PostgreSQL

9.3. Mathematical Functions and Operators
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9.3. Mathematical Functions and Operators #

Mathematical operators are provided for many PostgreSQL types. For types without standard mathematical conventions (e.g., date/time types) we describe the actual behavior in subsequent sections.

Table 9.4 shows the mathematical operators that are available for the standard numeric types. Unless otherwise noted, operators shown as accepting `numeric_type are available for all the types `smallint, integer, bigint, numeric, real, and double precision. Operators shown as accepting `integral_type are available for the types `smallint, integer, and bigint. Except where noted, each form of an operator returns the same data type as its argument(s). Calls involving multiple argument data types, such as integer + numeric, are resolved by using the type appearing later in these lists.

Table 9.4. Mathematical Operators

Operator Description Example(s)

Operator Description Example(s)
`numeric_type `+ `numeric_type `→` numeric_type` Addition `2 + 3` → `5`
`+` `numeric_type `→` numeric_type` Unary plus (no operation) `+ 3.5` → `3.5`
`numeric_type `- `numeric_type `→` numeric_type` Subtraction `2 - 3` → `-1`
`-` `numeric_type `→` numeric_type` Negation `- (-4)` → `4`
`numeric_type `* `numeric_type `→` numeric_type` Multiplication `2 * 3` → `6`
`numeric_type `/ `numeric_type `→` numeric_type` Division (for integral types, division truncates the result towards zero) `5.0 / 2` → `2.5000000000000000` `5 / 2` → `2` `(-5) / 2` → `-2`
`numeric_type `% `numeric_type `→` numeric_type` Modulo (remainder); available for `smallint`, `integer`, `bigint`, and `numeric` `5 % 4` → `1`
`numeric` `^` `numeric` → `numeric` `double precision` `^` `double precision` → `double precision` Exponentiation `2 ^ 3` → `8` Unlike typical mathematical practice, multiple uses of `^` will associate left to right by default: `2 ^ 3 ^ 3` → `512` `2 ^ (3 ^ 3)` → `134217728`
`+
/` `+double precision` → `double precision` Square root `+
/ 25.0+` → `5`
`+

/` `+double precision` → `double precision` Cube root `+

/ 64.0+` → `4`
`@` `numeric_type `→` numeric_type` Absolute value `@ -5.0` → `5.0`
`integral_type `& `integral_type `→` integral_type` Bitwise AND `91 & 15` → `11`
`integral_type` `+
` _`+integral_type`_ →` integral_type` Bitwise OR `+32
3+` → `35`
`integral_type `# `integral_type `→` integral_type` Bitwise exclusive OR `17 # 5` → `20`
`~` `integral_type `→` integral_type` Bitwise NOT `~1` → `-2`
`integral_type `<< `integer` → `integral_type` Bitwise shift left `1 << 4` → `16`
`integral_type `>> `integer` → `integral_type` Bitwise shift right `8 >> 2` → `2`

`numeric_type `+ `numeric_type → numeric_type`

Addition

2 + 3 → 5

+ `numeric_type → numeric_type`

Unary plus (no operation)

+ 3.5 → 3.5

`numeric_type `- `numeric_type → numeric_type`

Subtraction

2 - 3 → -1

- `numeric_type → numeric_type`

Negation

- (-4) → 4

`numeric_type `* `numeric_type → numeric_type`

Multiplication

2 * 3 → 6

`numeric_type `/ `numeric_type → numeric_type`

Division (for integral types, division truncates the result towards zero)

5.0 / 2 → 2.5000000000000000

5 / 2 → 2

(-5) / 2 → -2

`numeric_type `% `numeric_type → numeric_type`

Modulo (remainder); available for smallint, integer, bigint, and numeric

5 % 4 → 1

numeric ^ numeric → numeric

double precision ^ double precision → double precision

Exponentiation

2 ^ 3 → 8

Unlike typical mathematical practice, multiple uses of ^ will associate left to right by default:

