0.1.8 FastTrig

Author: RobTillaart

libraries/FastTrig/FastTrig.h

@@ -2,7 +2,7 @@
 //
 //    FILE: FastTrig.h
 //  AUTHOR: Rob Tillaart
-// VERSION: 0.1.7
+// VERSION: 0.1.8
 // PURPOSE: Arduino library for a faster approximation of sin() and cos()
 //    DATE: 2011-08-18
 //     URL: https://github.com/RobTillaart/FastTrig
@@ -19,8 +19,10 @@
 //  0.1.3   2020-09-07  initial release.
 //  0.1.4   2020-09-08  rewrite itan() + cleanup + examples
 //  0.1.5   2020-09-11  fixed optimize, new table, added iasin() and iacos()
-//  0.1.6   2020-12-23  arduino-CI + unit tests
+//  0.1.6   2020-12-23  Arduino-CI + unit tests
 //  0.1.7   2021-04-23  fix for PlatformIO
+//  0.1.8   2021-08-10  made % 180 conditional in itan() => performance gain
+//                      added icot() cotangent.
 
 
 #include "Arduino.h"
@@ -137,13 +139,14 @@ float itan(float f)
   // so no divide by 65535
 
   // FOLDING
-  bool neg = (f < 0);
   bool mir = false;
+  bool neg = (f < 0);
   if (neg) f = -f;
 
   long x = f;
   float rem = f - x;
-  float v = x % 180 + rem;  // normalised value 0..179.9999
+  if (x >= 180) x %= 180;
+  float v = rem + x;  // normalised value 0..179.9999
   if (v > 90)
   {
     v = 180 - v;
@@ -162,7 +165,7 @@ float itan(float f)
     if (mir) co = isinTable16[p - 1] + rem * delta;
     else     co = isinTable16[p]     - rem * delta;
   }
-  if (co == 0) return 0;
+  else if (co == 0) return 0;
 
   float si = isinTable16[d];
   if (rem != 0) si += rem * (isinTable16[d + 1]  - isinTable16[d]);
@@ -172,6 +175,21 @@ float itan(float f)
   return ta;
 }
 
+// some problem at 0 but at least we have a icot(x) cotangent.
+float icot(float f)
+{
+  float t = itan(f);
+  if (t == 0) return NAN;
+  return 1.0 / t;
+}
+
+
+// missing function...
+// float cot(float f)
+// {
+  // return 1.0/tan(f);
+// }
+
 
 ///////////////////////////////////////////////////////
 //

libraries/FastTrig/README.md

@@ -8,6 +8,7 @@
 
 Arduino library with interpolated lookup for sin() and cos(). Trades speed for accuracy.
 
+
 ## Description
 
 **Warning: The library trades speed for accuracy so use at own risk**
@@ -22,6 +23,7 @@ due to the folding.
 
 The **i** in the names stands for **int** and **interpolated** as the core is using integer math and lookuptable of 91 uint16_t = 182 bytes. By folding and mirroring the whole 360 degrees and beyond can be handled. When **isin(x)** is called and ```x == int(x)``` then the library will not interpolate and this will improve performance. When x is not a whole number the library will linear interpolate between **isin(int(x))** and **isin(int(x+1))**. Of course this introduces an error but it is quite fast (which was the goal).
 
+
 #### Lookup tables
 
 The lookup tables are optimized (sketch provided) to minimize the error when using the interpolation, this implies that the points in the table might not be optimal when you use only wholde degrees. A sketch that generates lookup tables is in the examples folder. This generator sketch can also generate tables with different resolution e.g. 24, 14, 12 or even 6, 5 or 4 bit lookup tables. So depending on the application these tables can be ideal, but verify they meet your requirements.
@@ -37,18 +39,20 @@ Although the tables can be written to, it is advised not to do so.
 
