2nthrt (problem 3.4.6) Percentage Accurate: 54.2% → 85.2%
Time: 18.2s
Alternatives: 18
Speedup: 70.3×
could not determine a ground truth (more) Specification ? \[\begin{array}{l}
t_0 := \frac{1}{n}\\
{\left(x + 1\right)}^{t_0} - {x}^{t_0}
\end{array}
\]
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: 85.2% accurate, 0.1× speedup? \[\begin{array}{l}
t_0 := \log \left(1 + x\right)\\
t_1 := {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{t_1}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\
\;\;\;\;\left(0.5 \cdot \frac{{t_0}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{t_0}^{4}}{{n}^{4}} + \left(0.16666666666666666 \cdot \frac{{t_0}^{3}}{{n}^{3}} + \frac{t_0}{n}\right)\right)\right) - \left(\frac{\log x}{n} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;e^{\frac{x}{n}} - t_1\\
\end{array}
\]
Derivation Split input into 3 regimes if (/.f64 1 n) < -1e-107 Initial program 81.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 93.8%
\[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}}
\]
Step-by-step derivation log-rec93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
distribute-frac-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
neg-mul-193.8%
\[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
remove-double-neg93.8%
\[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x}
\]
*-rgt-identity93.8%
\[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x}
\]
associate-*r/93.8%
\[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x}
\]
exp-to-pow93.8%
\[\leadsto \frac{\color{blue}{{x}^{\left({n}^{-1}\right)}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x}
\]
*-commutative93.8%
\[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}}
\]
Simplified93.8%
\[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}}
\]
if -1e-107 < (/.f64 1 n) < 9.9999999999999995e-8 Initial program 33.5%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 87.0%
\[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log \left(1 + x\right)}^{4}}{{n}^{4}} + \left(0.16666666666666666 \cdot \frac{{\log \left(1 + x\right)}^{3}}{{n}^{3}} + \frac{\log \left(1 + x\right)}{n}\right)\right)\right) - \left(\frac{\log x}{n} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right)\right)}
\]
if 9.9999999999999995e-8 < (/.f64 1 n) Initial program 55.5%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around 0 55.5%
\[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}} - e^{\frac{\log x}{n}}}
\]
Step-by-step derivation log1p-def97.7%
\[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - e^{\frac{\log x}{n}}
\]
*-rgt-identity97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\color{blue}{\log x \cdot 1}}{n}}
\]
associate-*r/97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\color{blue}{\log x \cdot \frac{1}{n}}}
\]
unpow-197.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\log x \cdot \color{blue}{{n}^{-1}}}
\]
exp-to-pow97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \color{blue}{{x}^{\left({n}^{-1}\right)}}
\]
/-rgt-identity97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{1}\right)}}
\]
metadata-eval97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{\frac{2}{2}}}\right)}
\]
associate-/l*97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1} \cdot 2}{2}\right)}}
\]
*-commutative97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{2 \cdot {n}^{-1}}}{2}\right)}
\]
*-commutative97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{{n}^{-1} \cdot 2}}{2}\right)}
\]
associate-/l*97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{\frac{2}{2}}\right)}}
\]
metadata-eval97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{1}}\right)}
\]
/-rgt-identity97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left({n}^{-1}\right)}}
\]
unpow-197.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{1}{n}\right)}}
\]
Simplified97.7%
\[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}
\]
Taylor expanded in x around 0 97.7%
\[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)}
\]
Recombined 3 regimes into one program. Final simplification91.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\
\;\;\;\;\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log \left(1 + x\right)}^{4}}{{n}^{4}} + \left(0.16666666666666666 \cdot \frac{{\log \left(1 + x\right)}^{3}}{{n}^{3}} + \frac{\log \left(1 + x\right)}{n}\right)\right)\right) - \left(\frac{\log x}{n} + \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\
\end{array}
\]
Alternative 2: 85.2% accurate, 0.2× speedup? \[\begin{array}{l}
t_0 := {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{t_0}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\
\;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\
\mathbf{else}:\\
\;\;\;\;e^{\frac{x}{n}} - t_0\\
\end{array}
\]
Derivation Split input into 3 regimes if (/.f64 1 n) < -1e-107 Initial program 81.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 93.8%
\[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}}
\]
Step-by-step derivation log-rec93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
distribute-frac-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
neg-mul-193.8%
\[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
remove-double-neg93.8%
\[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x}
\]
*-rgt-identity93.8%
\[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x}
\]
associate-*r/93.8%
\[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x}
\]
exp-to-pow93.8%
\[\leadsto \frac{\color{blue}{{x}^{\left({n}^{-1}\right)}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x}
\]
*-commutative93.8%
\[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}}
\]
Simplified93.8%
\[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}}
\]
if -1e-107 < (/.f64 1 n) < 9.9999999999999995e-8 Initial program 33.5%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around -inf 87.0%
\[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + -1 \cdot \frac{-1 \cdot \log \left(1 + x\right) - -1 \cdot \log x}{n}\right)\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}}
\]
Step-by-step derivation sub-neg87.0%
\[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + -1 \cdot \frac{-1 \cdot \log \left(1 + x\right) - -1 \cdot \log x}{n}\right)\right) + \left(-0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)}
\]
Simplified87.0%
\[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5}
\]
if 9.9999999999999995e-8 < (/.f64 1 n) Initial program 55.5%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around 0 55.5%
\[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}} - e^{\frac{\log x}{n}}}
\]
Step-by-step derivation log1p-def97.7%
\[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - e^{\frac{\log x}{n}}
\]
*-rgt-identity97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\color{blue}{\log x \cdot 1}}{n}}
\]
associate-*r/97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\color{blue}{\log x \cdot \frac{1}{n}}}
\]
unpow-197.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\log x \cdot \color{blue}{{n}^{-1}}}
\]
exp-to-pow97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \color{blue}{{x}^{\left({n}^{-1}\right)}}
\]
/-rgt-identity97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{1}\right)}}
\]
metadata-eval97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{\frac{2}{2}}}\right)}
\]
associate-/l*97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1} \cdot 2}{2}\right)}}
\]
*-commutative97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{2 \cdot {n}^{-1}}}{2}\right)}
\]
*-commutative97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{{n}^{-1} \cdot 2}}{2}\right)}
\]
associate-/l*97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{\frac{2}{2}}\right)}}
\]
metadata-eval97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{1}}\right)}
\]
/-rgt-identity97.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left({n}^{-1}\right)}}
\]
unpow-197.7%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{1}{n}\right)}}