2 ^ 3 ^ 3 → 512

2 ^ (3 ^ 3) → 134217728

/` `+double precision` → double precision

Square root

/ 25.0+` → 5

/` `+double precision` → double precision

Cube root

/ 64.0+` → 4

@ `numeric_type → numeric_type`

Absolute value

@ -5.0 → 5.0

`integral_type `& `integral_type → integral_type`

Bitwise AND

91 & 15 → 11

`integral_type` `+

` _`+integral_type_ → integral_type`

Bitwise OR

`+32

3+` → 35

`integral_type `# `integral_type → integral_type`

Bitwise exclusive OR

17 # 5 → 20

~ `integral_type → integral_type`

Bitwise NOT

~1 → -2

`integral_type `<< integer → `integral_type`

Bitwise shift left

1 << 4 → 16

`integral_type `>> integer → `integral_type`

Bitwise shift right

8 >> 2 → 2

Table 9.5 shows the available mathematical functions. Many of these functions are provided in multiple forms with different argument types. Except where noted, any given form of a function returns the same data type as its argument(s); cross-type cases are resolved in the same way as explained above for operators. The functions working with double precision data are mostly implemented on top of the host system’s C library; accuracy and behavior in boundary cases can therefore vary depending on the host system.

Table 9.5. Mathematical Functions

Function Description Example(s)

