Calculations

Expressions are left unevaluated when this prevents loss of accuracy.

(2^100-1)/2^99

((3+Pi)/2)^(-3!)

?Rationalize

Gamma[34/10+1]
3.4!

{Zeta[6],Zeta[Pi],Zeta[Pi+1.2]}

Plot[Abs[Zeta[1/2+I y]],{y,0,40}]

pA=2^-2000

numA=N[pA]

(pA-numA)/pA

numA40=N[pA,40]

(pA-numA40)/pA

Another example.

pB=Sin[Pi^(7/2)+ I]

numB35=N[pB,35]

(pB-numB35)/Abs[pB]

Approximations

2 vs 2.0

expA=2.0^-2000

40 digits?

N[expA,40]

(N[expA,40]-pA)/pA

3.00000000000000000000+3.0000000000000000000000000

(expA-pA)/pA

expA100=SetPrecision[2.^-2000,100]

Abs[pA-expA100]/pA

expA40=N[expA100,40]

Abs[pA-expA40]/pA

Second example

expB=Sin[Pi^3.5 +I]

Abs[(expB-pB)/pB]

N[expB,35]

Precision[expB]

Abs[(N[expB,35]-pB)/pB]

expB35=SetPrecision[expB,35]

Abs[(expB35-pB)/pB]

expC=SetPrecision[Sin[Pi^3.5+ I],50]

Abs[(expC-pB)/pB]

Where to put SetPrecision

expD=Sin[SetPrecision[Pi^3.5+I,35]]

Abs[(expD-pB)/pB]

expE=Sin[SetPrecision[Pi^3.5,35]+I]

Abs[(expE-pB)/pB]

expF=Sin[Pi^SetPrecision[3.5,50]+ I]

Abs[(expF-pB)/pB]

expG=Sin[Pi^3.5000000000000000000000000000000000 +I]

Abs[(expG-pB)/pB]

N*

Difference between the Mathematica functions "N*" and "*": numeric vs algebraic solvers!
Solve[...]//N is NOT the same as NSolve[...]. Same for Sum vs. NSum, Integrate vs. NIntegrate, etc.

Solve[a x^2 + b x + c ==0,x]

NSolve[Pi x^5 + E x^3 -x+2.0 ==0,x]

Sum[1/i^2,{i,1,Infinity}]

NSum[1/i^2,{i,1,Infinity}]

Integrate[Exp[x]Sin[x],x]

Integrate[Exp[x]Sin[x],{x,0,3}]//Timing

NIntegrate[Exp[x]Sin[x],{x,0,3}]//Timing

NProduct, Solve, NSolve, NDSolve, NMinimize, ...

Examples

sqsqrt[x_,n_]:=
Nest[#^2&,Nest[Sqrt,x,n],n]

sqrtsq[x_,n_]:=
Nest[Sqrt,Nest[#^2&,x,n],n]

n=30;Clear[t];
Map[
{#,sqsqrt[#,n],sqrtsq[#,n]}&,
{t,Pi,1/2,.5,1.0,1.1,2,2.}
]//TableForm

Play[If[t<2,Sin[600 2 Pi t],Sin[900 2 Pi t]],{t,0,4}]

Clear[f]

f[t_/;t<2]:=Sin[2 Pi 600 t]
f[t_/;t>2]:=Sin[2 Pi 1000 t]

g[t_]:=Sin[1000 t 2 Pi]

Play[ g[t],{t,0,4}]