\]
Simplified97.7%
\[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}
\]
Taylor expanded in x around 0 97.7%
\[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)}
\]
Recombined 3 regimes into one program. Final simplification91.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\
\;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\
\mathbf{else}:\\
\;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\
\end{array}
\]
Alternative 3: 85.2% accurate, 0.3× speedup? \[\begin{array}{l}
t_0 := {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{t_0}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\
\mathbf{else}:\\
\;\;\;\;e^{\frac{x}{n}} - t_0\\
\end{array}
\]
Derivation Split input into 3 regimes if (/.f64 1 n) < -1e-107 Initial program 81.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 93.8%
\[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}}
\]
Step-by-step derivation log-rec93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
distribute-frac-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
neg-mul-193.8%
\[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
remove-double-neg93.8%
\[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x}
\]
*-rgt-identity93.8%
\[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x}
\]
associate-*r/93.8%
\[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x}
\]
exp-to-pow93.8%
\[\leadsto \frac{\color{blue}{{x}^{\left({n}^{-1}\right)}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x}
\]
*-commutative93.8%
\[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}}
\]
Simplified93.8%
\[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}}
\]
if -1e-107 < (/.f64 1 n) < 5.0000000000000001e-9 Initial program 33.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 86.9%
\[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \left(\frac{\log x}{n} + 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)}
\]
Step-by-step derivation associate--r+81.7%
\[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}}
\]
sub-neg81.7%
\[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \frac{\log x}{n}\right) + \left(-0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)}
\]
Simplified86.9%
\[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5}
\]
if 5.0000000000000001e-9 < (/.f64 1 n) Initial program 56.0%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around 0 56.0%
\[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}} - e^{\frac{\log x}{n}}}
\]
Step-by-step derivation log1p-def97.2%
\[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - e^{\frac{\log x}{n}}
\]
*-rgt-identity97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\color{blue}{\log x \cdot 1}}{n}}
\]
associate-*r/97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\color{blue}{\log x \cdot \frac{1}{n}}}
\]
unpow-197.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\log x \cdot \color{blue}{{n}^{-1}}}
\]
exp-to-pow97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \color{blue}{{x}^{\left({n}^{-1}\right)}}
\]
/-rgt-identity97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{1}\right)}}
\]
metadata-eval97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{\frac{2}{2}}}\right)}
\]
associate-/l*97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1} \cdot 2}{2}\right)}}
\]
*-commutative97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{2 \cdot {n}^{-1}}}{2}\right)}
\]
*-commutative97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{{n}^{-1} \cdot 2}}{2}\right)}
\]
associate-/l*97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{\frac{2}{2}}\right)}}
\]
metadata-eval97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{1}}\right)}
\]
/-rgt-identity97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left({n}^{-1}\right)}}
\]
unpow-197.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{1}{n}\right)}}
\]
Simplified97.2%
\[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}
\]
Taylor expanded in x around 0 97.2%
\[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)}
\]
Recombined 3 regimes into one program. Final simplification91.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\
\mathbf{else}:\\
\;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\
\end{array}
\]
Alternative 4: 85.0% accurate, 0.3× speedup? \[\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-15}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{else}:\\
\;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}}\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if (/.f64 1 n) < -1e-107 Initial program 81.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 93.8%
\[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}}
\]
Step-by-step derivation log-rec93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
distribute-frac-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
neg-mul-193.8%
\[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
remove-double-neg93.8%
\[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x}
\]
*-rgt-identity93.8%
\[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x}
\]
associate-*r/93.8%
\[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x}
\]
exp-to-pow93.8%
\[\leadsto \frac{\color{blue}{{x}^{\left({n}^{-1}\right)}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x}
\]
*-commutative93.8%
\[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}}
\]
Simplified93.8%
\[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}}
\]
if -1e-107 < (/.f64 1 n) < 2.0000000000000002e-15 Initial program 32.9%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 86.7%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def86.7%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified86.7%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-udef86.7%
\[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n}
\]
diff-log86.7%
\[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n}
\]
+-commutative86.7%
\[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n}
\]
Applied egg-rr 86.7%
\[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n}
\]
if 2.0000000000000002e-15 < (/.f64 1 n) Initial program 56.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Step-by-step derivation add-log-exp56.2%
\[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}
\]
pow-to-exp56.2%
\[\leadsto \log \left(e^{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right)
\]
un-div-inv56.2%
\[\leadsto \log \left(e^{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right)
\]
+-commutative56.2%
\[\leadsto \log \left(e^{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)
\]
log1p-udef96.5%
\[\leadsto \log \left(e^{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)
\]
inv-pow96.5%
\[\leadsto \log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left({n}^{-1}\right)}}}\right)
\]
Applied egg-rr 96.5%
\[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}}\right)}
\]
Recombined 3 regimes into one program. Final simplification91.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-15}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{else}:\\
\;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}}\right)\\
\end{array}
\]
Alternative 5: 85.2% accurate, 1.0× speedup? \[\begin{array}{l}
t_0 := {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{t_0}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{else}:\\
\;\;\;\;e^{\frac{x}{n}} - t_0\\
\end{array}
\]
Derivation Split input into 3 regimes if (/.f64 1 n) < -1e-107 Initial program 81.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 93.8%
\[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}}
\]
Step-by-step derivation log-rec93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
distribute-frac-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
neg-mul-193.8%
\[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
remove-double-neg93.8%
\[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x}
\]
*-rgt-identity93.8%
\[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x}