Function Description Example(s)
`+abs+` ( _`+numeric_type+`_ ) → _`+numeric_type+`_ Absolute value `abs(-17.4)` → `17.4`
`+cbrt+` ( `+double precision+` ) → `+double precision+` Cube root `cbrt(64.0)` → `4`
`+ceil+` ( `+numeric+` ) → `+numeric+` `ceil` ( `double precision` ) → `double precision` Nearest integer greater than or equal to argument `ceil(42.2)` → `43` `ceil(-42.8)` → `-42`
`+ceiling+` ( `+numeric+` ) → `+numeric+` `ceiling` ( `double precision` ) → `double precision` Nearest integer greater than or equal to argument (same as `ceil`) `ceiling(95.3)` → `96`
`+degrees+` ( `+double precision+` ) → `+double precision+` Converts radians to degrees `degrees(0.5)` → `28.64788975654116`
`+div+` ( _`+y+`_ `+numeric+`, _`+x+`_ `+numeric+` ) → `+numeric+` Integer quotient of `y`/`x` (truncates towards zero) `div(9, 4)` → `2`
`+erf+` ( `+double precision+` ) → `+double precision+` Error function `erf(1.0)` → `0.8427007929497149`
`+erfc+` ( `+double precision+` ) → `+double precision+` Complementary error function (`1 - erf(x)`, without loss of precision for large inputs) `erfc(1.0)` → `0.15729920705028513`
`+exp+` ( `+numeric+` ) → `+numeric+` `exp` ( `double precision` ) → `double precision` Exponential (`e` raised to the given power) `exp(1.0)` → `2.7182818284590452`
[FUNCTION-FACTORIAL .indexterm]# `factorial` ( `bigint` ) → `numeric` Factorial `factorial(5)` → `120`
`+floor+` ( `+numeric+` ) → `+numeric+` `floor` ( `double precision` ) → `double precision` Nearest integer less than or equal to argument `floor(42.8)` → `42` `floor(-42.8)` → `-43`
`+gcd+` ( _`+numeric_type+`_, _`+numeric_type+`_ ) → _`+numeric_type+`_ Greatest common divisor (the largest positive number that divides both inputs with no remainder); returns `0` if both inputs are zero; available for `integer`, `bigint`, and `numeric` `gcd(1071, 462)` → `21`
`+lcm+` ( _`+numeric_type+`_, _`+numeric_type+`_ ) → _`+numeric_type+`_ Least common multiple (the smallest strictly positive number that is an integral multiple of both inputs); returns `0` if either input is zero; available for `integer`, `bigint`, and `numeric` `lcm(1071, 462)` → `23562`
`+ln+` ( `+numeric+` ) → `+numeric+` `ln` ( `double precision` ) → `double precision` Natural logarithm `ln(2.0)` → `0.6931471805599453`
`+log+` ( `+numeric+` ) → `+numeric+` `log` ( `double precision` ) → `double precision` Base 10 logarithm `log(100)` → `2`
`+log10+` ( `+numeric+` ) → `+numeric+` `log10` ( `double precision` ) → `double precision` Base 10 logarithm (same as `log`) `log10(1000)` → `3`
`log` ( `b `numeric, `x `numeric ) → `numeric` Logarithm of `x `to base` b` `log(2.0, 64.0)` → `6.0000000000000000`
`+min_scale+` ( `+numeric+` ) → `+integer+` Minimum scale (number of fractional decimal digits) needed to represent the supplied value precisely `min_scale(8.4100)` → `2`
`+mod+` ( _`+y+`_ _`+numeric_type+`_, _`+x+`_ _`+numeric_type+`_ ) → _`+numeric_type+`_ Remainder of `y`/`x; available for `smallint, `integer`, `bigint`, and `numeric` `mod(9, 4)` → `1`
`+pi+` ( ) → `+double precision+` Approximate value of π `pi()` → `3.141592653589793`
`+power+` ( _`+a+`_ `+numeric+`, _`+b+`_ `+numeric+` ) → `+numeric+` `power` ( `a `double precision, `b `double precision ) → `double precision` `a `raised to the power of` b` `power(9, 3)` → `729`
`+radians+` ( `+double precision+` ) → `+double precision+` Converts degrees to radians `radians(45.0)` → `0.7853981633974483`
`+round+` ( `+numeric+` ) → `+numeric+` `round` ( `double precision` ) → `double precision` Rounds to nearest integer. For `numeric`, ties are broken by rounding away from zero. For `double precision`, the tie-breaking behavior is platform dependent, but “[.quote]#round to nearest even”# is the most common rule. `round(42.4)` → `42`
`round` ( `v `numeric, `s `integer ) → `numeric` Rounds `v `to` s` decimal places. Ties are broken by rounding away from zero. `round(42.4382, 2)` → `42.44` `round(1234.56, -1)` → `1230`
`+scale+` ( `+numeric+` ) → `+integer+` Scale of the argument (the number of decimal digits in the fractional part) `scale(8.4100)` → `4`
`+sign+` ( `+numeric+` ) → `+numeric+` `sign` ( `double precision` ) → `double precision` Sign of the argument (-1, 0, or +1) `sign(-8.4)` → `-1`
`+sqrt+` ( `+numeric+` ) → `+numeric+` `sqrt` ( `double precision` ) → `double precision` Square root `sqrt(2)` → `1.4142135623730951`
`+trim_scale+` ( `+numeric+` ) → `+numeric+` Reduces the value’s scale (number of fractional decimal digits) by removing trailing zeroes `trim_scale(8.4100)` → `8.41`
`+trunc+` ( `+numeric+` ) → `+numeric+` `trunc` ( `double precision` ) → `double precision` Truncates to integer (towards zero) `trunc(42.8)` → `42` `trunc(-42.8)` → `-42`
`trunc` ( `v `numeric, `s `integer ) → `numeric` Truncates `v `to` s` decimal places `trunc(42.4382, 2)` → `42.43`
`+width_bucket+` ( _`+operand+`_ `+numeric+`, _`+low+`_ `+numeric+`, _`+high+`_ `+numeric+`, _`+count+`_ `+integer+` ) → `+integer+` `width_bucket` ( `operand `double precision, `low `double precision, `high `double precision, `count `integer ) → `integer` Returns the number of the bucket in which `operand `falls in a histogram having` count `equal-width buckets spanning the range` low `to` high. Returns `0 or `count+1` for an input outside that range. `width_bucket(5.35, 0.024, 10.06, 5)` → `3`
`width_bucket` ( `operand `anycompatible, `thresholds `anycompatiblearray ) → `integer` Returns the number of the bucket in which `operand falls given an array listing the lower bounds of the buckets. Returns `0 for an input less than the first lower bound. `operand `and the array elements can be of any type having standard comparison operators. The` thresholds` array must be sorted, smallest first, or unexpected results will be obtained. `width_bucket(now(), array['yesterday', 'today', 'tomorrow']::timestamptz[])` → `2`

