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vecLib Reference

The vecLib framework contains nine C header files (not counting vecLib.h which merely includes the others). Two of them, vDSP.h and vDSP_translate.h, are covered in vDSP Programming Guide and vDSP Reference.

Three of the header files are Apple’s versions of well-known libraries which are described in detail in external references:

This document describes the functions declared in the remaining header files: vecLibTypes.h, vfp.h, vForce.h, vBasicOps.h, vectorOps.h, and vBigNum.h. These files support the vector mathematical functions library (also called “vMathLib”), which runs on vector processing hardware if available.

This library abstracts the vector processing capability so that code written for it will execute appropriate instructions for the processor available at runtime. For this reason, unless you are writing specialized code that targets a single CPU, you should generally use these functions rather than directly using vector instructions. By using these functions, your code does not have to deal with subtle differences in vector instruction availability between different microarchitectures, freeing you to focus on the problem you are trying to solve. Also, code written using vecLib tends to be easier to port to different CPU architectures (porting your core libraries to ARM on iOS, for example).

vecLibTypes.h

The vecLibTypes.h header file defines a set of vector data types (vFloat, vUInt32, etc.), which represent 128-bit vectors containing values of type float, UInt32, etc. The vBasicOps.h and vfp.h headers make use of these types.

The type names all begin with the letter “v,” followed by a mnemonic for the scalar data type used for elements of the vector. For example, vUInt32, vSInt16, vFloat, etc.

vBasicOps.h

vBasicOps.h declares a set of basic arithmetic and logical functions on 128-bit vectors, using the integer types from vecLibTypes.h.

The function names begin with “v,” followed by a mnemonic for the type of operation, e.g. “S” or “U” for signed or unsigned, then the width of the operation, then the name of the operation. For example, vS8Divide performs division of signed 8-bit values packed into 128-bit vectors.

vfp.h

vfp.h declares a set of floating-point arithmetic, transcendental and trigonometric functions, on 128-bit vectors, using the floating-point types from vecLibTypes.h.

These functions are named with their customary mathematical names, prefixed with the letter “v”, and all except vtablelookup() have the suffix “f” to indicate that they work with single-precision floating-point data. For example, vcosf is the single-precision cosine function.

vForce.h

vForce.h declares a set of trigonometric and transcendental functions in terms of C arrays (double * or float *), which can be of any desired length. Internally, the C arrays are converted piecewise into collections of 128-bit vectors, if appropriate for the current architecture.

The functions declared in vForce.h are named with the customary mathematical names, but with the prefix “vv.” Each mathematical function is available in two variants: one for single-precision floating-point data and one for double-precision data. The single-precision forms have the suffix “f”, while the double-precision forms have no suffix. For example, vvcosf is the single-precision cosine function, while vvcos is the double-precision variant.

All of the vForce.h functions follow a common format:

  • The return type is void.

  • The first parameter points to an array to hold the results. (The only exceptions are vvsincosf() and vvsincos(), which have two result arrays pointed to by the first two parameters.)

  • One or more parameters point to operand arrays, the same length as the result array.

  • The last parameter is the array length.

vectorOps.h

vectorOps.h declares a set of vector and matrix BLAS functions on arrays of 128-bit vectors containing single-precision floating-point values. The arrays can be of any desired length, but the number of float elements must be a multiple of 4.

vBigNum.h

vBigNum.h provides arithmetic and logical operations on large integers, which may be 128, 256, 512, or 1024 bits in length. It defines types for these values, and internally processes them as collections of 128-bit vectors.

vBigNum.h defines its own set of data types to represent large integer quantities, such as vS128 for a signed, 128-bit integer or vU1024 for an unsigned, 1024-bit integer. The function names begin with the data type name, followed by the name of the operation. For example, vS512Add performs addition of two 128-bit signed integers.

The functions perform logical and arithmetic operations on scalar values that may be 128, 256, 512, or 1024 bits in width. These values are implemented as structures of one, two, four, or eight 128-bit vectors, and the operations execute on the available vector-processing hardware if possible.

The functions have names that are compatible with those in vBasicOps.h.

Threading Model

In versions of vecLib prior to OS X v10.7, many vecLib routines used multiple POSIX threads to improve performance. In some cases, this could cause performance problems in applications that use large numbers of POSIX threads.

