To pay for a spell with improvise, you can tap no more artifacts you control than the amount of "generic mana in [the] spell's total cost" (C.R. 702.125a; under that rule, you tap the artifacts "rather than pay" the corresponding generic mana). Therefore, if the total cost of the spell is 0 due to Sram's Expertise, improvise won't let you tap any artifacts to pay for that spell. Only if another effect adds generic mana to the spell's total cost can you do so (for example, an effect that makes spells "cost 1 more to cast") (see also C.R. 601.2f).
Moreover, improvise is not an "alternative cost" within the meaning of C.R. 118.9 (so is not governed by C.R. 118.9a, for example; see also C.R. 702.125b), but rather it changes the way a spell is paid, much like delve and convoke (review C.R. 702.65b, 702.50b).
EDIT (Jan. 20): With the Aether Revolt rules update, portions of some rules were moved.
EDIT (Feb. 10): In the case of Whir of Invention, moreover, since the spell has X in its mana cost and doesn't define the value of X in the text of the spell, "the only legal choice for X is 0" if you cast it without paying its mana cost (C.R. 107.3b). (Currently, Spoils of War is the only spell that defines the value of X in its text [see this thread].) Therefore, even if another effect adds generic mana to Whir of Invention's total cost, making improvise applicable to that spell, that won't change the value of X for that spell, which will still be 0.
EDIT (Apr. 5, 2017; Apr. 11, 2017; Apr. 5, 2018): Clarified note from Feb. 10.
EDIT (Nov. 4, 2019): Edited, including because some rules were renumbered with Core Set 2020.
Moreover, improvise is not an "alternative cost" within the meaning of C.R. 118.9 (so is not governed by C.R. 118.9a, for example; see also C.R. 702.125b), but rather it changes the way a spell is paid, much like delve and convoke (review C.R. 702.65b, 702.50b).
EDIT (Jan. 20): With the Aether Revolt rules update, portions of some rules were moved.
EDIT (Feb. 10): In the case of Whir of Invention, moreover, since the spell has X in its mana cost and doesn't define the value of X in the text of the spell, "the only legal choice for X is 0" if you cast it without paying its mana cost (C.R. 107.3b). (Currently, Spoils of War is the only spell that defines the value of X in its text [see this thread].) Therefore, even if another effect adds generic mana to Whir of Invention's total cost, making improvise applicable to that spell, that won't change the value of X for that spell, which will still be 0.
EDIT (Apr. 5, 2017; Apr. 11, 2017; Apr. 5, 2018): Clarified note from Feb. 10.
EDIT (Nov. 4, 2019): Edited, including because some rules were renumbered with Core Set 2020.