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kglr
kglr
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pW[f_, pW[n_n_, m_, min_: 0, max_: 1] := Piecewise[
  {Fit[Table[{x, Sqrtf @ x}, {x, ##, (#2 - #)/m}], {1, x, x^2, E^x}, x], # < x <= #2} & @@@
   Partition[Subdivide[min, max, n], 2, 1]]

Examples:

pW[4pW[Sqrt, 4, 10] // TeXForm

$\small\begin{cases} -149.176 x^2-246.803 x+251.787 e^x-251.768 & 0<x\leq \frac{1}{4} \\ -2.882 x^2-1.65139 x+3.18162 e^x-2.99226 & \frac{1}{4}<x\leq \frac{1}{2} \\ -0.876557 x^2+0.489058 x+0.663232 e^x-0.41176 & \frac{1}{2}<x\leq \frac{3}{4} \\ -0.418328 x^2+0.738036 x+0.220335 e^x+0.0813601 & \frac{3}{4}<x\leq 1 \end{cases}$

pW[5pW[Sqrt, 5, 50, 5, 15] // TeXForm

$\small\begin{cases} -0.010714 x^2+0.328728 x+\text{9.866783884779016$\grave{ }$*${}^{\wedge}$-6} e^x+0.858866 & 5<x\leq 7 \\ -0.00658734 x^2+0.280252 x+\text{6.475622629784991$\grave{ }$*${}^{\wedge}$-7} e^x+1.00608 & 7<x\leq 9 \\ -0.00455749 x^2+0.248165 x+\text{5.00746734942773$\grave{ }$*${}^{\wedge}$-8} e^x+1.13528 & 9<x\leq 11 \\ -0.00338884 x^2+0.224974 x+\text{4.292376170350944$\grave{ }$*${}^{\wedge}$-9} e^x+1.25171 & 11<x\leq 13 \\ -0.00264528 x^2+0.207225 x+\text{3.949544062291019$\grave{ }$*${}^{\wedge}$-10} e^x+1.3585 & 13<x\leq 15 \end{cases}$

 pW[n_, m_, min_: 0, max_: 1] := Piecewise[
  {Fit[Table[{x, Sqrt @ x}, {x, ##, (#2 - #)/m}], {1, x, x^2, E^x}, x], # < x <= #2} & @@@
   Partition[Subdivide[min, max, n], 2, 1]]

Examples:

pW[4, 10] // TeXForm

$\small\begin{cases} -149.176 x^2-246.803 x+251.787 e^x-251.768 & 0<x\leq \frac{1}{4} \\ -2.882 x^2-1.65139 x+3.18162 e^x-2.99226 & \frac{1}{4}<x\leq \frac{1}{2} \\ -0.876557 x^2+0.489058 x+0.663232 e^x-0.41176 & \frac{1}{2}<x\leq \frac{3}{4} \\ -0.418328 x^2+0.738036 x+0.220335 e^x+0.0813601 & \frac{3}{4}<x\leq 1 \end{cases}$

pW[5, 50, 5, 15] // TeXForm

$\small\begin{cases} -0.010714 x^2+0.328728 x+\text{9.866783884779016$\grave{ }$*${}^{\wedge}$-6} e^x+0.858866 & 5<x\leq 7 \\ -0.00658734 x^2+0.280252 x+\text{6.475622629784991$\grave{ }$*${}^{\wedge}$-7} e^x+1.00608 & 7<x\leq 9 \\ -0.00455749 x^2+0.248165 x+\text{5.00746734942773$\grave{ }$*${}^{\wedge}$-8} e^x+1.13528 & 9<x\leq 11 \\ -0.00338884 x^2+0.224974 x+\text{4.292376170350944$\grave{ }$*${}^{\wedge}$-9} e^x+1.25171 & 11<x\leq 13 \\ -0.00264528 x^2+0.207225 x+\text{3.949544062291019$\grave{ }$*${}^{\wedge}$-10} e^x+1.3585 & 13<x\leq 15 \end{cases}$

pW[f_, n_, m_, min_: 0, max_: 1] := Piecewise[
  {Fit[Table[{x, f @ x}, {x, ##, (#2 - #)/m}], {1, x, x^2, E^x}, x], # < x <= #2} & @@@
   Partition[Subdivide[min, max, n], 2, 1]]