An application can limit the number of threads used by vecLib by setting the environment variable VECLIB_MAXIMUM_THREADS as follows:

  1. #include <stdlib.h>
  2. setenv("VECLIB_MAXIMUM_THREADS", "1", true);

To remove the limit, you can unset the environment variable as follows:

  1. unsetenv("VECLIB_MAXIMUM_THREADS");

In OS X v10.7, these routines use Grand Central Dispatch, so the use of this environment variable should no longer be necessary.

For More Information

For information about membership in Apple’s developer program, go to this URL:

http://developer.apple.com/membership/

For information about the Velocity Engine, go to this URL:

http://developer.apple.com/hardwaredrivers/ve/index.html

For general technical support from Apple, go to this URL:

https://developer.apple.com/support/

Functions

  • Computes the ceiling of values in a vector of floating-point values.

    Declaration

    Objective-C

    vFloat vceilf ( vFloat );

    Parameters

    A

    The input vector.

    Return Value

    Returns a vector of floating-point values, each of which is the ceiling of the corresponding value in A.

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, produces a value with the magnitude of arg2 and sign arg1. Note that the order of the arguments matches the recommendation of the IEEE 754 floating-point standard, which is opposite from the SANE copysign function.

    Declaration

    Objective-C

    vFloat vcopysignf ( vFloat , vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates A/B.

    Declaration

    Objective-C

    vFloat vdivf ( vFloat , vFloat );

    Availability

    Available in iOS 6.0 and later.

  • Computes the floor of values in a vector of floating-point values.

    Declaration

    Objective-C

    vFloat vfloorf ( vFloat );

    Parameters

    A

    The input vector.

    Return Value

    Returns a vector of floating-point values, each of which is the floor of the corresponding value in A.

    Availability

    Available in iOS 6.0 and later.

  • Rounds to the nearest integer (nearest even for ties).

    Declaration

    Objective-C

    vFloat vnintf ( vFloat );

    Parameters

    A

    The input vector.

    Return Value

    Returns a vector of floating-point values, each of which is the result of rounding the corresponding value in A to the nearest integer, with ties rounded to the nearest even integer.

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the next representable value after x in the direction of y. If x is equal to y, then y is returned.

    Declaration

    Objective-C

    vFloat vnextafterf ( vFloat , vFloat );

    Availability

    Available in iOS 6.0 and later.

  • Computes the reciprocal of values in a vector.

    Declaration

    Objective-C

    vFloat vrecf ( vFloat );

    Parameters

    A

    The source vector

    Return Value

    Returns a vector containing the reciprocals of the values in the source vector.

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the inverse of the square root of X.

    Declaration

    Objective-C

    vFloat vrsqrtf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the square root of X.

    Declaration

    Objective-C

    vFloat vsqrtf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element of Index_Vect, returns the corresponding value from Table.

    Declaration

    Objective-C

    vUInt32 vtablelookup ( vSInt32 , uint32_t * );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the exponential of X.

    Declaration

    Objective-C

    vFloat vexpf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates ExpM1(x) = Exp(x) - 1. But, for small enough arguments, ExpM1(x) is expected to be more accurate than Exp(x) - 1.

    Declaration

    Objective-C

    vFloat vexpm1f ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the natural logarithm of X.

    Declaration

    Objective-C

    vFloat vlogf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates Log1P = Log(1 + x). But, for small enough arguments, Log1P is expected to be more accurate than Log(1 + x).

    Declaration

    Objective-C

    vFloat vlog1pf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • Computes the base-10 logarithm of values in a vector.

    Declaration

    Objective-C

    vFloat vlog10f ( vFloat );

    Parameters

    X

    The source vector.

    Return Value

    Returns a vector of floating point values, each of which is the base-10 logarithm of the corresponding value in the source vector.

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, extracts the exponent of X, as a signed integral value. A subnormal argument is treated as though it were first normalized. Thus: 1 <= x * 2^(-logb(x)) < 2.

    Declaration

    Objective-C

    vFloat vlogbf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates x * 2^n efficiently. This is not normally done by computing 2^n explicitly.

    Declaration

    Objective-C

    vFloat vscalbf ( vFloat , vSInt32 );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the sine.

    Declaration

    Objective-C

    vFloat vsinf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the cosine.

    Declaration

    Objective-C

    vFloat vcosf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • Simultaneously computes sine and cosine of values in a vector.