 *OK, the optimize example does a write to improve the table to minimize errors*
 
+
 ## Performance isin icos itan
 
 time in us - calls 0 - 360 step 1 degree and calls 720 - 1080 (lib version 0.1.5)
+(clock speeds in MHz)
 
-| function | UNO 16MHz | ESP32 240 MHz | UNO  (720-1080) | ESP (720-1080) |
-|:----:|:----:|:----:|:----:|:----:|
-|  sin |  120.43 | 10.90 | 124.19 | 10.91 |
-| isin |   44.24 |  1.09 |  85.00 |  1.11 |
-|  cos |  120.27 | 10.81 | 123.98 | 10.83 |
-| icos |   51.40 |  1.16 |  91.42 |  1.18 |
-|  tan |  147.59 | 18.07 | 151.39 | 18.07 |
-| itan |  126.73 |  1.31 | 129.93 |  1.29 |
+| function | UNO 16 | ESP32 240 | UNO (720-1080) | ESP (720-1080) |
+|:--------:|:------:|:---------:|:--------------:|:--------------:|
+|    sin   | 120.43 |   10.90   |    124.19      |   10.91        |
+|   isin   |  44.24 |    1.09   |     85.00      |    1.11        |
+|    cos   | 120.27 |   10.81   |    123.98      |   10.83        |
+|   icos   |  51.40 |    1.16   |     91.42      |    1.18        |
+|    tan   | 147.59 |   18.07   |    151.39      |   18.07        |
+|   itan   | 126.73 |    1.31   |    129.93      |    1.29        |
 
 *Note: itan() 0.1.3 was ( 131.23, 3.05 ) so it improved quite a bit on ESP32. *
 
@@ -72,19 +76,19 @@ errors - based upon example sketch - lib version 0.1.5
 
 ESP32 calls 0.0 - 360.0 step 0.1 degree
 
-| function | max abs  error | avg abs error | max rel error | avg rel error |
-|:----:|----:|----:|----:|----:|
-| isin | 0.00010264 | 0.00002059 | 0.02955145 | 0.00035180 |
-| icos | 0.00010264 | 0.00002031 | 0.02955145 | 0.00034868 |
-| itan | 0.69696045 | 0.00640957 | 0.00144703 | 0.00010100 |
+| function | max abs error | avg abs error | max rel error | avg rel error |
+|:--------:|--------------:|--------------:|--------------:|--------------:|
+|   isin   |   0.00010264  |  0.00002059   |  0.02955145   |  0.00035180   |
+|   icos   |   0.00010264  |  0.00002031   |  0.02955145   |  0.00034868   |
+|   itan   |   0.69696045  |  0.00640957   |  0.00144703   |  0.00010100   |
 
 UNO calls 0.0 - 360.0 step 0.1 degree
 
-| function | max abs  error | avg abs error | max rel error | avg rel error |
-|:----:|----:|----:|----:|----:|
-| isin | 0.00010270 | 0.00002059 | 0.02955145 | 0.00035171 |
-| icos | 0.00010264 | 0.00002032 | 0.02949960 | 0.00034869 |
-| itan | 0.72760009 | 0.00641527 | 0.00144703 | 0.00037889 | 
+| function | max abs error | avg abs error | max rel error | avg rel error |
+|:--------:|--------------:|--------------:|--------------:|--------------:|
+|   isin   |   0.00010270  |  0.00002059   |  0.02955145   |  0.00035171   |
+|   icos   |   0.00010264  |  0.00002032   |  0.02949960   |  0.00034869   |
+|   itan   |   0.72760009  |  0.00641527   |  0.00144703   |  0.00037889   | 
 
 *Note: 0.1.3 for AVR was bad:   17.41900634 , 0.02249339 , 0.02953807 for itan() *
 
@@ -99,14 +103,14 @@ Please, verify the performance to see if it meets your requirements.
 
 time in us - calls -1 ..+1 step 0.001 degree
 
-| function | UNO 16MHz | ESP32 240 MHz |
-|:----:|:----:|:----:|
-|  asin |  149.76 | 16.71 |
-| iasin |  107.70 |  2.58 |
-|  acos |  169.50 | 15.44 |
-| iacos |  114.65 |  2.67 |
-|  atan |  155.75 | 11.68 |
-| iatan |   NI    |  NI   |
+| function | UNO 16 -| ESP32 240 |
+|:--------:|:-------:|:---------:|
+|   asin   |  149.76 |  16.71    |
+|  iasin   |  107.70 |   2.58    |
+|   acos   |  169.50 |  15.44    |
+|  iacos   |  114.65 |   2.67    |
+|   atan   |  155.75 |  11.68    |
+|  iatan   |   NI    |   NI      |
 
 - the interpolated reverse lookup is around 30% faster on UNO an 80+% on ESP32
 - iatan is Not Implemented.
@@ -120,11 +124,11 @@ Please, verify the accuracy to see if it meets your requirements.
 