Examples:

pW[Sqrt, 4, 10] // TeXForm

$\small\begin{cases} -149.176 x^2-246.803 x+251.787 e^x-251.768 & 0<x\leq \frac{1}{4} \\ -2.882 x^2-1.65139 x+3.18162 e^x-2.99226 & \frac{1}{4}<x\leq \frac{1}{2} \\ -0.876557 x^2+0.489058 x+0.663232 e^x-0.41176 & \frac{1}{2}<x\leq \frac{3}{4} \\ -0.418328 x^2+0.738036 x+0.220335 e^x+0.0813601 & \frac{3}{4}<x\leq 1 \end{cases}$

pW[Sqrt, 5, 50, 5, 15] // TeXForm

$\small\begin{cases} -0.010714 x^2+0.328728 x+\text{9.866783884779016$\grave{ }$*${}^{\wedge}$-6} e^x+0.858866 & 5<x\leq 7 \\ -0.00658734 x^2+0.280252 x+\text{6.475622629784991$\grave{ }$*${}^{\wedge}$-7} e^x+1.00608 & 7<x\leq 9 \\ -0.00455749 x^2+0.248165 x+\text{5.00746734942773$\grave{ }$*${}^{\wedge}$-8} e^x+1.13528 & 9<x\leq 11 \\ -0.00338884 x^2+0.224974 x+\text{4.292376170350944$\grave{ }$*${}^{\wedge}$-9} e^x+1.25171 & 11<x\leq 13 \\ -0.00264528 x^2+0.207225 x+\text{3.949544062291019$\grave{ }$*${}^{\wedge}$-10} e^x+1.3585 & 13<x\leq 15 \end{cases}$

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kglr
kglr
  • 400.5k
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  • 929
 pW[n_, m_]m_, min_: 0, max_: 1] := Piecewise[
  {Fit[Table[{x, Sqrt @ x}, {x, ##, (#2 - #)/m}], {1, x, x^2, E^x},  x], # < x <= #2} & @@@
  Partition[Subdivide[n] Partition[Subdivide[min, max, n], 2, 1]] 
 

Examples:

pW[4, 10] // TeXForm

$\begin{cases} -149.176 x^2-246.803 x+251.787 e^x-251.768 & 0<x\leq \frac{1}{4} \\ -2.882 x^2-1.65139 x+3.18162 e^x-2.99226 & \frac{1}{4}<x\leq \frac{1}{2} \\ -0.876557 x^2+0.489058 x+0.663232 e^x-0.41176 & \frac{1}{2}<x\leq \frac{3}{4} \\ -0.418328 x^2+0.738036 x+0.220335 e^x+0.0813601 & \frac{3}{4}<x\leq 1 \end{cases}$$\small\begin{cases} -149.176 x^2-246.803 x+251.787 e^x-251.768 & 0<x\leq \frac{1}{4} \\ -2.882 x^2-1.65139 x+3.18162 e^x-2.99226 & \frac{1}{4}<x\leq \frac{1}{2} \\ -0.876557 x^2+0.489058 x+0.663232 e^x-0.41176 & \frac{1}{2}<x\leq \frac{3}{4} \\ -0.418328 x^2+0.738036 x+0.220335 e^x+0.0813601 & \frac{3}{4}<x\leq 1 \end{cases}$

pW[5, 50, 5, 15] // TeXForm

$\small\begin{cases} -0.010714 x^2+0.328728 x+\text{9.866783884779016$\grave{ }$*${}^{\wedge}$-6} e^x+0.858866 & 5<x\leq 7 \\ -0.00658734 x^2+0.280252 x+\text{6.475622629784991$\grave{ }$*${}^{\wedge}$-7} e^x+1.00608 & 7<x\leq 9 \\ -0.00455749 x^2+0.248165 x+\text{5.00746734942773$\grave{ }$*${}^{\wedge}$-8} e^x+1.13528 & 9<x\leq 11 \\ -0.00338884 x^2+0.224974 x+\text{4.292376170350944$\grave{ }$*${}^{\wedge}$-9} e^x+1.25171 & 11<x\leq 13 \\ -0.00264528 x^2+0.207225 x+\text{3.949544062291019$\grave{ }$*${}^{\wedge}$-10} e^x+1.3585 & 13<x\leq 15 \end{cases}$

