Falkner and Boettcher, Appendix A Percentage Accurate: 90.1% → 97.4%
Time: 10.9s
Alternatives: 16
Speedup: TODO×
23.2% of points produce a very large (infinite) output. You may want to add a precondition. (more) could not determine a ground truth (more) Specification ? \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
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? \[\begin{array}{l}
t_0 := a \cdot {k}^{m}\\
\mathbf{if}\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+215}:\\
\;\;\;\;\frac{a}{\frac{1 + \left(k \cdot k + k \cdot 10\right)}{{k}^{m}}}\\
\mathbf{else}:\\
\;\;\;\;t_0\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 9.99999999999999907e214 Initial program 97.5%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-/l*97.5%
\[\leadsto \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}}
\]
associate-+l+97.5%
\[\leadsto \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}}
\]
*-commutative97.5%
\[\leadsto \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}}
\]
Simplified97.5%
\[\leadsto \color{blue}{\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}}
\]
if 9.99999999999999907e214 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) Initial program 66.7%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/66.7%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+66.7%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative66.7%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out66.7%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def66.7%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative66.7%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified66.7%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in k around 0 60.3%
\[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a}
\]
Step-by-step derivation exp-to-pow100.0%
\[\leadsto \color{blue}{{k}^{m}} \cdot a
\]
*-commutative100.0%
\[\leadsto \color{blue}{a \cdot {k}^{m}}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot {k}^{m}}
\]
Recombined 2 regimes into one program. Final simplification98.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+215}:\\
\;\;\;\;\frac{a}{\frac{1 + \left(k \cdot k + k \cdot 10\right)}{{k}^{m}}}\\
\mathbf{else}:\\
\;\;\;\;a \cdot {k}^{m}\\
\end{array}
\]
Alternative 2? \[\begin{array}{l}
\mathbf{if}\;m \leq -1.05 \cdot 10^{-8}:\\
\;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\
\mathbf{elif}\;m \leq 1.4 \cdot 10^{-9}:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\mathbf{else}:\\
\;\;\;\;a \cdot {k}^{m}\\
\end{array}
\]
Derivation Split input into 3 regimes if m < -1.04999999999999997e-8 Initial program 98.7%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-/l*98.7%
\[\leadsto \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}}
\]
associate-+l+98.7%
\[\leadsto \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}}
\]
*-commutative98.7%
\[\leadsto \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}}
\]
Simplified98.7%
\[\leadsto \color{blue}{\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}}
\]
Taylor expanded in k around 0 98.7%
\[\leadsto \frac{a}{\frac{1 + \color{blue}{10 \cdot k}}{{k}^{m}}}
\]
Step-by-step derivation *-commutative98.7%
\[\leadsto \frac{a}{\frac{1 + \color{blue}{k \cdot 10}}{{k}^{m}}}
\]
Simplified98.7%
\[\leadsto \frac{a}{\frac{1 + \color{blue}{k \cdot 10}}{{k}^{m}}}
\]
if -1.04999999999999997e-8 < m < 1.39999999999999992e-9 Initial program 95.5%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/95.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified95.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 94.5%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
if 1.39999999999999992e-9 < m Initial program 79.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/79.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified79.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in k around 0 50.0%
\[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a}
\]
Step-by-step derivation exp-to-pow100.0%
\[\leadsto \color{blue}{{k}^{m}} \cdot a
\]
*-commutative100.0%
\[\leadsto \color{blue}{a \cdot {k}^{m}}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot {k}^{m}}
\]
Recombined 3 regimes into one program. Final simplification97.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;m \leq -1.05 \cdot 10^{-8}:\\
\;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\
\mathbf{elif}\;m \leq 1.4 \cdot 10^{-9}:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\mathbf{else}:\\
\;\;\;\;a \cdot {k}^{m}\\
\end{array}
\]
Alternative 3? \[\begin{array}{l}
\mathbf{if}\;m \leq 0.0066:\\
\;\;\;\;\frac{a}{\frac{1 + k \cdot k}{{k}^{m}}}\\
\mathbf{else}:\\
\;\;\;\;a \cdot {k}^{m}\\
\end{array}
\]
Derivation Split input into 2 regimes if m < 0.0066 Initial program 96.9%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-/l*96.9%
\[\leadsto \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}}
\]
associate-+l+96.9%
\[\leadsto \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}}
\]
*-commutative96.9%
\[\leadsto \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}}
\]
Simplified96.9%
\[\leadsto \color{blue}{\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}}
\]
Taylor expanded in k around inf 95.7%
\[\leadsto \frac{a}{\frac{1 + \color{blue}{{k}^{2}}}{{k}^{m}}}
\]
Step-by-step derivation unpow295.7%