\]
associate-*r/93.8%
\[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x}
\]
exp-to-pow93.8%
\[\leadsto \frac{\color{blue}{{x}^{\left({n}^{-1}\right)}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x}
\]
*-commutative93.8%
\[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}}
\]
Simplified93.8%
\[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}}
\]
if -1e-107 < (/.f64 1 n) < 5.0000000000000001e-9 Initial program 33.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 86.5%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def86.5%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified86.5%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-udef86.5%
\[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n}
\]
diff-log86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n}
\]
+-commutative86.5%
\[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n}
\]
Applied egg-rr 86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n}
\]
if 5.0000000000000001e-9 < (/.f64 1 n) Initial program 56.0%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around 0 56.0%
\[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}} - e^{\frac{\log x}{n}}}
\]
Step-by-step derivation log1p-def97.2%
\[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - e^{\frac{\log x}{n}}
\]
*-rgt-identity97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\color{blue}{\log x \cdot 1}}{n}}
\]
associate-*r/97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\color{blue}{\log x \cdot \frac{1}{n}}}
\]
unpow-197.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\log x \cdot \color{blue}{{n}^{-1}}}
\]
exp-to-pow97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \color{blue}{{x}^{\left({n}^{-1}\right)}}
\]
/-rgt-identity97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{1}\right)}}
\]
metadata-eval97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{\frac{2}{2}}}\right)}
\]
associate-/l*97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1} \cdot 2}{2}\right)}}
\]
*-commutative97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{2 \cdot {n}^{-1}}}{2}\right)}
\]
*-commutative97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{{n}^{-1} \cdot 2}}{2}\right)}
\]
associate-/l*97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{\frac{2}{2}}\right)}}
\]
metadata-eval97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{1}}\right)}
\]
/-rgt-identity97.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left({n}^{-1}\right)}}
\]
unpow-197.2%
\[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{1}{n}\right)}}
\]
Simplified97.2%
\[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}
\]
Taylor expanded in x around 0 97.2%
\[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)}
\]
Recombined 3 regimes into one program. Final simplification91.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{else}:\\
\;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\
\end{array}
\]
Alternative 6: 53.4% accurate, 1.5× speedup? \[\begin{array}{l}
t_0 := \frac{x - \log x}{n}\\
t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\
t_2 := \frac{1}{n \cdot x}\\
\mathbf{if}\;\frac{1}{n} \leq -1000000:\\
\;\;\;\;t_1\\
\mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-109}:\\
\;\;\;\;t_2\\
\mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-197}:\\
\;\;\;\;t_0\\
\mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-215}:\\
\;\;\;\;t_2\\
\mathbf{elif}\;\frac{1}{n} \leq 10^{-295}:\\
\;\;\;\;\frac{-\log x}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-214}:\\
\;\;\;\;\frac{\frac{1}{x}}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;t_0\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+154}:\\
\;\;\;\;t_1\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Derivation Split input into 6 regimes if (/.f64 1 n) < -1e6 or 5.0000000000000001e-9 < (/.f64 1 n) < 5.00000000000000004e154 Initial program 96.0%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 66.1%
\[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}}
\]
Step-by-step derivation *-rgt-identity66.1%
\[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}}
\]
associate-*r/66.1%
\[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}}
\]
unpow-166.1%
\[\leadsto 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}}
\]
exp-to-pow66.1%
\[\leadsto 1 - \color{blue}{{x}^{\left({n}^{-1}\right)}}
\]
unpow-166.1%
\[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}}
\]
Simplified66.1%
\[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}}
\]
if -1e6 < (/.f64 1 n) < -9.9999999999999999e-110 or -5.0000000000000002e-197 < (/.f64 1 n) < -4.99999999999999956e-215 Initial program 30.2%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 54.9%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def54.9%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified54.9%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 75.3%
\[\leadsto \color{blue}{\frac{1}{n \cdot x}}
\]
Step-by-step derivation *-commutative75.3%
\[\leadsto \frac{1}{\color{blue}{x \cdot n}}
\]
Simplified75.3%
\[\leadsto \color{blue}{\frac{1}{x \cdot n}}
\]
if -9.9999999999999999e-110 < (/.f64 1 n) < -5.0000000000000002e-197 or 1.99999999999999983e-214 < (/.f64 1 n) < 5.0000000000000001e-9 Initial program 19.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 5.0%
\[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 67.1%
\[\leadsto \color{blue}{\frac{x - \log x}{n}}
\]
if -4.99999999999999956e-215 < (/.f64 1 n) < 1.00000000000000006e-295 Initial program 40.0%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 40.0%
\[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}}
\]
Step-by-step derivation *-rgt-identity40.0%
\[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}}
\]
associate-*r/40.0%
\[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}}
\]
unpow-140.0%
\[\leadsto 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}}
\]
exp-to-pow40.0%
\[\leadsto 1 - \color{blue}{{x}^{\left({n}^{-1}\right)}}
\]
unpow-140.0%
\[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}}
\]
Simplified40.0%
\[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}}
\]
Taylor expanded in n around inf 68.0%
\[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}}
\]
Step-by-step derivation neg-mul-168.0%
\[\leadsto \color{blue}{-\frac{\log x}{n}}
\]
distribute-neg-frac68.0%
\[\leadsto \color{blue}{\frac{-\log x}{n}}
\]
Simplified68.0%
\[\leadsto \color{blue}{\frac{-\log x}{n}}
\]
if 1.00000000000000006e-295 < (/.f64 1 n) < 1.99999999999999983e-214 Initial program 73.0%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 94.5%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def94.5%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified94.5%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 78.6%
\[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n}
\]
if 5.00000000000000004e154 < (/.f64 1 n) Initial program 22.5%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 75.0%
\[\leadsto \color{blue}{\left(\frac{x}{n} + \left(1 + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Step-by-step derivation associate-+r+75.0%
\[\leadsto \color{blue}{\left(\left(\frac{x}{n} + 1\right) + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
+-commutative75.0%
\[\leadsto \left(\color{blue}{\left(1 + \frac{x}{n}\right)} + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow275.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow275.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
Simplified75.0%
\[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 75.0%
\[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}}
\]
Step-by-step derivation *-commutative75.0%
\[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}
\]
unpow275.0%
\[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)
\]
metadata-eval75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
unpow275.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)
\]