`+abs+` ( _`+numeric_type+`_ ) → _`+numeric_type+`_

Absolute value

abs(-17.4) → 17.4

`+cbrt+` ( `+double precision+` ) → `+double precision+`

Cube root

cbrt(64.0) → 4

`+ceil+` ( `+numeric+` ) → `+numeric+`

ceil ( double precision ) → double precision

Nearest integer greater than or equal to argument

ceil(42.2) → 43

ceil(-42.8) → -42

`+ceiling+` ( `+numeric+` ) → `+numeric+`

ceiling ( double precision ) → double precision

Nearest integer greater than or equal to argument (same as ceil)

ceiling(95.3) → 96

`+degrees+` ( `+double precision+` ) → `+double precision+`

Converts radians to degrees

degrees(0.5) → 28.64788975654116

`+div+` ( _`+y+`_ `+numeric+`, _`+x+`_ `+numeric+` ) → `+numeric+`

Integer quotient of `y/x` (truncates towards zero)

div(9, 4) → 2

`+erf+` ( `+double precision+` ) → `+double precision+`

Error function

erf(1.0) → 0.8427007929497149

`+erfc+` ( `+double precision+` ) → `+double precision+`

Complementary error function (1 - erf(x), without loss of precision for large inputs)

erfc(1.0) → 0.15729920705028513

`+exp+` ( `+numeric+` ) → `+numeric+`

exp ( double precision ) → double precision

Exponential (e raised to the given power)

exp(1.0) → 2.7182818284590452

[FUNCTION-FACTORIAL .indexterm]# factorial ( bigint ) → numeric

Factorial

factorial(5) → 120

`+floor+` ( `+numeric+` ) → `+numeric+`

floor ( double precision ) → double precision

Nearest integer less than or equal to argument

floor(42.8) → 42

floor(-42.8) → -43

`+gcd+` ( _`+numeric_type+`_, _`+numeric_type+`_ ) → _`+numeric_type+`_

Greatest common divisor (the largest positive number that divides both inputs with no remainder); returns 0 if both inputs are zero; available for integer, bigint, and numeric

gcd(1071, 462) → 21

`+lcm+` ( _`+numeric_type+`_, _`+numeric_type+`_ ) → _`+numeric_type+`_

Least common multiple (the smallest strictly positive number that is an integral multiple of both inputs); returns 0 if either input is zero; available for integer, bigint, and numeric

lcm(1071, 462) → 23562

`+ln+` ( `+numeric+` ) → `+numeric+`

ln ( double precision ) → double precision

Natural logarithm

ln(2.0) → 0.6931471805599453

`+log+` ( `+numeric+` ) → `+numeric+`

log ( double precision ) → double precision

Base 10 logarithm

log(100) → 2

`+log10+` ( `+numeric+` ) → `+numeric+`

log10 ( double precision ) → double precision

Base 10 logarithm (same as log)

log10(1000) → 3

log ( `b `numeric, `x `numeric ) → numeric

Logarithm of `x to base b`

log(2.0, 64.0) → 6.0000000000000000

`+min_scale+` ( `+numeric+` ) → `+integer+`

Minimum scale (number of fractional decimal digits) needed to represent the supplied value precisely

min_scale(8.4100) → 2

`+mod+` ( _`+y+`_ _`+numeric_type+`_, _`+x+`_ _`+numeric_type+`_ ) → _`+numeric_type+`_