\[\leadsto \frac{a}{\frac{1 + \color{blue}{k \cdot k}}{{k}^{m}}}
\]
Simplified95.7%
\[\leadsto \frac{a}{\frac{1 + \color{blue}{k \cdot k}}{{k}^{m}}}
\]
if 0.0066 < m Initial program 79.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/79.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified79.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in k around 0 50.0%
\[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a}
\]
Step-by-step derivation exp-to-pow100.0%
\[\leadsto \color{blue}{{k}^{m}} \cdot a
\]
*-commutative100.0%
\[\leadsto \color{blue}{a \cdot {k}^{m}}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot {k}^{m}}
\]
Recombined 2 regimes into one program. Final simplification97.4%
\[\leadsto \begin{array}{l}
\mathbf{if}\;m \leq 0.0066:\\
\;\;\;\;\frac{a}{\frac{1 + k \cdot k}{{k}^{m}}}\\
\mathbf{else}:\\
\;\;\;\;a \cdot {k}^{m}\\
\end{array}
\]
Alternative 4? \[\begin{array}{l}
\mathbf{if}\;m \leq -0.027:\\
\;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\
\mathbf{elif}\;m \leq 0.48:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\mathbf{elif}\;m \leq 6.9 \cdot 10^{+72}:\\
\;\;\;\;\sqrt{a \cdot a}\\
\mathbf{else}:\\
\;\;\;\;a + a \cdot \left(k \cdot -10 + \left(k \cdot k\right) \cdot 100\right)\\
\end{array}
\]
Derivation Split input into 4 regimes if m < -0.0269999999999999997 Initial program 100.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 32.3%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Step-by-step derivation clear-num34.2%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}}
\]
inv-pow34.2%
\[\leadsto \color{blue}{{\left(\frac{1 + k \cdot \left(k + 10\right)}{a}\right)}^{-1}}
\]
+-commutative34.2%
\[\leadsto {\left(\frac{\color{blue}{k \cdot \left(k + 10\right) + 1}}{a}\right)}^{-1}
\]
fma-udef34.2%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(k, k + 10, 1\right)}}{a}\right)}^{-1}
\]
Applied egg-rr 34.2%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}\right)}^{-1}}
\]
Step-by-step derivation unpow-134.2%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Simplified34.2%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Taylor expanded in k around inf 56.6%
\[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{a}}
\]
Step-by-step derivation unpow256.6%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
Simplified56.6%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
if -0.0269999999999999997 < m < 0.47999999999999998 Initial program 94.5%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/94.5%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+94.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative94.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out94.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def94.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative94.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified94.5%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 92.0%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
if 0.47999999999999998 < m < 6.90000000000000034e72 Initial program 82.8%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/82.8%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+82.8%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative82.8%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out82.8%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def82.8%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative82.8%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified82.8%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 3.4%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Step-by-step derivation clear-num3.4%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}}
\]
inv-pow3.4%
\[\leadsto \color{blue}{{\left(\frac{1 + k \cdot \left(k + 10\right)}{a}\right)}^{-1}}
\]
+-commutative3.4%
\[\leadsto {\left(\frac{\color{blue}{k \cdot \left(k + 10\right) + 1}}{a}\right)}^{-1}
\]
fma-udef3.4%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(k, k + 10, 1\right)}}{a}\right)}^{-1}
\]
Applied egg-rr 3.4%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}\right)}^{-1}}
\]
Step-by-step derivation unpow-13.4%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Simplified3.4%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Taylor expanded in k around 0 4.7%
\[\leadsto \frac{1}{\color{blue}{\frac{1}{a}}}
\]
Step-by-step derivation remove-double-div4.7%
\[\leadsto \color{blue}{a}
\]
add-sqr-sqrt3.1%
\[\leadsto \color{blue}{\sqrt{a} \cdot \sqrt{a}}
\]
sqrt-unprod45.9%
\[\leadsto \color{blue}{\sqrt{a \cdot a}}
\]
Applied egg-rr 45.9%
\[\leadsto \color{blue}{\sqrt{a \cdot a}}
\]
if 6.90000000000000034e72 < m Initial program 77.1%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/77.1%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+77.1%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative77.1%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out77.1%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def77.1%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative77.1%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified77.1%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 3.0%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Step-by-step derivation clear-num3.0%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}}
\]
inv-pow3.0%
