Sollya User's Manual

Name:

searchgal searches for a preimage of a function such that the rounding the image yields an error smaller than a constant

Usage:

searchgal(function, start, preimage precision, steps, format, error bound) : (function, constant, integer, integer, HP|halfprecision|SG|single|D|double|DE|doubleextended|DD|doubledouble|QD|quad|TD|tripledouble, constant) -> list searchgal(list of functions, start, preimage precision, steps, list of format, list of error bounds) : (list, constant, integer, integer, list, list) -> list

Parameters:

function represents the function to be considered
start represents a value around which the search is to be performed
preimage precision represents the precision (discretization) for the eligible preimage values
steps represents the binary logarithm (log2) of the number of search steps to be performed
format represents the format the image of the function is to be rounded to
error bound represents a upper bound on the relative rounding error when rounding the image
list of functions represents the functions to be considered
list of formats represents the respective formats the images of the functions are to be rounded to
list of error bounds represents a upper bound on the relative rounding error when rounding the image

Description:

The command searchgal searches for a preimage z of function function or a list of functions list of functions such that z is a floating-point number with preimage precision significant mantissa bits and the image y of the function, respectively each image yi of the functions, rounds to format format respectively to the corresponding format in list of format with a relative rounding error less than error bound respectively the corresponding value in list of error bounds. During this search, at most 2^steps attempts are made. The search starts with a preimage value equal to start. This value is then increased and decreased by 1 ulp in precision preimage precision until a value is found or the step limit is reached.

If the search finds an appropriate preimage z, searchgal evaluates to a list containing this value. Otherwise, searchgal evaluates to an empty list.

Example 1:

   > searchgal(log(x),2,53,15,DD,1b-112);
   [| |]
   > searchgal(log(x),2,53,18,DD,1b-112);
   [|2.0000000000384972054234822280704975128173828125|]

Example 2:

   > f = exp(x);
   > s = searchgal(f,2,53,18,DD,1b-112);
   > if (s != [||]) then {
        v = s[0];
        print("The rounding error is 2^(",evaluate(log2(abs(DD(f)/f - 1)),v),")");
     } else print("No value found");
   The rounding error is 2^( -1.12106878438809380148206984258358542322113874177832e2 )

Example 3:

> searchgal([|sin(x),cos(x)|],1,53,15,[|D,D|],[|1b-62,1b-60|]);
[|1.00000000000159494639717649988597258925437927246094|]

See also: round, double, doubledouble, tripledouble, evaluate, worstcase

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