\\
\;\;\;\;1\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -1.39999999999999991e69 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 46.3%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in x around 0 0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \color{blue}{\left(\left(\frac{1}{\varepsilon} + -1 \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)\right)\right) - 1\right)}}{2}
\]
Step-by-step derivation associate--l+0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1 \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)\right) - 1\right)\right)}}{2}
\]
associate-*r*0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\color{blue}{\left(-1 \cdot \left(\frac{1}{\varepsilon} - 1\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)} - 1\right)\right)}{2}
\]
neg-mul-10.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\color{blue}{\left(-\left(\frac{1}{\varepsilon} - 1\right)\right)} \cdot \left(\left(1 + \varepsilon\right) \cdot x\right) - 1\right)\right)}{2}
\]
sub-neg0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\left(-\color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)}\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right) - 1\right)\right)}{2}
\]
metadata-eval0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\left(-\left(\frac{1}{\varepsilon} + \color{blue}{-1}\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right) - 1\right)\right)}{2}
\]
+-commutative0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\left(-\color{blue}{\left(-1 + \frac{1}{\varepsilon}\right)}\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right) - 1\right)\right)}{2}
\]
*-commutative0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\left(-\left(-1 + \frac{1}{\varepsilon}\right)\right) \cdot \color{blue}{\left(x \cdot \left(1 + \varepsilon\right)\right)} - 1\right)\right)}{2}
\]
+-commutative0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\left(-\left(-1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(x \cdot \color{blue}{\left(\varepsilon + 1\right)}\right) - 1\right)\right)}{2}
\]
Simplified0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \color{blue}{\left(\frac{1}{\varepsilon} + \left(\left(-\left(-1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(x \cdot \left(\varepsilon + 1\right)\right) - 1\right)\right)}}{2}
\]
Taylor expanded in eps around inf 0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \color{blue}{\left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}}{2}
\]
Step-by-step derivation add-sqr-sqrt0.0%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) + 1\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
sqrt-unprod43.7%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot \color{blue}{\sqrt{x \cdot x}}\right) + 1\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
sqr-neg43.7%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}\right) + 1\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
sqrt-unprod40.9%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}\right) + 1\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
add-sqr-sqrt40.9%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot \color{blue}{\left(-x\right)}\right) + 1\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
distribute-rgt-neg-out40.9%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \color{blue}{\left(-\left(1 - \varepsilon\right) \cdot x\right)} + 1\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
*-commutative40.9%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(-\color{blue}{x \cdot \left(1 - \varepsilon\right)}\right) + 1\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
Applied egg-rr 40.9%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)} + 1\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
if -1.39999999999999991e69 < x < 13000 Initial program 59.7%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub59.7%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity59.7%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub59.7%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified59.7%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 60.1%
\[\leadsto \frac{\color{blue}{2}}{2}
\]
if 13000 < x Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}}
\]
Taylor expanded in eps around 0 46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
Step-by-step derivation div-sub46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
rec-exp46.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2}
\]
neg-mul-146.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2}
\]
+-inverses46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Simplified46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Recombined 3 regimes into one program. Final simplification53.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -1.4 \cdot 10^{+69}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \left(1 - \varepsilon\right) \cdot x\right) + \left(1 + \left(\left(x - x\right) - \varepsilon \cdot x\right)\right)}{2}\\
\mathbf{elif}\;x \leq 13000:\\
\;\;\;\;1\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Alternative 13: 60.4% accurate, 11.9× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -1.12 \cdot 10^{+69}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \left(\left(x - x\right) - \varepsilon \cdot x\right)\right)}{2}\\
\mathbf{elif}\;x \leq 13000:\\
\;\;\;\;1\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -1.12e69 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 46.3%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in x around 0 0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \color{blue}{\left(\left(\frac{1}{\varepsilon} + -1 \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)\right)\right) - 1\right)}}{2}