Remainder of `y/x; available for `smallint, integer, bigint, and numeric

mod(9, 4) → 1

`+pi+` ( ) → `+double precision+`

Approximate value of π

pi() → 3.141592653589793

`+power+` ( _`+a+`_ `+numeric+`, _`+b+`_ `+numeric+` ) → `+numeric+`

power ( `a `double precision, `b `double precision ) → double precision

`a raised to the power of b`

power(9, 3) → 729

`+radians+` ( `+double precision+` ) → `+double precision+`

Converts degrees to radians

radians(45.0) → 0.7853981633974483

`+round+` ( `+numeric+` ) → `+numeric+`

round ( double precision ) → double precision

Rounds to nearest integer. For numeric, ties are broken by rounding away from zero. For double precision, the tie-breaking behavior is platform dependent, but “[.quote]#round to nearest even”# is the most common rule.

round(42.4) → 42

round ( `v `numeric, `s `integer ) → numeric

Rounds `v to s` decimal places. Ties are broken by rounding away from zero.

round(42.4382, 2) → 42.44

round(1234.56, -1) → 1230

`+scale+` ( `+numeric+` ) → `+integer+`

Scale of the argument (the number of decimal digits in the fractional part)

scale(8.4100) → 4

`+sign+` ( `+numeric+` ) → `+numeric+`

sign ( double precision ) → double precision

Sign of the argument (-1, 0, or +1)

sign(-8.4) → -1

`+sqrt+` ( `+numeric+` ) → `+numeric+`

sqrt ( double precision ) → double precision

Square root

sqrt(2) → 1.4142135623730951

`+trim_scale+` ( `+numeric+` ) → `+numeric+`

Reduces the value’s scale (number of fractional decimal digits) by removing trailing zeroes

trim_scale(8.4100) → 8.41

`+trunc+` ( `+numeric+` ) → `+numeric+`

trunc ( double precision ) → double precision

Truncates to integer (towards zero)

trunc(42.8) → 42

trunc(-42.8) → -42

trunc ( `v `numeric, `s `integer ) → numeric

Truncates `v to s` decimal places

trunc(42.4382, 2) → 42.43

`+width_bucket+` ( _`+operand+`_ `+numeric+`, _`+low+`_ `+numeric+`, _`+high+`_ `+numeric+`, _`+count+`_ `+integer+` ) → `+integer+`

width_bucket ( `operand `double precision, `low `double precision, `high `double precision, `count `integer ) → integer

Returns the number of the bucket in which `operand falls in a histogram having count equal-width buckets spanning the range low to high. Returns `0 or `count+1` for an input outside that range.

width_bucket(5.35, 0.024, 10.06, 5) → 3

width_bucket ( `operand `anycompatible, `thresholds `anycompatiblearray ) → integer

Returns the number of the bucket in which `operand falls given an array listing the lower bounds of the buckets. Returns `0 for an input less than the first lower bound. `operand and the array elements can be of any type having standard comparison operators. The thresholds` array must be sorted, smallest first, or unexpected results will be obtained.

width_bucket(now(), array['yesterday', 'today', 'tomorrow']::timestamptz[]) → 2

Table 9.6 shows functions for generating random numbers.

Table 9.6. Random Functions

Function Description Example(s)

Function Description Example(s)
`+random+` ( ) → `+double precision+` Returns a random value in the range 0.0 ⇐ x < 1.0 `random()` → `0.897124072839091`
`+random_normal+` ( [ [.optional]#_`+mean+`_ `+double precision+` [[.optional]#, _`+stddev+`_ `+double precision+`# ]#] ) → `+double precision+` Returns a random value from the normal distribution with the given parameters; `mean `defaults to 0.0 and` stddev` defaults to 1.0 `random_normal(0.0, 1.0)` → `0.051285419`
`+setseed+` ( `+double precision+` ) → `+void+` Sets the seed for subsequent `random()` and `random_normal()` calls; argument must be between -1.0 and 1.0, inclusive `setseed(0.12345)`

