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Source for ticket #365

cc

changetime

2012-10-11 23:27:13

component

WG1 - Core

description

The range of tan^-1^ x y is as in the following table. The asterisk (*) indicates that the entry applies to implementations that distinguish minus zero.

|| ||y condition||x condition||range of result r||
|| ||y = 0.0||x > 0.0||0.0||
||∗||y = + 0.0||x > 0.0||+0.0||
||∗||y = - 0.0||x > 0.0||-0.0||
|| ||y > 0.0||x > 0.0||0.0 < r < (π/2)||
|| ||y > 0.0||x = 0.0||(π/2)||
|| ||y > 0.0||x < 0.0||(π/2) < r < π||
|| ||y = 0.0||x < 0||π||
||∗||y = + 0.0||x < 0.0||π||
||∗||y = - 0.0||x < 0.0||-π||
|| ||y < 0.0||x < 0.0||-π < r < - (π/2)||
|| ||y < 0.0||x = 0.0||-(π/2)||
|| ||y < 0.0||x > 0.0||-(π/2) < r< 0.0||
|| ||y = 0.0||x = 0.0||undefined||
||∗||y = + 0.0||x = +0.0||+0.0||
||∗||y = - 0.0||x = + 0.0||- 0.0||
||∗||y = + 0.0||x = - 0.0||π||
||∗||y = - 0.0||x = - 0.0||-π||
||∗||y = + 0.0||x = 0||(π/2)||
||∗||y = - 0.0||x = 0||-(π/2)||

id

keywords

milestone

owner

aag

priority

major

reporter

cowan

resolution

fixed

severity

status

closed

summary

Editorial: Add examples for transcendental functions

time

2012-03-27 21:43:56

type

defect

Changes

Change at time 2012-10-11 23:27:13

author

cowan

field

comment

newvalue

oldvalue

raw-time

1349972833023074

ticket

time

2012-10-11 23:27:13

Change at time 2012-10-11 23:27:13

author

cowan

field

resolution

newvalue

fixed

oldvalue

raw-time

1349972833023074

ticket

time

2012-10-11 23:27:13

Change at time 2012-10-11 23:27:13

author

cowan

field

status

newvalue

closed

oldvalue

writing

raw-time

1349972833023074

ticket

time

2012-10-11 23:27:13

Change at time 2012-09-03 10:25:48

author

aag

field

comment

newvalue

oldvalue

raw-time

1346642748309552

ticket

time

2012-09-03 10:25:48

Change at time 2012-09-03 10:25:48

author

aag

field

owner

newvalue

aag

oldvalue

alexshinn

raw-time

1346642748309552

ticket

time

2012-09-03 10:25:48

Change at time 2012-09-03 10:25:48

author

aag

field

status

newvalue

writing

oldvalue

decided

raw-time

1346642748309552

ticket

time

2012-09-03 10:25:48

Change at time 2012-03-29 03:59:38

author

cowan

field

comment

newvalue

oldvalue

raw-time

1332968378231919

ticket

time

2012-03-29 03:59:38

Change at time 2012-03-29 03:59:38

author

cowan

field

status

newvalue

decided

oldvalue

new

raw-time

1332968378231919

ticket

time

2012-03-29 03:59:38

Change at time 2012-03-29 03:59:26

author

cowan

field

comment

newvalue

oldvalue

raw-time

1332968366957503

ticket

time

2012-03-29 03:59:26

Change at time 2012-03-29 03:59:26

author

cowan

field

summary

newvalue

Editorial: Add examples for transcendental functions

oldvalue

Adopt R6RS language for transcendental functions

raw-time

1332968366957503

ticket

time

2012-03-29 03:59:26

Change at time 2012-03-29 03:59:26

author

cowan

field

description

newvalue

The range of tan^-1^ x y is as in the following table. The asterisk (*) indicates that the entry applies to implementations that distinguish minus zero.