metadata-eval75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right)
\]
associate-*l*85.5%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}
\]
sub-neg85.5%
\[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{0.5}{n \cdot n} + \left(-\frac{0.5}{n}\right)\right)}\right)
\]
associate-/r*85.5%
\[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{0.5}{n}}{n}} + \left(-\frac{0.5}{n}\right)\right)\right)
\]
distribute-neg-frac85.5%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \color{blue}{\frac{-0.5}{n}}\right)\right)
\]
metadata-eval85.5%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{\color{blue}{-0.5}}{n}\right)\right)
\]
Simplified85.5%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)}
\]
Recombined 6 regimes into one program. Final simplification69.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1000000:\\
\;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-109}:\\
\;\;\;\;\frac{1}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-197}:\\
\;\;\;\;\frac{x - \log x}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-215}:\\
\;\;\;\;\frac{1}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 10^{-295}:\\
\;\;\;\;\frac{-\log x}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-214}:\\
\;\;\;\;\frac{\frac{1}{x}}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{x - \log x}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+154}:\\
\;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Alternative 7: 81.5% accurate, 1.5× speedup? \[\begin{array}{l}
t_0 := \frac{1}{n \cdot x}\\
t_1 := {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{t_1}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 10^{+137}:\\
\;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - t_1\\
\mathbf{else}:\\
\;\;\;\;\sqrt[3]{t_0 \cdot \frac{t_0}{n \cdot x}}\\
\end{array}
\]
Derivation Split input into 4 regimes if (/.f64 1 n) < -1e-107 Initial program 81.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 93.8%
\[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}}
\]
Step-by-step derivation log-rec93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
distribute-frac-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
neg-mul-193.8%
\[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
remove-double-neg93.8%
\[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x}
\]
*-rgt-identity93.8%
\[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x}
\]
associate-*r/93.8%
\[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x}
\]
exp-to-pow93.8%
\[\leadsto \frac{\color{blue}{{x}^{\left({n}^{-1}\right)}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x}
\]
*-commutative93.8%
\[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}}
\]
Simplified93.8%
\[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}}
\]
if -1e-107 < (/.f64 1 n) < 5.0000000000000001e-9 Initial program 33.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 86.5%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def86.5%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified86.5%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-udef86.5%
\[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n}
\]
diff-log86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n}
\]
+-commutative86.5%
\[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n}
\]
Applied egg-rr 86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n}
\]
if 5.0000000000000001e-9 < (/.f64 1 n) < 1e137 Initial program 90.9%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 92.1%
\[\leadsto \color{blue}{\left(\frac{x}{n} + \left(1 + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Step-by-step derivation associate-+r+92.1%
\[\leadsto \color{blue}{\left(\left(\frac{x}{n} + 1\right) + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
+-commutative92.1%
\[\leadsto \left(\color{blue}{\left(1 + \frac{x}{n}\right)} + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/92.1%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval92.1%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow292.1%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/92.1%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval92.1%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow292.1%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
Simplified92.1%
\[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
if 1e137 < (/.f64 1 n) Initial program 24.2%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 6.7%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def6.7%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified6.7%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 55.0%
\[\leadsto \color{blue}{\frac{1}{n \cdot x}}
\]
Step-by-step derivation *-commutative55.0%
\[\leadsto \frac{1}{\color{blue}{x \cdot n}}
\]
Simplified55.0%
\[\leadsto \color{blue}{\frac{1}{x \cdot n}}
\]
Step-by-step derivation add-cbrt-cube83.2%
\[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}\right) \cdot \frac{1}{x \cdot n}}}
\]
Applied egg-rr 83.2%
\[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}\right) \cdot \frac{1}{x \cdot n}}}
\]
Step-by-step derivation associate-*r/83.2%
\[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}\right) \cdot 1}{x \cdot n}}}
\]
*-rgt-identity83.2%
\[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}}}{x \cdot n}}
\]
associate-*r/83.2%
\[\leadsto \sqrt[3]{\color{blue}{\frac{1}{x \cdot n} \cdot \frac{\frac{1}{x \cdot n}}{x \cdot n}}}
\]
Simplified83.2%
\[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot n} \cdot \frac{\frac{1}{x \cdot n}}{x \cdot n}}}
\]
Recombined 4 regimes into one program. Final simplification89.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 10^{+137}:\\
\;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{1}{n \cdot x} \cdot \frac{\frac{1}{n \cdot x}}{n \cdot x}}\\
\end{array}
\]
Alternative 8: 81.4% accurate, 1.7× speedup? \[\begin{array}{l}
t_0 := {x}^{\left(\frac{1}{n}\right)}\\
t_1 := \frac{1}{n \cdot x}\\
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{t_0}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 10^{+137}:\\
\;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\
\mathbf{else}:\\
\;\;\;\;\sqrt[3]{t_1 \cdot \frac{t_1}{n \cdot x}}\\
\end{array}
\]
Derivation Split input into 4 regimes if (/.f64 1 n) < -1e-107 Initial program 81.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 93.8%
\[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}}
\]
Step-by-step derivation log-rec93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
distribute-frac-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
neg-mul-193.8%
\[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
remove-double-neg93.8%
\[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x}
\]
*-rgt-identity93.8%
\[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x}
\]
associate-*r/93.8%
\[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x}
\]
exp-to-pow93.8%
\[\leadsto \frac{\color{blue}{{x}^{\left({n}^{-1}\right)}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x}
\]
*-commutative93.8%
\[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}}
\]
Simplified93.8%
\[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}}
\]
if -1e-107 < (/.f64 1 n) < 5.0000000000000001e-9 Initial program 33.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 86.5%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def86.5%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified86.5%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-udef86.5%
\[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n}
\]
diff-log86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n}
\]
+-commutative86.5%
\[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n}
\]
Applied egg-rr 86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n}
\]
if 5.0000000000000001e-9 < (/.f64 1 n) < 1e137 Initial program 90.9%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 91.4%
\[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