`+random+` ( ) → `+double precision+`

Returns a random value in the range 0.0 ⇐ x < 1.0

random() → 0.897124072839091

`+random_normal+` ( [ [.optional]#_`+mean+`_ `+double precision+` [[.optional]#, _`+stddev+`_ `+double precision+`# ]#] ) → `+double precision+`

Returns a random value from the normal distribution with the given parameters; `mean defaults to 0.0 and stddev` defaults to 1.0

random_normal(0.0, 1.0) → 0.051285419

`+setseed+` ( `+double precision+` ) → `+void+`

Sets the seed for subsequent random() and random_normal() calls; argument must be between -1.0 and 1.0, inclusive

setseed(0.12345)

The random() function uses a deterministic pseudo-random number generator. It is fast but not suitable for cryptographic applications; see the pgcrypto module for a more secure alternative. If setseed() is called, the series of results of subsequent random() calls in the current session can be repeated by re-issuing setseed() with the same argument. Without any prior setseed() call in the same session, the first random() call obtains a seed from a platform-dependent source of random bits. These remarks hold equally for random_normal().

Table 9.7 shows the available trigonometric functions. Each of these functions comes in two variants, one that measures angles in radians and one that measures angles in degrees.

Table 9.7. Trigonometric Functions

Function Description Example(s)

Function Description Example(s)
`+acos+` ( `+double precision+` ) → `+double precision+` Inverse cosine, result in radians `acos(1)` → `0`
`+acosd+` ( `+double precision+` ) → `+double precision+` Inverse cosine, result in degrees `acosd(0.5)` → `60`
`+asin+` ( `+double precision+` ) → `+double precision+` Inverse sine, result in radians `asin(1)` → `1.5707963267948966`
`+asind+` ( `+double precision+` ) → `+double precision+` Inverse sine, result in degrees `asind(0.5)` → `30`
`+atan+` ( `+double precision+` ) → `+double precision+` Inverse tangent, result in radians `atan(1)` → `0.7853981633974483`
`+atand+` ( `+double precision+` ) → `+double precision+` Inverse tangent, result in degrees `atand(1)` → `45`
`+atan2+` ( _`+y+`_ `+double precision+`, _`+x+`_ `+double precision+` ) → `+double precision+` Inverse tangent of `y`/`x`, result in radians `atan2(1, 0)` → `1.5707963267948966`
`+atan2d+` ( _`+y+`_ `+double precision+`, _`+x+`_ `+double precision+` ) → `+double precision+` Inverse tangent of `y`/`x`, result in degrees `atan2d(1, 0)` → `90`
`+cos+` ( `+double precision+` ) → `+double precision+` Cosine, argument in radians `cos(0)` → `1`
`+cosd+` ( `+double precision+` ) → `+double precision+` Cosine, argument in degrees `cosd(60)` → `0.5`
`+cot+` ( `+double precision+` ) → `+double precision+` Cotangent, argument in radians `cot(0.5)` → `1.830487721712452`
`+cotd+` ( `+double precision+` ) → `+double precision+` Cotangent, argument in degrees `cotd(45)` → `1`
`+sin+` ( `+double precision+` ) → `+double precision+` Sine, argument in radians `sin(1)` → `0.8414709848078965`
`+sind+` ( `+double precision+` ) → `+double precision+` Sine, argument in degrees `sind(30)` → `0.5`
`+tan+` ( `+double precision+` ) → `+double precision+` Tangent, argument in radians `tan(1)` → `1.5574077246549023`
`+tand+` ( `+double precision+` ) → `+double precision+` Tangent, argument in degrees `tand(45)` → `1`