\[\leadsto \color{blue}{{\left(\frac{1 + k \cdot \left(k + 10\right)}{a}\right)}^{-1}}
\]
+-commutative3.0%
\[\leadsto {\left(\frac{\color{blue}{k \cdot \left(k + 10\right) + 1}}{a}\right)}^{-1}
\]
fma-udef3.0%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(k, k + 10, 1\right)}}{a}\right)}^{-1}
\]
Applied egg-rr 3.0%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}\right)}^{-1}}
\]
Step-by-step derivation unpow-13.0%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Simplified3.0%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Taylor expanded in k around 0 2.8%
\[\leadsto \frac{1}{\color{blue}{10 \cdot \frac{k}{a} + \frac{1}{a}}}
\]
Taylor expanded in k around 0 25.6%
\[\leadsto \color{blue}{a + \left(-10 \cdot \left(k \cdot a\right) + 100 \cdot \left({k}^{2} \cdot a\right)\right)}
\]
Step-by-step derivation associate-*r*25.6%
\[\leadsto a + \left(\color{blue}{\left(-10 \cdot k\right) \cdot a} + 100 \cdot \left({k}^{2} \cdot a\right)\right)
\]
unpow225.6%
\[\leadsto a + \left(\left(-10 \cdot k\right) \cdot a + 100 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot a\right)\right)
\]
associate-*r*25.6%
\[\leadsto a + \left(\left(-10 \cdot k\right) \cdot a + \color{blue}{\left(100 \cdot \left(k \cdot k\right)\right) \cdot a}\right)
\]
distribute-rgt-out32.8%
\[\leadsto a + \color{blue}{a \cdot \left(-10 \cdot k + 100 \cdot \left(k \cdot k\right)\right)}
\]
*-commutative32.8%
\[\leadsto a + a \cdot \left(\color{blue}{k \cdot -10} + 100 \cdot \left(k \cdot k\right)\right)
\]
Simplified32.8%
\[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10 + 100 \cdot \left(k \cdot k\right)\right)}
\]
Recombined 4 regimes into one program. Final simplification60.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;m \leq -0.027:\\
\;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\
\mathbf{elif}\;m \leq 0.48:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\mathbf{elif}\;m \leq 6.9 \cdot 10^{+72}:\\
\;\;\;\;\sqrt{a \cdot a}\\
\mathbf{else}:\\
\;\;\;\;a + a \cdot \left(k \cdot -10 + \left(k \cdot k\right) \cdot 100\right)\\
\end{array}
\]
Alternative 5? \[\begin{array}{l}
\mathbf{if}\;m \leq -1.3 \cdot 10^{-14} \lor \neg \left(m \leq 3.2 \cdot 10^{-6}\right):\\
\;\;\;\;a \cdot {k}^{m}\\
\mathbf{else}:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\end{array}
\]
Derivation Split input into 2 regimes if m < -1.29999999999999998e-14 or 3.1999999999999999e-6 < m Initial program 87.2%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/87.2%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+87.2%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative87.2%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out87.2%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def87.2%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative87.2%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified87.2%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in k around 0 53.5%
\[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a}
\]
Step-by-step derivation exp-to-pow99.4%
\[\leadsto \color{blue}{{k}^{m}} \cdot a
\]
*-commutative99.4%
\[\leadsto \color{blue}{a \cdot {k}^{m}}
\]
Simplified99.4%
\[\leadsto \color{blue}{a \cdot {k}^{m}}
\]
if -1.29999999999999998e-14 < m < 3.1999999999999999e-6 Initial program 95.5%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/95.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified95.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 94.5%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Recombined 2 regimes into one program. Final simplification97.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;m \leq -1.3 \cdot 10^{-14} \lor \neg \left(m \leq 3.2 \cdot 10^{-6}\right):\\
\;\;\;\;a \cdot {k}^{m}\\
\mathbf{else}:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\end{array}
\]
Alternative 6? \[\begin{array}{l}
\mathbf{if}\;m \leq -2.4 \cdot 10^{-8}:\\
\;\;\;\;\frac{a}{\frac{1}{{k}^{m}}}\\
\mathbf{elif}\;m \leq 1.3 \cdot 10^{-9}:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\mathbf{else}:\\
\;\;\;\;a \cdot {k}^{m}\\
\end{array}
\]
Derivation Split input into 3 regimes if m < -2.39999999999999998e-8 Initial program 98.7%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-/l*98.7%
\[\leadsto \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}}
\]
associate-+l+98.7%
\[\leadsto \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}}
\]
*-commutative98.7%
\[\leadsto \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}}
\]
Simplified98.7%
\[\leadsto \color{blue}{\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}}
\]
Taylor expanded in k around 0 58.4%
\[\leadsto \frac{a}{\color{blue}{\frac{1}{e^{\log k \cdot m}}}}
\]
Step-by-step derivation exp-to-pow98.6%
\[\leadsto \frac{a}{\frac{1}{\color{blue}{{k}^{m}}}}
\]
Simplified98.6%
\[\leadsto \frac{a}{\color{blue}{\frac{1}{{k}^{m}}}}
\]
if -2.39999999999999998e-8 < m < 1.3000000000000001e-9 Initial program 95.5%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/95.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative95.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified95.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 94.5%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