\]
Step-by-step derivation associate--l+0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1 \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)\right) - 1\right)\right)}}{2}
\]
associate-*r*0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\color{blue}{\left(-1 \cdot \left(\frac{1}{\varepsilon} - 1\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)} - 1\right)\right)}{2}
\]
neg-mul-10.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\color{blue}{\left(-\left(\frac{1}{\varepsilon} - 1\right)\right)} \cdot \left(\left(1 + \varepsilon\right) \cdot x\right) - 1\right)\right)}{2}
\]
sub-neg0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\left(-\color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)}\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right) - 1\right)\right)}{2}
\]
metadata-eval0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\left(-\left(\frac{1}{\varepsilon} + \color{blue}{-1}\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right) - 1\right)\right)}{2}
\]
+-commutative0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\left(-\color{blue}{\left(-1 + \frac{1}{\varepsilon}\right)}\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right) - 1\right)\right)}{2}
\]
*-commutative0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\left(-\left(-1 + \frac{1}{\varepsilon}\right)\right) \cdot \color{blue}{\left(x \cdot \left(1 + \varepsilon\right)\right)} - 1\right)\right)}{2}
\]
+-commutative0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \left(\frac{1}{\varepsilon} + \left(\left(-\left(-1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(x \cdot \color{blue}{\left(\varepsilon + 1\right)}\right) - 1\right)\right)}{2}
\]
Simplified0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \color{blue}{\left(\frac{1}{\varepsilon} + \left(\left(-\left(-1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(x \cdot \left(\varepsilon + 1\right)\right) - 1\right)\right)}}{2}
\]
Taylor expanded in eps around inf 0.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \color{blue}{\left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}}{2}
\]
Taylor expanded in x around 0 40.9%
\[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
Step-by-step derivation unpow-140.9%
\[\leadsto \frac{\left(\color{blue}{{\varepsilon}^{-1}} + 1\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
+-commutative40.9%
\[\leadsto \frac{\color{blue}{\left(1 + {\varepsilon}^{-1}\right)} - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
remove-double-neg40.9%
\[\leadsto \frac{\left(1 + \color{blue}{\left(-\left(-{\varepsilon}^{-1}\right)\right)}\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
sub-neg40.9%
\[\leadsto \frac{\color{blue}{\left(1 - \left(-{\varepsilon}^{-1}\right)\right)} - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
unpow-140.9%
\[\leadsto \frac{\left(1 - \left(-\color{blue}{\frac{1}{\varepsilon}}\right)\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
distribute-neg-frac40.9%
\[\leadsto \frac{\left(1 - \color{blue}{\frac{-1}{\varepsilon}}\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
metadata-eval40.9%
\[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
Simplified40.9%
\[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\left(\varepsilon \cdot x + \left(-1 \cdot x + x\right)\right) - 1\right)}{2}
\]
if -1.12e69 < x < 13000 Initial program 59.7%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub59.7%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity59.7%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub59.7%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified59.7%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 60.1%
\[\leadsto \frac{\color{blue}{2}}{2}
\]
if 13000 < x Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}}
\]
Taylor expanded in eps around 0 46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
Step-by-step derivation div-sub46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
rec-exp46.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2}
\]
neg-mul-146.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2}
\]
+-inverses46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Simplified46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Recombined 3 regimes into one program. Final simplification53.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -1.12 \cdot 10^{+69}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \left(\left(x - x\right) - \varepsilon \cdot x\right)\right)}{2}\\
\mathbf{elif}\;x \leq 13000:\\
\;\;\;\;1\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Alternative 14: 59.8% accurate, 20.5× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq 6.4 \cdot 10^{-12}:\\
\;\;\;\;\frac{2 - \left(1 - \varepsilon\right) \cdot x}{2}\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Derivation Split input into 2 regimes if x < 6.4000000000000002e-12 Initial program 66.6%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub66.6%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity66.6%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub66.6%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified66.6%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 39.4%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2}
\]
Taylor expanded in eps around inf 70.4%
\[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + 1}}{2}
\]
Step-by-step derivation associate-*r*70.4%
\[\leadsto \frac{e^{\color{blue}{\left(-1 \cdot \left(1 - \varepsilon\right)\right) \cdot x}} + 1}{2}
\]
sub-neg70.4%