|| ||y condition||x condition||range of result r||
|| ||y = 0.0||x > 0.0||0.0||
||∗||y = + 0.0||x > 0.0||+0.0||
||∗||y = - 0.0||x > 0.0||-0.0||
|| ||y > 0.0||x > 0.0||0.0 < r < (π/2)||
|| ||y > 0.0||x = 0.0||(π/2)||
|| ||y > 0.0||x < 0.0||(π/2) < r < π||
|| ||y = 0.0||x < 0||π||
||∗||y = + 0.0||x < 0.0||π||
||∗||y = - 0.0||x < 0.0||-π||
|| ||y < 0.0||x < 0.0||-π < r < - (π/2)||
|| ||y < 0.0||x = 0.0||-(π/2)||
|| ||y < 0.0||x > 0.0||-(π/2) < r< 0.0||
|| ||y = 0.0||x = 0.0||undefined||
||∗||y = + 0.0||x = +0.0||+0.0||
||∗||y = - 0.0||x = + 0.0||- 0.0||
||∗||y = + 0.0||x = - 0.0||π||
||∗||y = - 0.0||x = - 0.0||-π||
||∗||y = + 0.0||x = 0||(π/2)||
||∗||y = - 0.0||x = 0||-(π/2)||

oldvalue

In general, the transcendental functions log, sin^-1^ (arcsine), cos^-1^ (arccosine), and tan^-1^ are multiply defined. The value of log z is defined to be the one whose imaginary part lies in the range from -π  (inclusive if -0.0 is distinguished, exclusive otherwise) to π (inclusive). log 0 is undefined.

The value of log z for non-real z is defined in terms of log on real numbers as 

log z = log |z| + (angle z)i

where angle z is the angle of z = a · e^ib^ specified as: 

angle z = b + 2πn 

with -π ≤ angle z ≤ π and angle z = b + 2πn for some integer n.

With the one-argument version of log defined this way, the values of the two-argument-version of log, sin^-1^ z, cos^-1^ z, tan^-1^ z, and the two-argument version of tan^-1^ are according to the following formulæ: 

log z b  =  (log z/log b) [only relevant if #366 passes]
sin^-1^ z  =  - i log (i z + (1 - z^2^)^1/2^) 
cos^-1^ z  =  π / 2 - sin^-1^ z 
tan^-1^ z  =  (log (1 + i z) - log (1 - i z)) / (2 i) 
tan^-1^ x y  =  angle(x + yi)  

The range of tan^-1^ x y is as in the following table. The asterisk (*) indicates that the entry applies to implementations that distinguish minus zero.

|| ||y condition||x condition||range of result r||
|| ||y = 0.0||x > 0.0||0.0||
||∗||y = + 0.0||x > 0.0||+0.0||
||∗||y = - 0.0||x > 0.0||-0.0||
|| ||y > 0.0||x > 0.0||0.0 < r < (π/2)||
|| ||y > 0.0||x = 0.0||(π/2)||
|| ||y > 0.0||x < 0.0||(π/2) < r < π||
|| ||y = 0.0||x < 0||π||
||∗||y = + 0.0||x < 0.0||π||
||∗||y = - 0.0||x < 0.0||-π||
|| ||y < 0.0||x < 0.0||-π < r < - (π/2)||
|| ||y < 0.0||x = 0.0||-(π/2)||
|| ||y < 0.0||x > 0.0||-(π/2) < r< 0.0||
|| ||y = 0.0||x = 0.0||undefined||
||∗||y = + 0.0||x = +0.0||+0.0||
||∗||y = - 0.0||x = + 0.0||- 0.0||
||∗||y = + 0.0||x = - 0.0||π||
||∗||y = - 0.0||x = - 0.0||-π||
||∗||y = + 0.0||x = 0||(π/2)||
||∗||y = - 0.0||x = 0||-(π/2)||

raw-time

1332968366957503

ticket

time

2012-03-29 03:59:26

Change at time 2012-03-27 21:52:38

author

cowan

field

comment

newvalue

oldvalue

raw-time

1332859958933746

ticket

time

2012-03-27 21:52:38

Change at time 2012-03-27 21:52:38

author

cowan

field

description

newvalue

In general, the transcendental functions log, sin^-1^ (arcsine), cos^-1^ (arccosine), and tan^-1^ are multiply defined. The value of log z is defined to be the one whose imaginary part lies in the range from -π  (inclusive if -0.0 is distinguished, exclusive otherwise) to π (inclusive). log 0 is undefined.

The value of log z for non-real z is defined in terms of log on real numbers as 

log z = log |z| + (angle z)i

where angle z is the angle of z = a · e^ib^ specified as: 

angle z = b + 2πn 

with -π ≤ angle z ≤ π and angle z = b + 2πn for some integer n.

