# [git] CMU Common Lisp branch master updated. snapshot-2013-12-a-35-g4e3589d

Raymond Toy rtoy at common-lisp.net
Mon Dec 23 06:41:14 UTC 2013

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- Log -----------------------------------------------------------------
commit 4e3589d9d175382fe85b6efe2e3e94ccf598a88e
Author: Raymond Toy <toy.raymond at gmail.com>
Date:   Sun Dec 22 22:41:03 2013 -0800

More documentation.

o Derive the value of asin(2), lest we forget again why it is the way
it is.
o Use the formulas from the CLHS in the docstrings.

diff --git a/src/code/irrat.lisp b/src/code/irrat.lisp
index 078e56f..ff23a5e 100644
--- a/src/code/irrat.lisp
+++ b/src/code/irrat.lisp
@@ -1916,8 +1916,43 @@ Z may be any number, but the result is always a complex."
(realpart sqrt-z+1))))))))

+;; What is the value of asin(2)?  Here is a derivation.
+;;
+;; asin(2) = -i*log(i*2+sqrt(-3))
+;;         = -i*log(2*i+sqrt(3)*i)
+;;         = -i*(log(2+sqrt(3)) + i*pi/2)
+;;         = pi/2 - i*log(2+sqrt(3))
+;;
+;; Note that this differs from asin(2+0.0*i) because we support signed
+;; zeroes.
+;;
+;; asin(2+0*i) = -i*log(i*(2+0*i) + sqrt(1-(4+0*i)))
+;;             = -i*log((2*i - 0) + sqrt(-3-0*i))
+;;             = -i*log(-0 + 2*i - sqrt(3)*i)
+;;             = -i*log(-0 + i*(2-sqrt(3)))
+;;             = -i*(log(2-sqrt(3)) + i*pi/2)
+;;             = pi/2 - i*log(2-sqrt(3))
+;;             = pi/2 + i*log(2+sqrt(3))
+;;
+;; The last equation follows because (2-sqrt(3)) = 1/(2+sqrt(3)).
+;; Hence asin(2) /= asin(2+0*i).
+;;
+;; Also
+;;
+;; asin(2-0*i) = -i*log(i*(2-0*i) + sqrt(1-(4-0*i)))
+;;             = -i*log((2*i + 0) + sqrt(-3+0*i))
+;;             = -i*log(0 + 2*i + sqrt(3)*i)
+;;             = -i*log(0 + i*(2+sqrt(3)))
+;;             = -i*(log(2+sqrt(3)) + i*pi/2)
+;;             = pi/2 + i*log(2+sqrt(3))
+;;
+;; Hence asin(2) = asin(2-0.0*i).
+;;
+;; Similar derivations will show that asin(-2) = asin(-2 + 0.0*i) and
+;; asin(-2+0.0*i) is different from asin(-2-0.0*i) because of the
+;; branch cut, of course.
(defun complex-asin (z)
-  "Compute asin z = asinh(i*z)/i
+  "Compute asin z = -i*log(i*z + sqrt(1-z^2))

Z may be any number, but the result is always a complex."
(declare (number z))
@@ -1950,7 +1985,7 @@ Z may be any number, but the result is always a complex."
(- (realpart result)))))

(defun complex-atan (z)
-  "Compute atan z = atanh (i*z) / i
+  "Compute atan z = (log(1+i*z) - log(1-i*z))/(2*i)

Z may be any number, but the result is always a complex."
(declare (number z))

commit 1ba45c8177e4a6ef66f7fac3a3870f477a3d03eb
Author: Raymond Toy <toy.raymond at gmail.com>
Date:   Sun Dec 22 22:39:37 2013 -0800

Use the formulas from the CLHS in the docstrings.

diff --git a/src/code/irrat-dd.lisp b/src/code/irrat-dd.lisp
index 2eabb58..024ec93 100644
--- a/src/code/irrat-dd.lisp
+++ b/src/code/irrat-dd.lisp
@@ -1854,7 +1854,7 @@ Z may be any number, but the result is always a complex."

(defun dd-complex-asin (z)
-  _N"Compute asin z = asinh(i*z)/i
+  _N"Compute asin z = -i*log(i*z + sqrt(1-z^2))

Z may be any number, but the result is always a complex."
(declare (number z))
@@ -1894,7 +1894,7 @@ Z may be any number, but the result is always a complex."
(- (realpart result)))))

(defun dd-complex-atan (z)
-  _N"Compute atan z = atanh (i*z) / i
+  _N"Compute atan z = (log(1+i*z) - log(1-i*z))/(2*i)

Z may be any number, but the result is always a complex."
(declare (number z))

-----------------------------------------------------------------------

Summary of changes:
src/code/irrat-dd.lisp |    4 ++--
src/code/irrat.lisp    |   39 +++++++++++++++++++++++++++++++++++++--
2 files changed, 39 insertions(+), 4 deletions(-)