\[\leadsto \frac{e^{\left(-1 \cdot \color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right) \cdot x} + 1}{2}
\]
neg-mul-170.4%
\[\leadsto \frac{e^{\left(-1 \cdot \left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot x} + 1}{2}
\]
associate-*r*70.4%
\[\leadsto \frac{e^{\color{blue}{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2}
\]
+-commutative70.4%
\[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}}}{2}
\]
*-commutative70.4%
\[\leadsto \frac{1 + e^{-1 \cdot \color{blue}{\left(x \cdot \left(1 + -1 \cdot \varepsilon\right)\right)}}}{2}
\]
neg-mul-170.4%
\[\leadsto \frac{1 + e^{-1 \cdot \left(x \cdot \left(1 + \color{blue}{\left(-\varepsilon\right)}\right)\right)}}{2}
\]
sub-neg70.4%
\[\leadsto \frac{1 + e^{-1 \cdot \left(x \cdot \color{blue}{\left(1 - \varepsilon\right)}\right)}}{2}
\]
neg-mul-170.4%
\[\leadsto \frac{1 + e^{\color{blue}{-x \cdot \left(1 - \varepsilon\right)}}}{2}
\]
distribute-lft-neg-in70.4%
\[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Simplified70.4%
\[\leadsto \frac{\color{blue}{1 + e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2}
\]
Taylor expanded in x around 0 55.2%
\[\leadsto \frac{\color{blue}{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 2}}{2}
\]
if 6.4000000000000002e-12 < x Initial program 98.8%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Simplified98.8%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}}
\]
Taylor expanded in eps around 0 44.0%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
Step-by-step derivation div-sub44.0%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
rec-exp44.0%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2}
\]
neg-mul-144.0%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2}
\]
+-inverses44.0%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Simplified44.0%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Recombined 2 regimes into one program. Final simplification51.7%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq 6.4 \cdot 10^{-12}:\\
\;\;\;\;\frac{2 - \left(1 - \varepsilon\right) \cdot x}{2}\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Alternative 15: 60.6% accurate, 25.0× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -1.5 \cdot 10^{+160}:\\
\;\;\;\;\frac{-0.25 \cdot \left(x \cdot x\right)}{\varepsilon}\\
\mathbf{elif}\;x \leq 13000:\\
\;\;\;\;1\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -1.4999999999999999e160 Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity100.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub100.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 43.5%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in eps around 0 47.6%
\[\leadsto \frac{\color{blue}{\frac{\left(1 + -1 \cdot x\right) - e^{-1 \cdot x}}{\varepsilon}}}{2}
\]
Step-by-step derivation mul-1-neg47.6%
\[\leadsto \frac{\frac{\left(1 + \color{blue}{\left(-x\right)}\right) - e^{-1 \cdot x}}{\varepsilon}}{2}
\]
unsub-neg47.6%
\[\leadsto \frac{\frac{\color{blue}{\left(1 - x\right)} - e^{-1 \cdot x}}{\varepsilon}}{2}
\]
mul-1-neg47.6%
\[\leadsto \frac{\frac{\left(1 - x\right) - e^{\color{blue}{-x}}}{\varepsilon}}{2}
\]
Simplified47.6%
\[\leadsto \frac{\color{blue}{\frac{\left(1 - x\right) - e^{-x}}{\varepsilon}}}{2}
\]
Taylor expanded in x around 0 47.6%
\[\leadsto \color{blue}{-0.25 \cdot \frac{{x}^{2}}{\varepsilon}}
\]
Step-by-step derivation associate-*r/47.6%
\[\leadsto \color{blue}{\frac{-0.25 \cdot {x}^{2}}{\varepsilon}}
\]
unpow247.6%
\[\leadsto \frac{-0.25 \cdot \color{blue}{\left(x \cdot x\right)}}{\varepsilon}
\]
Simplified47.6%
\[\leadsto \color{blue}{\frac{-0.25 \cdot \left(x \cdot x\right)}{\varepsilon}}
\]
if -1.4999999999999999e160 < x < 13000 Initial program 62.7%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub62.7%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity62.7%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub62.7%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified62.7%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 55.9%
\[\leadsto \frac{\color{blue}{2}}{2}
\]
if 13000 < x Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}}
\]
Taylor expanded in eps around 0 46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
Step-by-step derivation div-sub46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
rec-exp46.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2}
\]
neg-mul-146.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2}
\]
+-inverses46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Simplified46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Recombined 3 regimes into one program. Final simplification52.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -1.5 \cdot 10^{+160}:\\
\;\;\;\;\frac{-0.25 \cdot \left(x \cdot x\right)}{\varepsilon}\\
\mathbf{elif}\;x \leq 13000:\\
\;\;\;\;1\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Alternative 16: 60.7% accurate, 32.1× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -1:\\
\;\;\;\;\frac{\varepsilon \cdot x}{2}\\
\mathbf{elif}\;x \leq 13000:\\
\;\;\;\;1\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -1 Initial program 93.5%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub93.5%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity93.5%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub93.5%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified93.4%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 41.6%
\[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}
\]
Taylor expanded in x around inf 21.9%
\[\leadsto \frac{\color{blue}{-1 \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)\right)}}{2}
\]