if 1e137 < (/.f64 1 n) Initial program 24.2%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 6.7%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def6.7%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified6.7%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 55.0%
\[\leadsto \color{blue}{\frac{1}{n \cdot x}}
\]
Step-by-step derivation *-commutative55.0%
\[\leadsto \frac{1}{\color{blue}{x \cdot n}}
\]
Simplified55.0%
\[\leadsto \color{blue}{\frac{1}{x \cdot n}}
\]
Step-by-step derivation add-cbrt-cube83.2%
\[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}\right) \cdot \frac{1}{x \cdot n}}}
\]
Applied egg-rr 83.2%
\[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}\right) \cdot \frac{1}{x \cdot n}}}
\]
Step-by-step derivation associate-*r/83.2%
\[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}\right) \cdot 1}{x \cdot n}}}
\]
*-rgt-identity83.2%
\[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}}}{x \cdot n}}
\]
associate-*r/83.2%
\[\leadsto \sqrt[3]{\color{blue}{\frac{1}{x \cdot n} \cdot \frac{\frac{1}{x \cdot n}}{x \cdot n}}}
\]
Simplified83.2%
\[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot n} \cdot \frac{\frac{1}{x \cdot n}}{x \cdot n}}}
\]
Recombined 4 regimes into one program. Final simplification89.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 10^{+137}:\\
\;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{1}{n \cdot x} \cdot \frac{\frac{1}{n \cdot x}}{n \cdot x}}\\
\end{array}
\]
Alternative 9: 66.9% accurate, 1.7× speedup? \[\begin{array}{l}
t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{if}\;\frac{1}{n} \leq -1000000:\\
\;\;\;\;t_0\\
\mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{1}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+154}:\\
\;\;\;\;t_0\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Derivation Split input into 4 regimes if (/.f64 1 n) < -1e6 or 5.0000000000000001e-9 < (/.f64 1 n) < 5.00000000000000004e154 Initial program 96.0%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 66.1%
\[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}}
\]
Step-by-step derivation *-rgt-identity66.1%
\[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}}
\]
associate-*r/66.1%
\[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}}
\]
unpow-166.1%
\[\leadsto 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}}
\]
exp-to-pow66.1%
\[\leadsto 1 - \color{blue}{{x}^{\left({n}^{-1}\right)}}
\]
unpow-166.1%
\[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}}
\]
Simplified66.1%
\[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}}
\]
if -1e6 < (/.f64 1 n) < -1e-107 Initial program 15.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 44.0%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def44.0%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified44.0%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 71.0%
\[\leadsto \color{blue}{\frac{1}{n \cdot x}}
\]
Step-by-step derivation *-commutative71.0%
\[\leadsto \frac{1}{\color{blue}{x \cdot n}}
\]
Simplified71.0%
\[\leadsto \color{blue}{\frac{1}{x \cdot n}}
\]
if -1e-107 < (/.f64 1 n) < 5.0000000000000001e-9 Initial program 33.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 86.5%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def86.5%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified86.5%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-udef86.5%
\[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n}
\]
diff-log86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n}
\]
+-commutative86.5%
\[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n}
\]
Applied egg-rr 86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n}
\]
if 5.00000000000000004e154 < (/.f64 1 n) Initial program 22.5%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 75.0%
\[\leadsto \color{blue}{\left(\frac{x}{n} + \left(1 + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Step-by-step derivation associate-+r+75.0%
\[\leadsto \color{blue}{\left(\left(\frac{x}{n} + 1\right) + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
+-commutative75.0%
\[\leadsto \left(\color{blue}{\left(1 + \frac{x}{n}\right)} + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow275.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow275.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
Simplified75.0%
\[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 75.0%
\[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}}
\]
Step-by-step derivation *-commutative75.0%
\[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}
\]
unpow275.0%
\[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)
\]
metadata-eval75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
unpow275.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)
\]
metadata-eval75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right)
\]
associate-*l*85.5%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}
\]
sub-neg85.5%
\[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{0.5}{n \cdot n} + \left(-\frac{0.5}{n}\right)\right)}\right)
\]
associate-/r*85.5%
\[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{0.5}{n}}{n}} + \left(-\frac{0.5}{n}\right)\right)\right)
\]
distribute-neg-frac85.5%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \color{blue}{\frac{-0.5}{n}}\right)\right)
\]
metadata-eval85.5%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{\color{blue}{-0.5}}{n}\right)\right)
\]
Simplified85.5%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)}
\]
Recombined 4 regimes into one program. Final simplification77.5%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1000000:\\
\;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{1}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+154}:\\
\;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Alternative 10: 81.5% accurate, 1.7× speedup? \[\begin{array}{l}
t_0 := {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{t_0}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+167}:\\
\;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Derivation Split input into 4 regimes if (/.f64 1 n) < -1e-107 Initial program 81.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 93.8%
\[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}}
\]
Step-by-step derivation log-rec93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
distribute-frac-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
neg-mul-193.8%
\[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
remove-double-neg93.8%
\[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x}
\]
*-rgt-identity93.8%
\[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x}
\]
associate-*r/93.8%
\[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x}
\]
exp-to-pow93.8%
\[\leadsto \frac{\color{blue}{{x}^{\left({n}^{-1}\right)}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x}
\]
*-commutative93.8%
\[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}}
\]
Simplified93.8%
\[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}}
\]
if -1e-107 < (/.f64 1 n) < 5.0000000000000001e-9 Initial program 33.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 86.5%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def86.5%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified86.5%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-udef86.5%
\[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n}
\]
diff-log86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n}
\]
+-commutative86.5%
\[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n}
\]
Applied egg-rr 86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n}
\]
if 5.0000000000000001e-9 < (/.f64 1 n) < 2.0000000000000001e167 Initial program 81.4%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 82.0%
\[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
if 2.0000000000000001e167 < (/.f64 1 n) Initial program 19.3%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 77.8%