pW[n_, m_] := Piecewise[
 {Fit[Table[{x, Sqrt @ x}, {x, ##, (#2 - #)/m}], {1, x, x^2, E^x},  x], # < x <= #2} & @@@
  Partition[Subdivide[n], 2, 1]] 
 
pW[4, 10] // TeXForm

$\begin{cases} -149.176 x^2-246.803 x+251.787 e^x-251.768 & 0<x\leq \frac{1}{4} \\ -2.882 x^2-1.65139 x+3.18162 e^x-2.99226 & \frac{1}{4}<x\leq \frac{1}{2} \\ -0.876557 x^2+0.489058 x+0.663232 e^x-0.41176 & \frac{1}{2}<x\leq \frac{3}{4} \\ -0.418328 x^2+0.738036 x+0.220335 e^x+0.0813601 & \frac{3}{4}<x\leq 1 \end{cases}$

 pW[n_, m_, min_: 0, max_: 1] := Piecewise[
  {Fit[Table[{x, Sqrt @ x}, {x, ##, (#2 - #)/m}], {1, x, x^2, E^x}, x], # < x <= #2} & @@@
   Partition[Subdivide[min, max, n], 2, 1]]

Examples:

pW[4, 10] // TeXForm

$\small\begin{cases} -149.176 x^2-246.803 x+251.787 e^x-251.768 & 0<x\leq \frac{1}{4} \\ -2.882 x^2-1.65139 x+3.18162 e^x-2.99226 & \frac{1}{4}<x\leq \frac{1}{2} \\ -0.876557 x^2+0.489058 x+0.663232 e^x-0.41176 & \frac{1}{2}<x\leq \frac{3}{4} \\ -0.418328 x^2+0.738036 x+0.220335 e^x+0.0813601 & \frac{3}{4}<x\leq 1 \end{cases}$

pW[5, 50, 5, 15] // TeXForm

$\small\begin{cases} -0.010714 x^2+0.328728 x+\text{9.866783884779016$\grave{ }$*${}^{\wedge}$-6} e^x+0.858866 & 5<x\leq 7 \\ -0.00658734 x^2+0.280252 x+\text{6.475622629784991$\grave{ }$*${}^{\wedge}$-7} e^x+1.00608 & 7<x\leq 9 \\ -0.00455749 x^2+0.248165 x+\text{5.00746734942773$\grave{ }$*${}^{\wedge}$-8} e^x+1.13528 & 9<x\leq 11 \\ -0.00338884 x^2+0.224974 x+\text{4.292376170350944$\grave{ }$*${}^{\wedge}$-9} e^x+1.25171 & 11<x\leq 13 \\ -0.00264528 x^2+0.207225 x+\text{3.949544062291019$\grave{ }$*${}^{\wedge}$-10} e^x+1.3585 & 13<x\leq 15 \end{cases}$

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kglr
kglr
  • 400.5k
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  • 488
  • 929

pW[n_, m_] := Piecewise[
 {Fit[Table[{x, Sqrt @ x}, {x, ##, (#2 - #)/m}], {1, x, x^2, E^x},  x], # < x <= #2} & @@@
  Partition[Subdivide[n], 2, 1]] 

pW[4, 10] // TeXForm

$\begin{cases} -149.176 x^2-246.803 x+251.787 e^x-251.768 & 0<x\leq \frac{1}{4} \\ -2.882 x^2-1.65139 x+3.18162 e^x-2.99226 & \frac{1}{4}<x\leq \frac{1}{2} \\ -0.876557 x^2+0.489058 x+0.663232 e^x-0.41176 & \frac{1}{2}<x\leq \frac{3}{4} \\ -0.418328 x^2+0.738036 x+0.220335 e^x+0.0813601 & \frac{3}{4}<x\leq 1 \end{cases}$