`+acos+` ( `+double precision+` ) → `+double precision+`

Inverse cosine, result in radians

acos(1) → 0

`+acosd+` ( `+double precision+` ) → `+double precision+`

Inverse cosine, result in degrees

acosd(0.5) → 60

`+asin+` ( `+double precision+` ) → `+double precision+`

Inverse sine, result in radians

asin(1) → 1.5707963267948966

`+asind+` ( `+double precision+` ) → `+double precision+`

Inverse sine, result in degrees

asind(0.5) → 30

`+atan+` ( `+double precision+` ) → `+double precision+`

Inverse tangent, result in radians

atan(1) → 0.7853981633974483

`+atand+` ( `+double precision+` ) → `+double precision+`

Inverse tangent, result in degrees

atand(1) → 45

`+atan2+` ( _`+y+`_ `+double precision+`, _`+x+`_ `+double precision+` ) → `+double precision+`

Inverse tangent of `y/x`, result in radians

atan2(1, 0) → 1.5707963267948966

`+atan2d+` ( _`+y+`_ `+double precision+`, _`+x+`_ `+double precision+` ) → `+double precision+`

Inverse tangent of `y/x`, result in degrees

atan2d(1, 0) → 90

`+cos+` ( `+double precision+` ) → `+double precision+`

Cosine, argument in radians

cos(0) → 1

`+cosd+` ( `+double precision+` ) → `+double precision+`

Cosine, argument in degrees

cosd(60) → 0.5

`+cot+` ( `+double precision+` ) → `+double precision+`

Cotangent, argument in radians

cot(0.5) → 1.830487721712452

`+cotd+` ( `+double precision+` ) → `+double precision+`

Cotangent, argument in degrees

cotd(45) → 1

`+sin+` ( `+double precision+` ) → `+double precision+`

Sine, argument in radians

sin(1) → 0.8414709848078965

`+sind+` ( `+double precision+` ) → `+double precision+`

Sine, argument in degrees

sind(30) → 0.5

`+tan+` ( `+double precision+` ) → `+double precision+`

Tangent, argument in radians

tan(1) → 1.5574077246549023

`+tand+` ( `+double precision+` ) → `+double precision+`

Tangent, argument in degrees

tand(45) → 1

Note

Another way to work with angles measured in degrees is to use the unit transformation functions radians() and degrees() shown earlier. However, using the degree-based trigonometric functions is preferred, as that way avoids round-off error for special cases such as sind(30).

Table 9.8 shows the available hyperbolic functions.

Table 9.8. Hyperbolic Functions

Function Description Example(s)

Function Description Example(s)
`+sinh+` ( `+double precision+` ) → `+double precision+` Hyperbolic sine `sinh(1)` → `1.1752011936438014`
`+cosh+` ( `+double precision+` ) → `+double precision+` Hyperbolic cosine `cosh(0)` → `1`
`+tanh+` ( `+double precision+` ) → `+double precision+` Hyperbolic tangent `tanh(1)` → `0.7615941559557649`
`+asinh+` ( `+double precision+` ) → `+double precision+` Inverse hyperbolic sine `asinh(1)` → `0.881373587019543`
`+acosh+` ( `+double precision+` ) → `+double precision+` Inverse hyperbolic cosine `acosh(1)` → `0`
`+atanh+` ( `+double precision+` ) → `+double precision+` Inverse hyperbolic tangent `atanh(0.5)` → `0.5493061443340548`

`+sinh+` ( `+double precision+` ) → `+double precision+`

Hyperbolic sine

sinh(1) → 1.1752011936438014

`+cosh+` ( `+double precision+` ) → `+double precision+`

Hyperbolic cosine

cosh(0) → 1

`+tanh+` ( `+double precision+` ) → `+double precision+`

Hyperbolic tangent

tanh(1) → 0.7615941559557649

`+asinh+` ( `+double precision+` ) → `+double precision+`

Inverse hyperbolic sine

asinh(1) → 0.881373587019543

`+acosh+` ( `+double precision+` ) → `+double precision+`

Inverse hyperbolic cosine

acosh(1) → 0

`+atanh+` ( `+double precision+` ) → `+double precision+`

Inverse hyperbolic tangent

atanh(0.5) → 0.5493061443340548

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