In Apex you can use "describe" calls. If all other options fail - you could build a custom service that returns this data to you.
Schema.DescribeFieldResult dfr = Account.Address__c.getDescribe();
System.debug(dfr.getCalculatedFormula());
// BillingStreet & BR() & BillingCity & BR() & BillingPostalCode & BR() & BillingCountry & BR() & Street2__c & BR() & Street3__c
You've tagged metadata API so if you really use that - similar info should be available in it.
<CustomField>
<fullName>Address__c</fullName>
<externalId>false</externalId>
<formula>BillingStreet & BR() & BillingCity & BR() & BillingPostalCode & BR() & BillingCountry & BR() & Street2__c & BR() & Street3__c</formula>
<formulaTreatBlanksAs>BlankAsZero</formulaTreatBlanksAs>
<label>Address</label>
<required>false</required>
<trackHistory>false</trackHistory>
<type>Text</type>
<unique>false</unique>
</CustomField>
In REST API it's similar, a call to /services/data/v48.0/sobjects/Account/describe will return (among others)
{
"aggregatable" : true,
"aiPredictionField" : false,
"autoNumber" : false,
"byteLength" : 3900,
"calculated" : true,
"calculatedFormula" : "BillingStreet & BR() & BillingCity & BR() & BillingPostalCode & BR() & BillingCountry & BR() & Street2__c & BR() & Street3__c",
"cascadeDelete" : false,
(...)
"formulaTreatNullNumberAsZero" : true,
"groupable" : false,
"highScaleNumber" : false,
"htmlFormatted" : true,
"idLookup" : false,
"inlineHelpText" : null,
"label" : "Address",
"length" : 1300,
"mask" : null,
"maskType" : null,
"name" : "Address__c",
(...)
"type" : "string",
"unique" : false,
"updateable" : false,
"writeRequiresMasterRead" : false
}
And finally there's Tooling API where you can query the metadata like you'd query normal database tables. But the core of what you need will be hidden in a JSON field you'd have to parse. If your tool is some ETL - check if it can query FieldDefinition table.
/services/data/v48.0/tooling/query?q=SELECT+Metadata+FROM+FieldDefinition+WHERE+EntityDefinitionId+=+'Account'+AND+QualifiedApiName+=+'Address__c'
Hope this helps!
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Thanks!