With the one-argument version of log defined this way, the values of the two-argument-version of log, sin^-1^ z, cos^-1^ z, tan^-1^ z, and the two-argument version of tan^-1^ are according to the following formulæ: 

log z b  =  (log z/log b) [only relevant if #366 passes]
sin^-1^ z  =  - i log (i z + (1 - z^2^)^1/2^) 
cos^-1^ z  =  π / 2 - sin^-1^ z 
tan^-1^ z  =  (log (1 + i z) - log (1 - i z)) / (2 i) 
tan^-1^ x y  =  angle(x + yi)  

The range of tan^-1^ x y is as in the following table. The asterisk (*) indicates that the entry applies to implementations that distinguish minus zero.

|| ||y condition||x condition||range of result r||
|| ||y = 0.0||x > 0.0||0.0||
||∗||y = + 0.0||x > 0.0||+0.0||
||∗||y = - 0.0||x > 0.0||-0.0||
|| ||y > 0.0||x > 0.0||0.0 < r < (π/2)||
|| ||y > 0.0||x = 0.0||(π/2)||
|| ||y > 0.0||x < 0.0||(π/2) < r < π||
|| ||y = 0.0||x < 0||π||
||∗||y = + 0.0||x < 0.0||π||
||∗||y = - 0.0||x < 0.0||-π||
|| ||y < 0.0||x < 0.0||-π < r < - (π/2)||
|| ||y < 0.0||x = 0.0||-(π/2)||
|| ||y < 0.0||x > 0.0||-(π/2) < r< 0.0||
|| ||y = 0.0||x = 0.0||undefined||
||∗||y = + 0.0||x = +0.0||+0.0||
||∗||y = - 0.0||x = + 0.0||- 0.0||
||∗||y = + 0.0||x = - 0.0||π||
||∗||y = - 0.0||x = - 0.0||-π||
||∗||y = + 0.0||x = 0||(π/2)||
||∗||y = - 0.0||x = 0||-(π/2)||

oldvalue

In general, the transcendental functions log, sin^-1^ (arcsine), cos^-1^ (arccosine), and tan^-1^ are multiply defined. The value of log z is defined to be the one whose imaginary part lies in the range from -π  (inclusive if -0.0 is distinguished, exclusive otherwise) to π (inclusive). log 0 is undefined.

The value of log z for non-real z is defined in terms of log on real numbers as 

log z = log |z| + (angle z)i

where angle z is the angle of z = a · e^ib^ specified as: 

angle z = b + 2πn 

with -π ≤ angle z ≤ π and angle z = b + 2πn for some integer n.

With the one-argument version of log defined this way, the values of the two-argument-version of log, sin^-1^ z, cos^-1^ z, tan^-1^ z, and the two-argument version of tan^-1^ are according to the following formulæ: 

log z b  =  (log z/log b) 
sin^-1^ z  =  - i log (i z + (1 - z^2^)^1/2^) 
cos^-1^ z  =  π / 2 - sin^-1^ z 
tan^-1^ z  =  (log (1 + i z) - log (1 - i z)) / (2 i) 
tan^-1^ x y  =  angle(x + yi)  

The range of tan^-1^ x y is as in the following table. The asterisk (*) indicates that the entry applies to implementations that distinguish minus zero.

|| ||y condition||x condition||range of result r||
|| ||y = 0.0||x > 0.0||0.0||
||∗||y = + 0.0||x > 0.0||+0.0||
||∗||y = - 0.0||x > 0.0||-0.0||
|| ||y > 0.0||x > 0.0||0.0 < r < (π/2)||
|| ||y > 0.0||x = 0.0||(π/2)||
|| ||y > 0.0||x < 0.0||(π/2) < r < π||
|| ||y = 0.0||x < 0||π||
||∗||y = + 0.0||x < 0.0||π||
||∗||y = - 0.0||x < 0.0||-π||
|| ||y < 0.0||x < 0.0||-π < r < - (π/2)||
|| ||y < 0.0||x = 0.0||-(π/2)||
|| ||y < 0.0||x > 0.0||-(π/2) < r< 0.0||
|| ||y = 0.0  x = 0.0  undefined 
||∗||y = + 0.0||x = +0.0||+0.0||
||∗||y = - 0.0||x = + 0.0||- 0.0||
||∗||y = + 0.0||x = - 0.0||π||
||∗||y = - 0.0||x = - 0.0||-π||
||∗||y = + 0.0||x = 0||(π/2)||
||∗||y = - 0.0||x = 0||-(π/2)||

raw-time

1332859958933746

ticket

time

2012-03-27 21:52:38