    Declaration

    Objective-C

    vFloat vsincosf ( vFloat , vFloat * );

    Parameters

    arg

    The source vector.

    sine_result

    An output vector. On return, this contains the result of sin(x) for each value (x) in the source vector.

    Return Value

    Returns a vector that contains the result of cos(x) for each value (x) in the source vector.

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the tangent.

    Declaration

    Objective-C

    vFloat vtanf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the arcsine. Results are in the interval [-pi/2, pi/2].

    Declaration

    Objective-C

    vFloat vasinf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the arccosine. Results are in the interval [0, pi].

    Declaration

    Objective-C

    vFloat vacosf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the arctangent. Results are in the interval [-pi/2, pi/2].

    Declaration

    Objective-C

    vFloat vatanf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the arctangent of arg2/arg1 in the interval [-pi,pi] using the sign of both arguments to determine the quadrant of the computed value.

    Declaration

    Objective-C

    vFloat vatan2f ( vFloat , vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the hyperbolic sine of X.

    Declaration

    Objective-C

    vFloat vsinhf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the hyperbolic cosine of X.

    Declaration

    Objective-C

    vFloat vcoshf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the hyperbolic tangent of X.

    Declaration

    Objective-C

    vFloat vtanhf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the inverse hyperbolic sine of X.

    Declaration

    Objective-C

    vFloat vasinhf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the inverse hyperbolic cosine of X.

    Declaration

    Objective-C

    vFloat vacoshf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the inverse hyperbolic tangent of X.

    Declaration

    Objective-C

    vFloat vatanhf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates X to the integer power of Y.

    Declaration

    Objective-C

    vFloat vipowf ( vFloat , vSInt32 );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates X to the floating-point power of Y. The result is more accurate than using exp(log(X)*Y).

    Declaration

    Objective-C

    vFloat vpowf ( vFloat , vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates X modulo Y.

    Declaration

    Objective-C

    vFloat vfmodf ( vFloat , vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the remainder of X/Y, according to the IEEE 754 floating-point standard.

    Declaration

    Objective-C

    vFloat vremainderf ( vFloat , vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, calculates the remainder of X/Y, according to the SANE standard. It stores into QUO the 7 low-order bits of the integer quotient, such that -127 <= QUO <= 127.

    Declaration

    Objective-C

    vFloat vremquof ( vFloat , vFloat, vUInt32 * );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, returns the class of the argument (one of the FP_ ... constants defined in math.h).

    Declaration

    Objective-C

    vUInt32 vclassifyf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each vector element, returns a non-zero value if and only if the sign of arg is negative. This includes NaNs, infinities and zeros.

    Declaration

    Objective-C

    vUInt32 vsignbitf ( vFloat );

    Availability

    Available in iOS 6.0 and later.

  • For each double-precision array element, sets y to the exponential of x.

    Declaration

    Swift

    func vvexp(_ _: UnsafeMutablePointer<Double>, _ _: UnsafePointer<Double>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvexp ( double *, const double *, const int * );

    Availability

    Available in iOS 5.0 and later.

  • For each single-precision array element, sets y to the exponential of x.

    Declaration

    Swift

    func vvexpf(_ _: UnsafeMutablePointer<Float>, _ _: UnsafePointer<Float>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvexpf ( float *, const float *, const int * );

    Availability

    Available in iOS 5.0 and later.

  • Sets each element y[i] to 2 raised to (x[i]) for i in 0..n-1.

    Declaration

    Swift

    func vvexp2(_ _: UnsafeMutablePointer<Double>, _ _: UnsafePointer<Double>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvexp2 ( double *, const double *, const int * );

    Parameters

    y

    An input array of double-precision floating point numbers

    x

    An output array of double-precision floating point numbers.

    n

    A pointer to an integer containing the number of floating point elements to set.

    Availability

    Available in iOS 5.0 and later.

  • Sets each element y[i] to 2 raised to (x[i]) for i in 0..n-1.

    Declaration

    Swift

    func vvexp2f(_ _: UnsafeMutablePointer<Float>, _ _: UnsafePointer<Float>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvexp2f ( float *, const float *, const int * );

    Parameters

    y

    An input array of single-precision floating point values.

    x

    An output array of single-precision floating point values.

    n

    A pointer to an integer containing the number of floating point elements to set.

    Availability

    Available in iOS 5.0 and later.

  • Returns (e^x)-1 for each element y[i].