\[\leadsto \color{blue}{\left(\frac{x}{n} + \left(1 + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Step-by-step derivation associate-+r+77.8%
\[\leadsto \color{blue}{\left(\left(\frac{x}{n} + 1\right) + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
+-commutative77.8%
\[\leadsto \left(\color{blue}{\left(1 + \frac{x}{n}\right)} + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/77.8%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval77.8%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow277.8%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/77.8%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval77.8%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow277.8%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
Simplified77.8%
\[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 77.8%
\[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}}
\]
Step-by-step derivation *-commutative77.8%
\[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}
\]
unpow277.8%
\[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/77.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)
\]
metadata-eval77.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
unpow277.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/77.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)
\]
metadata-eval77.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right)
\]
associate-*l*89.2%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}
\]
sub-neg89.2%
\[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{0.5}{n \cdot n} + \left(-\frac{0.5}{n}\right)\right)}\right)
\]
associate-/r*89.2%
\[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{0.5}{n}}{n}} + \left(-\frac{0.5}{n}\right)\right)\right)
\]
distribute-neg-frac89.2%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \color{blue}{\frac{-0.5}{n}}\right)\right)
\]
metadata-eval89.2%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{\color{blue}{-0.5}}{n}\right)\right)
\]
Simplified89.2%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)}
\]
Recombined 4 regimes into one program. Final simplification88.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+167}:\\
\;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Alternative 11: 81.3% accurate, 1.8× speedup? \[\begin{array}{l}
t_0 := {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{t_0}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+154}:\\
\;\;\;\;1 - t_0\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Derivation Split input into 4 regimes if (/.f64 1 n) < -1e-107 Initial program 81.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 93.8%
\[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}}
\]
Step-by-step derivation log-rec93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x}
\]
mul-1-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x}
\]
distribute-frac-neg93.8%
\[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
neg-mul-193.8%
\[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x}
\]
remove-double-neg93.8%
\[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x}
\]
*-rgt-identity93.8%
\[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x}
\]
associate-*r/93.8%
\[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x}
\]
exp-to-pow93.8%
\[\leadsto \frac{\color{blue}{{x}^{\left({n}^{-1}\right)}}}{n \cdot x}
\]
unpow-193.8%
\[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x}
\]
*-commutative93.8%
\[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}}
\]
Simplified93.8%
\[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}}
\]
if -1e-107 < (/.f64 1 n) < 5.0000000000000001e-9 Initial program 33.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 86.5%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def86.5%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified86.5%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-udef86.5%
\[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n}
\]
diff-log86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n}
\]
+-commutative86.5%
\[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n}
\]
Applied egg-rr 86.5%
\[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n}
\]
if 5.0000000000000001e-9 < (/.f64 1 n) < 5.00000000000000004e154 Initial program 83.9%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 83.9%
\[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}}
\]
Step-by-step derivation *-rgt-identity83.9%
\[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}}
\]
associate-*r/83.9%
\[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}}
\]
unpow-183.9%
\[\leadsto 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}}
\]
exp-to-pow83.9%
\[\leadsto 1 - \color{blue}{{x}^{\left({n}^{-1}\right)}}
\]
unpow-183.9%
\[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}}
\]
Simplified83.9%
\[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}}
\]
if 5.00000000000000004e154 < (/.f64 1 n) Initial program 22.5%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 75.0%
\[\leadsto \color{blue}{\left(\frac{x}{n} + \left(1 + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Step-by-step derivation associate-+r+75.0%
\[\leadsto \color{blue}{\left(\left(\frac{x}{n} + 1\right) + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
+-commutative75.0%
\[\leadsto \left(\color{blue}{\left(1 + \frac{x}{n}\right)} + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow275.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval75.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow275.0%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
Simplified75.0%
\[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 75.0%
\[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}}
\]
Step-by-step derivation *-commutative75.0%
\[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}
\]
unpow275.0%
\[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)
\]
metadata-eval75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
unpow275.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)
\]
metadata-eval75.0%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right)
\]
associate-*l*85.5%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}
\]
sub-neg85.5%
\[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{0.5}{n \cdot n} + \left(-\frac{0.5}{n}\right)\right)}\right)
\]
associate-/r*85.5%
\[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{0.5}{n}}{n}} + \left(-\frac{0.5}{n}\right)\right)\right)
\]
distribute-neg-frac85.5%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \color{blue}{\frac{-0.5}{n}}\right)\right)
\]
metadata-eval85.5%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{\color{blue}{-0.5}}{n}\right)\right)
\]
Simplified85.5%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)}
\]
Recombined 4 regimes into one program. Final simplification88.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-107}:\\
\;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+154}:\\
\;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Alternative 12: 59.3% accurate, 2.0× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq 1:\\
\;\;\;\;\frac{x - \log x}{n}\\
\mathbf{elif}\;x \leq 1.3 \cdot 10^{+87}:\\
\;\;\;\;\frac{1}{n \cdot x}\\
\mathbf{elif}\;x \leq 6.2 \cdot 10^{+153}:\\
\;\;\;\;\frac{0}{n}\\
\mathbf{elif}\;x \leq 7.2 \cdot 10^{+189}:\\
\;\;\;\;\frac{\frac{1}{x}}{n}\\
\mathbf{else}:\\
\;\;\;\;\frac{0}{n}\\
\end{array}
\]
Derivation Split input into 4 regimes if x < 1 Initial program 45.0%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 44.6%
\[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 50.0%
\[\leadsto \color{blue}{\frac{x - \log x}{n}}
\]
if 1 < x < 1.29999999999999999e87 Initial program 48.5%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 48.5%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def48.5%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified48.5%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 73.4%
\[\leadsto \color{blue}{\frac{1}{n \cdot x}}
\]
Step-by-step derivation *-commutative73.4%