if 1.3000000000000001e-9 < m Initial program 79.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/79.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified79.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in k around 0 50.0%
\[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a}
\]
Step-by-step derivation exp-to-pow100.0%
\[\leadsto \color{blue}{{k}^{m}} \cdot a
\]
*-commutative100.0%
\[\leadsto \color{blue}{a \cdot {k}^{m}}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot {k}^{m}}
\]
Recombined 3 regimes into one program. Final simplification97.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;m \leq -2.4 \cdot 10^{-8}:\\
\;\;\;\;\frac{a}{\frac{1}{{k}^{m}}}\\
\mathbf{elif}\;m \leq 1.3 \cdot 10^{-9}:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\mathbf{else}:\\
\;\;\;\;a \cdot {k}^{m}\\
\end{array}
\]
Alternative 7? \[\begin{array}{l}
\mathbf{if}\;m \leq -0.62:\\
\;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\
\mathbf{elif}\;m \leq 0.0066:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\mathbf{else}:\\
\;\;\;\;a + a \cdot \left(k \cdot -10 + \left(k \cdot k\right) \cdot 100\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if m < -0.619999999999999996 Initial program 100.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 32.3%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Step-by-step derivation clear-num34.2%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}}
\]
inv-pow34.2%
\[\leadsto \color{blue}{{\left(\frac{1 + k \cdot \left(k + 10\right)}{a}\right)}^{-1}}
\]
+-commutative34.2%
\[\leadsto {\left(\frac{\color{blue}{k \cdot \left(k + 10\right) + 1}}{a}\right)}^{-1}
\]
fma-udef34.2%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(k, k + 10, 1\right)}}{a}\right)}^{-1}
\]
Applied egg-rr 34.2%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}\right)}^{-1}}
\]
Step-by-step derivation unpow-134.2%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Simplified34.2%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Taylor expanded in k around inf 56.6%
\[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{a}}
\]
Step-by-step derivation unpow256.6%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
Simplified56.6%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
if -0.619999999999999996 < m < 0.0066 Initial program 94.4%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/94.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+94.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative94.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out94.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def94.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative94.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified94.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 92.8%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
if 0.0066 < m Initial program 79.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/79.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative79.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified79.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 3.2%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Step-by-step derivation clear-num3.2%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}}
\]
inv-pow3.2%
\[\leadsto \color{blue}{{\left(\frac{1 + k \cdot \left(k + 10\right)}{a}\right)}^{-1}}
\]
+-commutative3.2%
\[\leadsto {\left(\frac{\color{blue}{k \cdot \left(k + 10\right) + 1}}{a}\right)}^{-1}
\]
fma-udef3.2%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(k, k + 10, 1\right)}}{a}\right)}^{-1}
\]
Applied egg-rr 3.2%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}\right)}^{-1}}
\]
Step-by-step derivation unpow-13.2%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Simplified3.2%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Taylor expanded in k around 0 3.1%
\[\leadsto \frac{1}{\color{blue}{10 \cdot \frac{k}{a} + \frac{1}{a}}}
\]
Taylor expanded in k around 0 24.0%
\[\leadsto \color{blue}{a + \left(-10 \cdot \left(k \cdot a\right) + 100 \cdot \left({k}^{2} \cdot a\right)\right)}
\]
Step-by-step derivation associate-*r*24.0%
\[\leadsto a + \left(\color{blue}{\left(-10 \cdot k\right) \cdot a} + 100 \cdot \left({k}^{2} \cdot a\right)\right)
\]
unpow224.0%
\[\leadsto a + \left(\left(-10 \cdot k\right) \cdot a + 100 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot a\right)\right)
\]
associate-*r*24.0%
\[\leadsto a + \left(\left(-10 \cdot k\right) \cdot a + \color{blue}{\left(100 \cdot \left(k \cdot k\right)\right) \cdot a}\right)
\]
distribute-rgt-out32.0%
\[\leadsto a + \color{blue}{a \cdot \left(-10 \cdot k + 100 \cdot \left(k \cdot k\right)\right)}
\]
*-commutative32.0%
\[\leadsto a + a \cdot \left(\color{blue}{k \cdot -10} + 100 \cdot \left(k \cdot k\right)\right)
\]
Simplified32.0%
\[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10 + 100 \cdot \left(k \cdot k\right)\right)}
\]
Recombined 3 regimes into one program. Final simplification59.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;m \leq -0.62:\\
\;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\
\mathbf{elif}\;m \leq 0.0066:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\mathbf{else}:\\
\;\;\;\;a + a \cdot \left(k \cdot -10 + \left(k \cdot k\right) \cdot 100\right)\\
\end{array}
\]
Alternative 8? \[\begin{array}{l}
\mathbf{if}\;m \leq -0.23:\\
\;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\