    Declaration

    Swift

    func vvexpm1(_ _: UnsafeMutablePointer<Double>, _ _: UnsafePointer<Double>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvexpm1 ( double *, const double *, const int * );

    Parameters

    y

    An input array of double-precision floating point numbers

    x

    An output array of double-precision floating point numbers.

    n

    A pointer to an integer containing the number of floating point elements to set.

    Discussion

    This is accurate for very small values of x[i].

    Availability

    Available in iOS 5.0 and later.

  • Computes (e^x)-1 for each element in an array.

    Declaration

    Swift

    func vvexpm1f(_ _: UnsafeMutablePointer<Float>, _ _: UnsafePointer<Float>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvexpm1f ( float *, const float *, const int * );

    Parameters

    result

    An array, modified on return, where the results are stored.

    src

    The input array.

    count

    The number of values in the input and result arrays.

    Discussion

    Provides additional accuracy for values of x near -1.

    Availability

    Available in iOS 5.0 and later.

  • For each double-precision array element, sets y to the natural logarithm of x.

    Declaration

    Swift

    func vvlog(_ _: UnsafeMutablePointer<Double>, _ _: UnsafePointer<Double>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvlog ( double *, const double *, const int * );

    Availability

    Available in iOS 5.0 and later.

  • For each single-precision array element, sets y to the natural logarithm of x.

    Declaration

    Swift

    func vvlogf(_ _: UnsafeMutablePointer<Float>, _ _: UnsafePointer<Float>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvlogf ( float *, const float *, const int * );

    Availability

    Available in iOS 5.0 and later.

  • Sets each element y[i] to log(1+x[i]) for i in 0..n-1.

    Declaration

    Swift

    func vvlog1p(_ _: UnsafeMutablePointer<Double>, _ _: UnsafePointer<Double>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvlog1p ( double *, const double *, const int * );

    Parameters

    y

    An output array of double-precision floating point numbers.

    x

    An input array of double-precision floating point numbers.

    n

    A pointer to an integer containing the number of floating point elements to set.

    Availability

    Available in iOS 5.0 and later.

  • Computes the natural logarithm of a value plus 1.

    Declaration

    Swift

    func vvlog1pf(_ _: UnsafeMutablePointer<Float>, _ _: UnsafePointer<Float>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvlog1pf ( float *, const float *, const int * );

    Parameters

    results

    An array where results are stored on return.

    values

    The source array.

    count

    A pointer to an integer containing the number of elements to process.

    Discussion

    This algorithm computes

    log(1+x)

    for each element x in the source array.

    This algorithm is tuned to provide increased accuracy for values of x near 1.

    Availability

    Available in iOS 5.0 and later.

  • Sets each element y[i] to the base-2 logarithm of x[i] for i in 0..n-1.

    Declaration

    Swift

    func vvlog2(_ _: UnsafeMutablePointer<Double>, _ _: UnsafePointer<Double>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvlog2 ( double *, const double *, const int * );

    Parameters

    y

    An output array of double-precision floating point numbers.

    x

    An input array of double-precision floating point numbers.

    n

    A pointer to an integer containing the number of floating point elements to set.

    Availability

    Available in iOS 5.0 and later.

  • Sets each element y[i] to the base 2 logarithm of x[i] for i in 0..n-1.

    Declaration

    Swift

    func vvlog2f(_ _: UnsafeMutablePointer<Float>, _ _: UnsafePointer<Float>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvlog2f ( float *, const float *, const int * );

    Parameters

    y

    An output array of single-precision floating point numbers.

    x

    An input array of single-precision floating point numbers.

    n

    A pointer to an integer containing the number of floating point elements to set.

    Availability

    Available in iOS 5.0 and later.

  • For each double-precision array element, sets y to the base 10 logarithm of x.

    Declaration

    Swift

    func vvlog10(_ _: UnsafeMutablePointer<Double>, _ _: UnsafePointer<Double>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvlog10 ( double *, const double *, const int * );

    Availability

    Available in iOS 5.0 and later.

  • For each single-precision array element, sets y to the base 10 logarithm of x.

    Declaration

    Swift

    func vvlog10f(_ _: UnsafeMutablePointer<Float>, _ _: UnsafePointer<Float>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvlog10f ( float *, const float *, const int * );

    Availability

    Available in iOS 5.0 and later.

  • Sets each element y[i] to the exponent of x[i] for i in 0..n-1.