\[\leadsto \frac{1}{\color{blue}{x \cdot n}}
\]
Simplified73.4%
\[\leadsto \color{blue}{\frac{1}{x \cdot n}}
\]
if 1.29999999999999999e87 < x < 6.2e153 or 7.20000000000000017e189 < x Initial program 86.8%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 86.8%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def86.8%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified86.8%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-udef86.8%
\[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n}
\]
diff-log86.8%
\[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n}
\]
+-commutative86.8%
\[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n}
\]
Applied egg-rr 86.8%
\[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n}
\]
Taylor expanded in x around inf 86.8%
\[\leadsto \frac{\log \color{blue}{1}}{n}
\]
if 6.2e153 < x < 7.20000000000000017e189 Initial program 50.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 50.7%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def50.7%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified50.7%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 83.7%
\[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n}
\]
Recombined 4 regimes into one program. Final simplification62.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq 1:\\
\;\;\;\;\frac{x - \log x}{n}\\
\mathbf{elif}\;x \leq 1.3 \cdot 10^{+87}:\\
\;\;\;\;\frac{1}{n \cdot x}\\
\mathbf{elif}\;x \leq 6.2 \cdot 10^{+153}:\\
\;\;\;\;\frac{0}{n}\\
\mathbf{elif}\;x \leq 7.2 \cdot 10^{+189}:\\
\;\;\;\;\frac{\frac{1}{x}}{n}\\
\mathbf{else}:\\
\;\;\;\;\frac{0}{n}\\
\end{array}
\]
Alternative 13: 59.1% accurate, 2.0× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq 0.55:\\
\;\;\;\;\frac{-\log x}{n}\\
\mathbf{elif}\;x \leq 2.2 \cdot 10^{+87}:\\
\;\;\;\;\frac{1}{n \cdot x}\\
\mathbf{elif}\;x \leq 1.35 \cdot 10^{+153}:\\
\;\;\;\;\frac{0}{n}\\
\mathbf{elif}\;x \leq 10^{+190}:\\
\;\;\;\;\frac{\frac{1}{x}}{n}\\
\mathbf{else}:\\
\;\;\;\;\frac{0}{n}\\
\end{array}
\]
Derivation Split input into 4 regimes if x < 0.55000000000000004 Initial program 45.0%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 44.2%
\[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}}
\]
Step-by-step derivation *-rgt-identity44.2%
\[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}}
\]
associate-*r/44.2%
\[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}}
\]
unpow-144.2%
\[\leadsto 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}}
\]
exp-to-pow44.2%
\[\leadsto 1 - \color{blue}{{x}^{\left({n}^{-1}\right)}}
\]
unpow-144.2%
\[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}}
\]
Simplified44.2%
\[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}}
\]
Taylor expanded in n around inf 49.8%
\[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}}
\]
Step-by-step derivation neg-mul-149.8%
\[\leadsto \color{blue}{-\frac{\log x}{n}}
\]
distribute-neg-frac49.8%
\[\leadsto \color{blue}{\frac{-\log x}{n}}
\]
Simplified49.8%
\[\leadsto \color{blue}{\frac{-\log x}{n}}
\]
if 0.55000000000000004 < x < 2.2000000000000001e87 Initial program 48.5%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 48.5%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def48.5%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified48.5%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 73.4%
\[\leadsto \color{blue}{\frac{1}{n \cdot x}}
\]
Step-by-step derivation *-commutative73.4%
\[\leadsto \frac{1}{\color{blue}{x \cdot n}}
\]
Simplified73.4%
\[\leadsto \color{blue}{\frac{1}{x \cdot n}}
\]
if 2.2000000000000001e87 < x < 1.35e153 or 1.0000000000000001e190 < x Initial program 86.8%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 86.8%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def86.8%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified86.8%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-udef86.8%
\[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n}
\]
diff-log86.8%
\[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n}
\]
+-commutative86.8%
\[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n}
\]
Applied egg-rr 86.8%
\[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n}
\]
Taylor expanded in x around inf 86.8%
\[\leadsto \frac{\log \color{blue}{1}}{n}
\]
if 1.35e153 < x < 1.0000000000000001e190 Initial program 50.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 50.7%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def50.7%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified50.7%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 83.7%
\[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n}
\]
Recombined 4 regimes into one program. Final simplification62.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq 0.55:\\
\;\;\;\;\frac{-\log x}{n}\\
\mathbf{elif}\;x \leq 2.2 \cdot 10^{+87}:\\
\;\;\;\;\frac{1}{n \cdot x}\\
\mathbf{elif}\;x \leq 1.35 \cdot 10^{+153}:\\
\;\;\;\;\frac{0}{n}\\
\mathbf{elif}\;x \leq 10^{+190}:\\
\;\;\;\;\frac{\frac{1}{x}}{n}\\
\mathbf{else}:\\
\;\;\;\;\frac{0}{n}\\
\end{array}
\]
Alternative 14: 49.0% accurate, 8.4× speedup? \[\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{+239}:\\
\;\;\;\;\frac{1}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq -100000000:\\
\;\;\;\;\frac{0}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 2:\\
\;\;\;\;\frac{\frac{1}{x}}{n}\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Derivation Split input into 4 regimes if (/.f64 1 n) < -3.99999999999999996e239 Initial program 100.0%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 34.6%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def34.6%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified34.6%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 72.8%
\[\leadsto \color{blue}{\frac{1}{n \cdot x}}
\]
Step-by-step derivation *-commutative72.8%
\[\leadsto \frac{1}{\color{blue}{x \cdot n}}
\]
Simplified72.8%
\[\leadsto \color{blue}{\frac{1}{x \cdot n}}
\]
if -3.99999999999999996e239 < (/.f64 1 n) < -1e8 Initial program 100.0%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 51.1%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def51.1%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified51.1%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-udef51.1%
\[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n}
\]
diff-log51.1%
\[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n}
\]
+-commutative51.1%
\[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n}
\]
Applied egg-rr 51.1%
\[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n}
\]
Taylor expanded in x around inf 49.8%
\[\leadsto \frac{\log \color{blue}{1}}{n}
\]
if -1e8 < (/.f64 1 n) < 2 Initial program 32.8%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 78.3%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def78.3%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified78.3%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 47.3%
\[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n}
\]
if 2 < (/.f64 1 n) Initial program 52.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 76.6%
\[\leadsto \color{blue}{\left(\frac{x}{n} + \left(1 + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Step-by-step derivation associate-+r+76.6%
\[\leadsto \color{blue}{\left(\left(\frac{x}{n} + 1\right) + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
+-commutative76.6%
\[\leadsto \left(\color{blue}{\left(1 + \frac{x}{n}\right)} + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/76.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval76.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow276.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/76.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval76.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow276.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
Simplified76.6%
\[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 39.8%
\[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}}