\mathbf{elif}\;m \leq 1.2 \cdot 10^{+14}:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\mathbf{else}:\\
\;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if m < -0.23000000000000001 Initial program 100.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 32.3%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Step-by-step derivation clear-num34.2%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}}
\]
inv-pow34.2%
\[\leadsto \color{blue}{{\left(\frac{1 + k \cdot \left(k + 10\right)}{a}\right)}^{-1}}
\]
+-commutative34.2%
\[\leadsto {\left(\frac{\color{blue}{k \cdot \left(k + 10\right) + 1}}{a}\right)}^{-1}
\]
fma-udef34.2%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(k, k + 10, 1\right)}}{a}\right)}^{-1}
\]
Applied egg-rr 34.2%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}\right)}^{-1}}
\]
Step-by-step derivation unpow-134.2%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Simplified34.2%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Taylor expanded in k around inf 56.6%
\[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{a}}
\]
Step-by-step derivation unpow256.6%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
Simplified56.6%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
if -0.23000000000000001 < m < 1.2e14 Initial program 93.5%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/93.5%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+93.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative93.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out93.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def93.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative93.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified93.5%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 89.9%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
if 1.2e14 < m Initial program 79.4%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/79.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+79.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative79.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out79.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def79.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative79.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified79.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 3.1%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around 0 11.1%
\[\leadsto \color{blue}{a + -10 \cdot \left(k \cdot a\right)}
\]
Recombined 3 regimes into one program. Final simplification50.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;m \leq -0.23:\\
\;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\
\mathbf{elif}\;m \leq 1.2 \cdot 10^{+14}:\\
\;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\
\mathbf{else}:\\
\;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\
\end{array}
\]
Alternative 9? \[\begin{array}{l}
\mathbf{if}\;k \leq 1.25 \cdot 10^{-278}:\\
\;\;\;\;\frac{a}{k \cdot k}\\
\mathbf{elif}\;k \leq 1:\\
\;\;\;\;a\\
\mathbf{else}:\\
\;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\
\end{array}
\]
Derivation Split input into 3 regimes if k < 1.24999999999999996e-278 Initial program 86.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/86.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified86.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 14.1%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around inf 21.6%
\[\leadsto \color{blue}{\frac{a}{{k}^{2}}}
\]
Step-by-step derivation unpow221.6%
\[\leadsto \frac{a}{\color{blue}{k \cdot k}}
\]
Simplified21.6%
\[\leadsto \color{blue}{\frac{a}{k \cdot k}}
\]
if 1.24999999999999996e-278 < k < 1 Initial program 100.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 46.3%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around 0 45.9%
\[\leadsto \color{blue}{a}
\]
if 1 < k Initial program 84.4%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/84.3%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified84.3%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 63.1%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around inf 61.3%
\[\leadsto \color{blue}{\frac{a}{{k}^{2}}}
\]
Step-by-step derivation unpow261.3%
\[\leadsto \frac{a}{\color{blue}{k \cdot k}}
\]
Simplified61.3%
\[\leadsto \color{blue}{\frac{a}{k \cdot k}}
\]
Step-by-step derivation *-un-lft-identity61.3%
\[\leadsto \frac{\color{blue}{1 \cdot a}}{k \cdot k}
\]
times-frac63.5%
\[\leadsto \color{blue}{\frac{1}{k} \cdot \frac{a}{k}}
\]
Applied egg-rr 63.5%
\[\leadsto \color{blue}{\frac{1}{k} \cdot \frac{a}{k}}
\]
Recombined 3 regimes into one program. Final simplification43.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;k \leq 1.25 \cdot 10^{-278}:\\
\;\;\;\;\frac{a}{k \cdot k}\\
\mathbf{elif}\;k \leq 1:\\
\;\;\;\;a\\
\mathbf{else}:\\
\;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\
\end{array}
\]
Alternative 10? \[\begin{array}{l}
\mathbf{if}\;k \leq 5.8 \cdot 10^{-279}:\\
\;\;\;\;\frac{a}{k \cdot k}\\
\mathbf{elif}\;k \leq 0.06:\\
\;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\
\end{array}
\]
Derivation Split input into 3 regimes if k < 5.8e-279 Initial program 86.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/86.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified86.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 14.1%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around inf 21.6%
