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Exp

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Exp[z]

gives the exponential E^z.

Notes Exp is Listable. Exp\[0\] = 1, Exp\[Log\[x\]\] -\> x, Exp\[I Pi\] = -1. Numeric inputs route to libm / MPFR.

Examples (7)

Every input below was run against the current Mathilda build and its output recorded.

Applications (7)

In[1]:= Exp[0]
Out[1]= 1

In[2]:= Exp[I Pi]
Out[2]= -1

In[3]:= Exp[Log[x]]
Out[3]= x

In[4]:= Exp[2 Log[x]]
Out[4]= x^2

In[5]:= D[Exp[Sin[x]], x]
Out[5]= Cos[x] E^Sin[x]

In[6]:= Series[Exp[x], {x, 0, 6}]
Out[6]= 1 + x + 1/2 x^2 + 1/6 x^3 + 1/24 x^4 + 1/120 x^5 + 1/720 x^6 + O[x]^7

In[7]:= N[Exp[1], 50]
Out[7]= 2.71828182845904523536028747135266249775724709369996

Performance

Against other systems, from the benchmark suite (same input, results cross-checked for agreement):

case Mathilda Wolfram Python
NI 50-digit Gaussian 6.87 s 2.11 s 1.1 s
v^2.5 over 4x10^6 6.66 s 3.79 s 16.3 s
integrate Sin[x] Exp[x] 6.62 s 0.726 s 5.47 s
Sin[Exp[Log[v]]] fused? 6.52 s 5.61 s 42.5 s
integrate Exp[x]/x (non-elementary) 5.38 s 0.318 s 33.5 s
Sqrt over 4x10^6 5.35 s 0.444 s 0.925 s

Implementation notes

Algorithm. builtin_exp (1-arg). Exact special values: Exp[0] = 1, Exp[-Infinity] = 0, Exp[Infinity] = Infinity. The notable closed-form path is Euler's formula for Exp[I q Pi]: when the argument is Times[Complex[0,q], Pi] with q an integer or rational, it rewrites to Cos[q Pi] + I Sin[q Pi] (which the trig kernels then evaluate to exact algebraic values). Numeric arguments go through MPFR when USE_MPFR ā€” mpfr_exp for pure reals, the exp(a)(cos b + i sin b) complex helper otherwise ā€” or cexp on a double complex for machine inputs, collapsing to a real result when the imaginary part is zero. Anything that matches none of these is returned as Power[E, z] (the canonical internal form for an unevaluated exponential), so Exp is effectively a thin front-end that produces E^z and lets the Power machinery carry the symbolic case.

Data structures. Expr* trees throughout; numeric evaluation uses double complex (machine) or mpfr_t/complex-MPFR helpers (arbitrary precision). The Euler path scans the Times argument list for a Pi factor and a single pure-imaginary Complex[0, q] coefficient.

Attributes: Listable, NumericFunction, Protected.

References

Notes & additional examples

Notes

Exp[z] is E^z; it inverts Log and evaluates Euler's identity exactly. Logarithmic arguments collapse (Exp[2 Log[x]] -> x^2), it differentiates and series-expands symbolically, and numeric inputs route to libm / MPFR for arbitrary precision. Exp is Listable.