Step-by-step derivation mul-1-neg21.9%
\[\leadsto \frac{\color{blue}{-\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)}}{2}
\]
*-commutative21.9%
\[\leadsto \frac{-\left(\frac{1}{\varepsilon} + 1\right) \cdot \color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right)}}{2}
\]
*-commutative21.9%
\[\leadsto \frac{-\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2}
\]
unpow-121.9%
\[\leadsto \frac{-\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(\color{blue}{{\varepsilon}^{-1}} + 1\right)}{2}
\]
+-commutative21.9%
\[\leadsto \frac{-\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \color{blue}{\left(1 + {\varepsilon}^{-1}\right)}}{2}
\]
associate-*l*21.9%
\[\leadsto \frac{-\color{blue}{x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + {\varepsilon}^{-1}\right)\right)}}{2}
\]
+-commutative21.9%
\[\leadsto \frac{-x \cdot \left(\left(1 - \varepsilon\right) \cdot \color{blue}{\left({\varepsilon}^{-1} + 1\right)}\right)}{2}
\]
unpow-121.9%
\[\leadsto \frac{-x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(\color{blue}{\frac{1}{\varepsilon}} + 1\right)\right)}{2}
\]
*-commutative21.9%
\[\leadsto \frac{-x \cdot \color{blue}{\left(\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(1 - \varepsilon\right)\right)}}{2}
\]
distribute-rgt-neg-in21.9%
\[\leadsto \frac{\color{blue}{x \cdot \left(-\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(1 - \varepsilon\right)\right)}}{2}
\]
unpow-121.9%
\[\leadsto \frac{x \cdot \left(-\left(\color{blue}{{\varepsilon}^{-1}} + 1\right) \cdot \left(1 - \varepsilon\right)\right)}{2}
\]
+-commutative21.9%
\[\leadsto \frac{x \cdot \left(-\color{blue}{\left(1 + {\varepsilon}^{-1}\right)} \cdot \left(1 - \varepsilon\right)\right)}{2}
\]
distribute-rgt-neg-in21.9%
\[\leadsto \frac{x \cdot \color{blue}{\left(\left(1 + {\varepsilon}^{-1}\right) \cdot \left(-\left(1 - \varepsilon\right)\right)\right)}}{2}
\]
Simplified21.9%
\[\leadsto \frac{\color{blue}{x \cdot \left(\left(1 - \frac{-1}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right)}}{2}
\]
Taylor expanded in eps around inf 21.7%
\[\leadsto \frac{x \cdot \color{blue}{\varepsilon}}{2}
\]
if -1 < x < 13000 Initial program 58.3%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub58.3%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity58.3%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub58.3%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified58.3%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 65.1%
\[\leadsto \frac{\color{blue}{2}}{2}
\]
if 13000 < x Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}}
\]
Taylor expanded in eps around 0 46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
Step-by-step derivation div-sub46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
rec-exp46.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2}
\]
neg-mul-146.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2}
\]
+-inverses46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Simplified46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Recombined 3 regimes into one program. Final simplification52.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -1:\\
\;\;\;\;\frac{\varepsilon \cdot x}{2}\\
\mathbf{elif}\;x \leq 13000:\\
\;\;\;\;1\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Alternative 17: 57.3% accurate, 74.1× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq 13000:\\
\;\;\;\;1\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Derivation Split input into 2 regimes if x < 13000 Initial program 67.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Step-by-step derivation div-sub67.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
+-rgt-identity67.0%
\[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
div-sub67.0%
\[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}}
\]
Simplified67.0%
\[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}}
\]
Taylor expanded in x around 0 49.8%
\[\leadsto \frac{\color{blue}{2}}{2}
\]
if 13000 < x Initial program 100.0%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Simplified100.0%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}}
\]
Taylor expanded in eps around 0 46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
Step-by-step derivation div-sub46.8%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
rec-exp46.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2}
\]
neg-mul-146.8%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2}
\]
+-inverses46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Simplified46.8%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Recombined 2 regimes into one program. Final simplification48.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq 13000:\\
\;\;\;\;1\\
\mathbf{else}:\\
\;\;\;\;0\\
\end{array}
\]
Alternative 18: 15.9% accurate, 227.0× speedup? \[0
\]
Derivation Initial program 76.5%
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
\]
Simplified69.8%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}}
\]
Taylor expanded in eps around 0 15.0%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
Step-by-step derivation div-sub15.0%
\[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2}
\]
rec-exp14.9%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2}
\]
neg-mul-114.9%
\[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2}
\]
+-inverses15.2%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Simplified15.2%
\[\leadsto \frac{\color{blue}{0}}{2}
\]
Final simplification15.2%
\[\leadsto 0
\]
Reproduce ? herbie shell --seed 2023167
(FPCore (x eps)
:name "NMSE Section 6.1 mentioned, A"
:precision binary64
(/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))