\[\leadsto \color{blue}{\frac{a}{{k}^{2}}}
\]
Step-by-step derivation unpow221.6%
\[\leadsto \frac{a}{\color{blue}{k \cdot k}}
\]
Simplified21.6%
\[\leadsto \color{blue}{\frac{a}{k \cdot k}}
\]
if 5.8e-279 < k < 0.059999999999999998 Initial program 100.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 46.3%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around 0 46.3%
\[\leadsto \color{blue}{a + -10 \cdot \left(k \cdot a\right)}
\]
if 0.059999999999999998 < k Initial program 84.4%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/84.3%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified84.3%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 63.1%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around inf 61.3%
\[\leadsto \color{blue}{\frac{a}{{k}^{2}}}
\]
Step-by-step derivation unpow261.3%
\[\leadsto \frac{a}{\color{blue}{k \cdot k}}
\]
Simplified61.3%
\[\leadsto \color{blue}{\frac{a}{k \cdot k}}
\]
Step-by-step derivation *-un-lft-identity61.3%
\[\leadsto \frac{\color{blue}{1 \cdot a}}{k \cdot k}
\]
times-frac63.5%
\[\leadsto \color{blue}{\frac{1}{k} \cdot \frac{a}{k}}
\]
Applied egg-rr 63.5%
\[\leadsto \color{blue}{\frac{1}{k} \cdot \frac{a}{k}}
\]
Recombined 3 regimes into one program. Final simplification43.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;k \leq 5.8 \cdot 10^{-279}:\\
\;\;\;\;\frac{a}{k \cdot k}\\
\mathbf{elif}\;k \leq 0.06:\\
\;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\
\end{array}
\]
Alternative 11? \[\begin{array}{l}
\mathbf{if}\;k \leq 7 \cdot 10^{-279}:\\
\;\;\;\;\frac{a}{k \cdot k}\\
\mathbf{elif}\;k \leq 0.1:\\
\;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\
\end{array}
\]
Derivation Split input into 3 regimes if k < 7.00000000000000019e-279 Initial program 86.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/86.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified86.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 14.1%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around inf 21.6%
\[\leadsto \color{blue}{\frac{a}{{k}^{2}}}
\]
Step-by-step derivation unpow221.6%
\[\leadsto \frac{a}{\color{blue}{k \cdot k}}
\]
Simplified21.6%
\[\leadsto \color{blue}{\frac{a}{k \cdot k}}
\]
if 7.00000000000000019e-279 < k < 0.10000000000000001 Initial program 100.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 46.3%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around 0 46.3%
\[\leadsto \color{blue}{a + -10 \cdot \left(k \cdot a\right)}
\]
if 0.10000000000000001 < k Initial program 84.4%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/84.3%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified84.3%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 63.1%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Step-by-step derivation clear-num63.1%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}}
\]
inv-pow63.1%
\[\leadsto \color{blue}{{\left(\frac{1 + k \cdot \left(k + 10\right)}{a}\right)}^{-1}}
\]
+-commutative63.1%
\[\leadsto {\left(\frac{\color{blue}{k \cdot \left(k + 10\right) + 1}}{a}\right)}^{-1}
\]
fma-udef63.1%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(k, k + 10, 1\right)}}{a}\right)}^{-1}
\]
Applied egg-rr 63.1%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}\right)}^{-1}}
\]
Step-by-step derivation unpow-163.1%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Simplified63.1%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Taylor expanded in k around inf 61.3%
\[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2}}{a}}}
\]
Step-by-step derivation unpow261.3%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
associate-/l*63.5%
\[\leadsto \frac{1}{\color{blue}{\frac{k}{\frac{a}{k}}}}
\]
Simplified63.5%
\[\leadsto \frac{1}{\color{blue}{\frac{k}{\frac{a}{k}}}}
\]
Recombined 3 regimes into one program. Final simplification43.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;k \leq 7 \cdot 10^{-279}:\\
\;\;\;\;\frac{a}{k \cdot k}\\
\mathbf{elif}\;k \leq 0.1:\\
\;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\
\end{array}
\]
Alternative 12? \[\begin{array}{l}
\mathbf{if}\;k \leq 5.8 \cdot 10^{-279}:\\
\;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\
\mathbf{elif}\;k \leq 0.1:\\
\;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\
\end{array}
\]
Derivation Split input into 3 regimes if k < 5.8e-279 Initial program 86.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/86.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative86.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified86.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 14.1%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Step-by-step derivation clear-num15.6%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}}
\]
inv-pow15.6%
\[\leadsto \color{blue}{{\left(\frac{1 + k \cdot \left(k + 10\right)}{a}\right)}^{-1}}
\]
+-commutative15.6%
\[\leadsto {\left(\frac{\color{blue}{k \cdot \left(k + 10\right) + 1}}{a}\right)}^{-1}
\]
fma-udef15.6%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(k, k + 10, 1\right)}}{a}\right)}^{-1}
\]
Applied egg-rr 15.6%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}\right)}^{-1}}
\]