\]
Step-by-step derivation *-commutative39.8%
\[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}
\]
unpow239.8%
\[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/39.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)
\]
metadata-eval39.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
unpow239.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/39.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)
\]
metadata-eval39.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right)
\]
associate-*l*45.2%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}
\]
sub-neg45.2%
\[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{0.5}{n \cdot n} + \left(-\frac{0.5}{n}\right)\right)}\right)
\]
associate-/r*45.2%
\[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{0.5}{n}}{n}} + \left(-\frac{0.5}{n}\right)\right)\right)
\]
distribute-neg-frac45.2%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \color{blue}{\frac{-0.5}{n}}\right)\right)
\]
metadata-eval45.2%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{\color{blue}{-0.5}}{n}\right)\right)
\]
Simplified45.2%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)}
\]
Recombined 4 regimes into one program. Final simplification49.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{+239}:\\
\;\;\;\;\frac{1}{n \cdot x}\\
\mathbf{elif}\;\frac{1}{n} \leq -100000000:\\
\;\;\;\;\frac{0}{n}\\
\mathbf{elif}\;\frac{1}{n} \leq 2:\\
\;\;\;\;\frac{\frac{1}{x}}{n}\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Alternative 15: 42.7% accurate, 12.4× speedup? \[\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq 2:\\
\;\;\;\;\frac{\frac{1}{x}}{n}\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 1 n) < 2 Initial program 55.2%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 67.6%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def67.6%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified67.6%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 43.7%
\[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n}
\]
if 2 < (/.f64 1 n) Initial program 52.1%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around 0 76.6%
\[\leadsto \color{blue}{\left(\frac{x}{n} + \left(1 + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Step-by-step derivation associate-+r+76.6%
\[\leadsto \color{blue}{\left(\left(\frac{x}{n} + 1\right) + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
+-commutative76.6%
\[\leadsto \left(\color{blue}{\left(1 + \frac{x}{n}\right)} + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/76.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval76.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow276.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
associate-*r/76.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
metadata-eval76.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
unpow276.6%
\[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) - {x}^{\left(\frac{1}{n}\right)}
\]
Simplified76.6%
\[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right)\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in x around inf 39.8%
\[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}}
\]
Step-by-step derivation *-commutative39.8%
\[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}
\]
unpow239.8%
\[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/39.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)
\]
metadata-eval39.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)
\]
unpow239.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right)
\]
associate-*r/39.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)
\]
metadata-eval39.8%
\[\leadsto \left(x \cdot x\right) \cdot \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right)
\]
associate-*l*45.2%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}
\]
sub-neg45.2%
\[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{0.5}{n \cdot n} + \left(-\frac{0.5}{n}\right)\right)}\right)
\]
associate-/r*45.2%
\[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{0.5}{n}}{n}} + \left(-\frac{0.5}{n}\right)\right)\right)
\]
distribute-neg-frac45.2%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \color{blue}{\frac{-0.5}{n}}\right)\right)
\]
metadata-eval45.2%
\[\leadsto x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{\color{blue}{-0.5}}{n}\right)\right)
\]
Simplified45.2%
\[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)}
\]
Recombined 2 regimes into one program. Final simplification43.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{1}{n} \leq 2:\\
\;\;\;\;\frac{\frac{1}{x}}{n}\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \left(\frac{\frac{0.5}{n}}{n} + \frac{-0.5}{n}\right)\right)\\
\end{array}
\]
Alternative 16: 40.3% accurate, 42.2× speedup? \[\frac{1}{n \cdot x}
\]
Derivation Initial program 54.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 58.1%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def58.1%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified58.1%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 41.6%
\[\leadsto \color{blue}{\frac{1}{n \cdot x}}
\]
Step-by-step derivation *-commutative41.6%
\[\leadsto \frac{1}{\color{blue}{x \cdot n}}
\]
Simplified41.6%
\[\leadsto \color{blue}{\frac{1}{x \cdot n}}
\]
Final simplification41.6%
\[\leadsto \frac{1}{n \cdot x}
\]
Alternative 17: 40.8% accurate, 42.2× speedup? \[\frac{\frac{1}{x}}{n}
\]
Derivation Initial program 54.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 58.1%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def58.1%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified58.1%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 42.0%
\[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n}
\]
Final simplification42.0%
\[\leadsto \frac{\frac{1}{x}}{n}
\]
Alternative 18: 4.5% accurate, 70.3× speedup? \[\frac{x}{n}
\]
Derivation Initial program 54.7%
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
\]
Taylor expanded in n around inf 58.1%
\[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}}
\]
Step-by-step derivation log1p-def58.1%
\[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n}
\]
Simplified58.1%
\[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}}
\]
Taylor expanded in x around inf 42.0%
\[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n}
\]
Step-by-step derivation expm1-log1p-u31.9%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{x}}{n}\right)\right)}
\]
expm1-udef27.5%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{x}}{n}\right)} - 1}
\]
add-exp-log27.5%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{e^{\log \left(\frac{1}{x}\right)}}}{n}\right)} - 1
\]
neg-log27.5%
\[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{-\log x}}}{n}\right)} - 1
\]
add-sqr-sqrt6.6%
\[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\sqrt{-\log x} \cdot \sqrt{-\log x}}}}{n}\right)} - 1
\]
sqrt-unprod13.4%
\[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\sqrt{\left(-\log x\right) \cdot \left(-\log x\right)}}}}{n}\right)} - 1
\]
sqr-neg13.4%
\[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\sqrt{\color{blue}{\log x \cdot \log x}}}}{n}\right)} - 1
\]
sqrt-unprod6.8%
\[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}}{n}\right)} - 1
\]
add-sqr-sqrt9.2%
\[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\log x}}}{n}\right)} - 1
\]
add-exp-log9.2%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{n}\right)} - 1
\]
Applied egg-rr 9.2%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{n}\right)} - 1}
\]
Step-by-step derivation expm1-def3.7%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{n}\right)\right)}
\]
expm1-log1p4.5%
\[\leadsto \color{blue}{\frac{x}{n}}
\]
Simplified4.5%
\[\leadsto \color{blue}{\frac{x}{n}}
\]
Final simplification4.5%
\[\leadsto \frac{x}{n}
\]
Reproduce ? herbie shell --seed 2023167
(FPCore (x n)
:name "2nthrt (problem 3.4.6)"
:precision binary64
(- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