    Declaration

    Swift

    func vvlogb(_ _: UnsafeMutablePointer<Double>, _ _: UnsafePointer<Double>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvlogb ( double *, const double *, const int * );

    Parameters

    y

    An output array of double-precision floating point numbers.

    x

    An input array of double-precision floating point numbers.

    n

    A pointer to an integer containing the number of floating point elements to set.

    Discussion

    Floating point numbers are internally represented as a mantissa multiplied times a base raised to an exponent. That exponent is returned in the output array.

    Availability

    Available in iOS 5.0 and later.

  • Sets each element y[i] to the exponent of x[i] for i in 0..n-1.

    Declaration

    Swift

    func vvlogbf(_ _: UnsafeMutablePointer<Float>, _ _: UnsafePointer<Float>, _ _: UnsafePointer<Int32>)

    Objective-C

    void vvlogbf ( float *, const float *, const int * );

    Parameters

    y

    An array where results are stored on return.

    x

    The source array.

    n

    A pointer to an integer containing the number of elements to process.

    Discussion

    Floating point numbers are internally represented as a mantissa multiplied times a base raised to an exponent. That exponent is returned in the output array.

    Availability

    Available in iOS 5.0 and later.

Data Types

All data structures passed to vecLib must be naturally aligned. For example, a 32-bit floating-point value must be aligned to a four-byte boundary. Many routines in vecLib generate exceptions on unaligned data.

  • A 128-bit vector packed with unsigned char values.

    Declaration

    Objective-C

    typedef vector unsigned char vUInt8;

    Import Statement

    Objective-C

    @import Accelerate;

    Availability

    Available in iOS 4.0 through iOS 8.4.

  • A 128-bit vector packed with signed char values.

    Declaration

    Objective-C

    typedef vector signed char vSInt8;

    Import Statement

    Objective-C

    @import Accelerate;

    Availability

    Available in iOS 4.0 through iOS 8.4.

  • A 128-bit vector packed with unsigned short values.

    Declaration

    Objective-C

    typedef vector unsigned short vUInt16;

    Import Statement

    Objective-C

    @import Accelerate;

    Availability

    Available in iOS 4.0 through iOS 8.4.

  • A 128-bit vector packed with signed short values.

    Declaration

    Objective-C

    typedef vector signed short vSInt16;

    Import Statement

    Objective-C

    @import Accelerate;

    Availability

    Available in iOS 4.0 through iOS 8.4.

  • A 128-bit vector packed with unsigned int values.

    Declaration

    Objective-C

    typedef vector unsigned int vUInt32;

    Import Statement

    Objective-C

    @import Accelerate;

    Availability

    Available in iOS 4.0 through iOS 8.4.

  • A 128-bit vector packed with signed int values.

    Declaration

    Objective-C

    typedef vector signed int vSInt32;

    Import Statement

    Objective-C

    @import Accelerate;

    Availability

    Available in iOS 4.0 through iOS 8.4.

  • A 128-bit vector packed with uint64_t values.

    Declaration

    Objective-C

    typedef vector unsigned int vUInt64;

    Import Statement

    Objective-C

    @import Accelerate;

    Availability

    Available in iOS 7.0 through iOS 7.1.

  • A 128-bit vector packed with int64_t values.

    Declaration

    Objective-C

    typedef vector signed int vSInt64;

    Import Statement

    Objective-C

    @import Accelerate;

    Availability

    Available in iOS 7.0 through iOS 7.1.

  • A 128-bit vector packed with float values.

    Declaration

    Objective-C

    typedef vector float vFloat;

    Import Statement

    Objective-C

    @import Accelerate;

    Availability

    Available in iOS 4.0 through iOS 8.4.

  • A 128-bit vector packed with double values.

    Declaration

    Objective-C

    typedef double vDouble __attribute__((__vector_size__ (16)));

    Import Statement

  • A 128-bit vector packed with bool int values.

    Declaration

    Objective-C

    typedef vector bool int vBool32;

    Import Statement

    Objective-C

    @import Accelerate;

    Availability

    Available in iOS 4.0 through iOS 8.4.

  • A single-precision complex number type.

    Declaration

    Objective-C

    typedef complex float __float_complex_t;

    Import Statement

  • A double-precision complex number type.

    Declaration

    Objective-C

    typedef complex double __double_complex_t;

    Import Statement