Step-by-step derivation unpow-115.6%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Simplified15.6%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Taylor expanded in k around inf 23.1%
\[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{a}}
\]
Step-by-step derivation unpow223.1%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
Simplified23.1%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
if 5.8e-279 < k < 0.10000000000000001 Initial program 100.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 46.3%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around 0 46.3%
\[\leadsto \color{blue}{a + -10 \cdot \left(k \cdot a\right)}
\]
if 0.10000000000000001 < k Initial program 84.4%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/84.3%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative84.3%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified84.3%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 63.1%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Step-by-step derivation clear-num63.1%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}}
\]
inv-pow63.1%
\[\leadsto \color{blue}{{\left(\frac{1 + k \cdot \left(k + 10\right)}{a}\right)}^{-1}}
\]
+-commutative63.1%
\[\leadsto {\left(\frac{\color{blue}{k \cdot \left(k + 10\right) + 1}}{a}\right)}^{-1}
\]
fma-udef63.1%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(k, k + 10, 1\right)}}{a}\right)}^{-1}
\]
Applied egg-rr 63.1%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}\right)}^{-1}}
\]
Step-by-step derivation unpow-163.1%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Simplified63.1%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Taylor expanded in k around inf 61.3%
\[\leadsto \frac{1}{\color{blue}{\frac{{k}^{2}}{a}}}
\]
Step-by-step derivation unpow261.3%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
associate-/l*63.5%
\[\leadsto \frac{1}{\color{blue}{\frac{k}{\frac{a}{k}}}}
\]
Simplified63.5%
\[\leadsto \frac{1}{\color{blue}{\frac{k}{\frac{a}{k}}}}
\]
Recombined 3 regimes into one program. Final simplification44.4%
\[\leadsto \begin{array}{l}
\mathbf{if}\;k \leq 5.8 \cdot 10^{-279}:\\
\;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\
\mathbf{elif}\;k \leq 0.1:\\
\;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\
\end{array}
\]
Alternative 13? \[\begin{array}{l}
\mathbf{if}\;m \leq -0.195:\\
\;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\
\mathbf{elif}\;m \leq 1.2 \cdot 10^{+14}:\\
\;\;\;\;\frac{a}{1 + k \cdot k}\\
\mathbf{else}:\\
\;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if m < -0.19500000000000001 Initial program 100.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 32.3%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Step-by-step derivation clear-num34.2%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}}
\]
inv-pow34.2%
\[\leadsto \color{blue}{{\left(\frac{1 + k \cdot \left(k + 10\right)}{a}\right)}^{-1}}
\]
+-commutative34.2%
\[\leadsto {\left(\frac{\color{blue}{k \cdot \left(k + 10\right) + 1}}{a}\right)}^{-1}
\]
fma-udef34.2%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(k, k + 10, 1\right)}}{a}\right)}^{-1}
\]
Applied egg-rr 34.2%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}\right)}^{-1}}
\]
Step-by-step derivation unpow-134.2%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Simplified34.2%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Taylor expanded in k around inf 56.6%
\[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2}}}{a}}
\]
Step-by-step derivation unpow256.6%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
Simplified56.6%
\[\leadsto \frac{1}{\frac{\color{blue}{k \cdot k}}{a}}
\]
if -0.19500000000000001 < m < 1.2e14 Initial program 93.5%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/93.5%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+93.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative93.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out93.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def93.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative93.5%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified93.5%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 89.9%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around inf 87.7%
\[\leadsto \frac{a}{1 + \color{blue}{{k}^{2}}}
\]
Step-by-step derivation unpow287.7%
\[\leadsto \frac{a}{1 + \color{blue}{k \cdot k}}
\]
Simplified87.7%
\[\leadsto \frac{a}{1 + \color{blue}{k \cdot k}}
\]
if 1.2e14 < m Initial program 79.4%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/79.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+79.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative79.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out79.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def79.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative79.4%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified79.4%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 3.1%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around 0 11.1%
\[\leadsto \color{blue}{a + -10 \cdot \left(k \cdot a\right)}
\]
Recombined 3 regimes into one program. Final simplification50.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;m \leq -0.195:\\
\;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\
\mathbf{elif}\;m \leq 1.2 \cdot 10^{+14}:\\
\;\;\;\;\frac{a}{1 + k \cdot k}\\
