Inverse trig functions and atan2

packages/math/math.grnd

@@ -158,6 +158,140 @@ fun -double !tan -double &in
     return $store
 endfun
 
+fun -double !asin -double &in
+    pusharg $in
+    call !abs &store
+    greater $store 1 &store
+    if $store %error
+
+    # asin(a) is the root of sin(x)-a=0, -pi/2 <= x <= pi/2
+    # Using Newton's Method: x_k+1 = x_k - (sin(x_k)-a)/cos(x_k)
+
+    stod "0" &ans
+    @loop
+        pusharg $ans
+        call !sin &sin
+        subtract $sin $in &sin
+        pusharg $ans
+        call !cos &cos
+        divide $sin $cos &store
+        subtract $ans $store &ans
+        equal $store 0 &store
+        if $store %return
+        jump %loop
+
+    @return
+    return $ans
+
+    @error
+    error "Cannot find sin of a double with magnetude greater than 1" "mathError"
+endfun
+
+fun -double !acos -double &in
+    # asin(x) + acos(x) = pi/2
+    pusharg $in
+    call !asin &store
+    call !pi &pi
+    divide $pi 2 &pi
+    subtract $pi $store &store
+    return $store
+endfun
+
+fun -double !atan -double &in
+    # atan(a) is the root of tan(x)-a=0, -pi/2 <= x <= pi/2
+    # Using Newton's Method: x_k+1 = x_k - (tan(x_k)-a)/sec^2(x_k) = x_k - (tan(x_k)-a)cos^2(x_k)
+
+    stod "0" &ans
+    @loop
+
+        pusharg $ans
+        call !tan &num
+        subtract $num $in &num
+        
+        pusharg $ans
+        call !cos &dom
+        multiply $dom $dom &dom
+
+        multiply $num $dom &store
+        subtract $ans $store &ans
+
+        pusharg $store
+        call !abs &store
+        lesser $store 0.000000000001 &store
+        if $store %return
+
+        jump %loop
+
+    @return
+    # Make answer in range [-pi/2, pi/2]
+    call !pi &pi
+    pusharg $ans
+    pusharg $pi
+    call !mod &ans
+    divide $pi 2 &pi
+    greater $ans $pi &store
+    if $store %negans
+    return $ans
+    
+    @negans
+    call !pi &pi
+    subtract $ans $pi &ans
+    return $ans
+endfun
+
+fun -double !atan2 -double &y -double &x
+    call !pi &pi
+    divide $pi 2 &pi/2
+    
+    # catch "mathError" &divideByZero
+    set &divideByZero true
+    divide $y $x &ratio
+    if $divideByZero %continue
+    jump %error
+
+    @continue
+    # When x > 0, return atan(y/x). When x < 0, return pi+atan(y/x) mod 2pi
+    greater $x 0 &store
+    if $store %atan
+
+    # When x < 0, check sign of y
+    lesser $y 0 &store
+    if $store %quadrant3
+
+    pusharg $ratio
+    call !atan &store
+    add $store $pi &store
+    return $store
+
+    @quadrant3
+    pusharg $ratio
+    call !atan &store
+    subtract $store $pi &store
+    return $store
+
+    @atan
+    pusharg $ratio
+    call !atan &store
+    return $store
+
+    @error
+    greater $y 0 &store
+    if $store %90deg
+
+    lesser $y 0 &store
+    if $store %neg90deg
+
+    error "Cannot find atan of (0,0)" "mathError"
+    divide $pi 2 &pi/2
+
+    @90deg
+    return $pi/2
+
+    @neg90deg
+    multiply $pi/2 -1 &store
+    return $store
+endfun
+
 fun -double !random -int &seed
     set &m 2147483648
     set &a 1103515245

packages/math/readme.md

@@ -18,6 +18,22 @@ Gets the cos of input.
 
 Gets the tan of input.
 
+### fun -double !asin -double &input
+
+Finds the angle *x* in the range [-pi/2, pi/2] such that sin(x) equals the input.
+
+### fun -double !acos -double &input
+
+Finds the angle *x* in the range [0, pi] such that cos(x) equals the input.
+
+### fun -double !atan -double &input
+
+Finds the angle *x* in the range [-pi/2, pi/2] such that tan(x) equals the input.
+
+### fun -double !atan2 -double &y -double &x
+
+Finds the angle in the range (-pi, pi] between the positive x axis and the point (x,y). This is equivalent to atan(y/x) for positive values of x.
+
 ### fun -double !intexp -double &base -int &power
 
 Returns base^power, for positive integer values of power. Returns 1 if power is non-positive (and thus is inaccurate for negative values of power).