\mathbf{else}:\\
\;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\
\end{array}
\]
Alternative 14? \[\begin{array}{l}
\mathbf{if}\;k \leq 3.3 \cdot 10^{-276} \lor \neg \left(k \leq 1\right):\\
\;\;\;\;\frac{a}{k \cdot k}\\
\mathbf{else}:\\
\;\;\;\;a\\
\end{array}
\]
Derivation Split input into 2 regimes if k < 3.29999999999999991e-276 or 1 < k Initial program 85.2%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/85.2%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+85.2%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative85.2%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out85.2%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def85.2%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative85.2%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified85.2%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 38.9%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around inf 41.7%
\[\leadsto \color{blue}{\frac{a}{{k}^{2}}}
\]
Step-by-step derivation unpow241.7%
\[\leadsto \frac{a}{\color{blue}{k \cdot k}}
\]
Simplified41.7%
\[\leadsto \color{blue}{\frac{a}{k \cdot k}}
\]
if 3.29999999999999991e-276 < k < 1 Initial program 100.0%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative100.0%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified100.0%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 46.3%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around 0 45.9%
\[\leadsto \color{blue}{a}
\]
Recombined 2 regimes into one program. Final simplification43.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;k \leq 3.3 \cdot 10^{-276} \lor \neg \left(k \leq 1\right):\\
\;\;\;\;\frac{a}{k \cdot k}\\
\mathbf{else}:\\
\;\;\;\;a\\
\end{array}
\]
Alternative 15? \[\begin{array}{l}
\mathbf{if}\;m \leq -3.5 \cdot 10^{-53}:\\
\;\;\;\;\frac{a}{k} \cdot 0.1\\
\mathbf{else}:\\
\;\;\;\;a\\
\end{array}
\]
Derivation Split input into 2 regimes if m < -3.49999999999999993e-53 Initial program 98.7%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/98.7%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+98.7%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative98.7%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out98.7%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def98.7%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative98.7%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified98.7%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 36.5%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Step-by-step derivation clear-num38.1%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}}
\]
inv-pow38.1%
\[\leadsto \color{blue}{{\left(\frac{1 + k \cdot \left(k + 10\right)}{a}\right)}^{-1}}
\]
+-commutative38.1%
\[\leadsto {\left(\frac{\color{blue}{k \cdot \left(k + 10\right) + 1}}{a}\right)}^{-1}
\]
fma-udef38.1%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(k, k + 10, 1\right)}}{a}\right)}^{-1}
\]
Applied egg-rr 38.1%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}\right)}^{-1}}
\]
Step-by-step derivation unpow-138.1%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Simplified38.1%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}}
\]
Taylor expanded in k around 0 18.4%
\[\leadsto \frac{1}{\color{blue}{10 \cdot \frac{k}{a} + \frac{1}{a}}}
\]
Taylor expanded in k around inf 23.6%
\[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}}
\]
if -3.49999999999999993e-53 < m Initial program 86.1%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/86.1%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+86.1%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative86.1%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out86.1%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def86.1%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative86.1%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified86.1%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 43.4%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around 0 23.8%
\[\leadsto \color{blue}{a}
\]
Recombined 2 regimes into one program. Final simplification23.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;m \leq -3.5 \cdot 10^{-53}:\\
\;\;\;\;\frac{a}{k} \cdot 0.1\\
\mathbf{else}:\\
\;\;\;\;a\\
\end{array}
\]
Alternative 16? \[a
\]
Derivation Initial program 89.9%
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
\]
Step-by-step derivation associate-*r/89.9%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}}
\]
associate-+l+89.9%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}
\]
+-commutative89.9%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}
\]
distribute-rgt-out89.9%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1}
\]
fma-def89.9%
\[\leadsto a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}
\]
+-commutative89.9%
\[\leadsto a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}
\]
Simplified89.9%
\[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}}
\]
Taylor expanded in m around 0 41.3%
\[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}}
\]
Taylor expanded in k around 0 18.3%
\[\leadsto \color{blue}{a}
\]
Final simplification18.3%
\[\leadsto a
\]
Reproduce ? herbie shell --seed 2023166
(FPCore (a k m)
:name "Falkner and Boettcher, Appendix A"
:precision binary64
(/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))