 ESP32 calls -1 ..+1 step 0.001 degree
 
-| function | max abs  error | avg abs error | max rel error | avg rel error |
-|:----:|----:|----:|----:|----:|
-| iasin | 0.22498322 | 0.00195790 | 0.00456106 | 0.00005727 |
-| iacos | 0.22498587 | 0.00195794 | 0.64284271 | 0.00021902 |
-| iatan |   NI       | NI         | NI         | NI        |
+| function | max abs error | avg abs error | max rel error | avg rel error |
+|:--------:|--------------:|--------------:|--------------:|--------------:|
+|  iasin   |  0.22498322   |  0.00195790   |  0.00456106   |  0.00005727   |
+|  iacos   |  0.22498587   |  0.00195794   |  0.64284271   |  0.00021902   |
+|  iatan   |    NI         |  NI           |  NI           |  NI           |
 
 - largest error at 0.999981 - second largest error 0.052841 at -0.999000
 - iatan is Not Implemented
@@ -133,26 +137,27 @@ ESP32 calls -1 ..+1 step 0.001 degree
 
 UNO calls -1 ..+1 step 0.001 degree
 
-| function | max abs  error | avg abs error | max rel error | avg rel error |
-|:----:|----:|----:|----:|----:|
-| iasin | 0.22499084 | 0.00195719 | 0.00456125 | 0.00005725 |
-| iacos | 0.22498588 | 0.00195740 | 0.64284276 | 0.00021901 |
-| iatan |   NI       | NI         | NI         | NI        |
+| function | max abs error | avg abs error | max rel error | avg rel error |
+|:--------:|--------------:|--------------:|--------------:|--------------:|
+|  iasin   |  0.22499084   |  0.00195719   |  0.00456125   |  0.00005725   |
+|  iacos   |  0.22498588   |  0.00195740   |  0.64284276   |  0.00021901   |
+|  iatan   |    NI         |  NI           |  NI           |  NI           |
 
 - largest error at 0.999981 - second largest error 0.052841 at -0.999000
 - max rel error is high as it occured near zero.
 - iatan is Not Implemented
 
 
-
 Please, verify the accuracy to see if it meets your requirements.
 
+
 ## 0.1.4
 
 The library (0.1.4) provides an **itan()** which improved accuracy 
 upon the (0.1.3) version and performance for the ESP32.  
 Performance on AVR (UNO) is still an issue, accuracy is OK.
 
+
 ## 0.1.5
 
 In (0.1.4) an error was found in the optimize algorithm, so for 0.1.5 
@@ -168,11 +173,25 @@ is added, as it uses the same **isintable16\[\]** interpolation table.
 There is no **atan()** or **atan2()** replacement.
 
 
+## 0.1.6
+
+- Arduino-CI + unit tests
+
+## 0.1.7
+
+- fix for PlatformIO
+
+## 0.1.8
+
+- Made the % 180 in the **itan()** conditional.
+- added **icot(f)**
+
+
+
 ## TODO
 
 - How to improve the accuracy of the whole degrees, as now the table is optimized for interpolation.
-- investigate **itan()** lookup table with interpolation (see notes.txt). Interpolation is not trivial for angles between say 60-90 degrees 
-- investigate if **atan() / atan(()** can be made.
+
 
 ## Operation
 

libraries/FastTrig/examples/fastTrig_test2/fastTrig_test2.ino

@@ -0,0 +1,122 @@
+//
+//    FILE: fastTrig_test2.ino
+//  AUTHOR: Rob Tillaart
+// PURPOSE: testing the itan functions
+//    DATE: 2021-08-10
+//    (c) : MIT
+//
+
+
+#include "FastTrig.h"
+
+uint32_t start, d1, d2;
+
+volatile float x;
+int i;
+
+void setup()
+{
+  Serial.begin(115200);
+  Serial.println("start");
+
+  test_tan(0);
+  test_itan_error_1(false);
+
+  Serial.println("done...\n");
+}
+
+
+
+void test_tan(int n)
+{
+  Serial.println(__FUNCTION__);
+  Serial.print("TAN\titan\t360 calls - offset: " );
+  Serial.println(n);
+  delay(10);
+
+  start = micros();
+  for ( i = n; i < n + 360; i++)
+  {
+    x = tan(i);
+  }
+  float t1 = (micros() - start) / 360.0;
+  Serial.println(t1);
+  delay(10);
+
+  start = micros();
+  for ( i = n; i < n + 360; i++)
+  {
+    x = itan(i);
+  }
+  float t2 = (micros() - start) / 360.0;
+  Serial.println(t2);
+  Serial.print(t2 / t1, 3);
+  Serial.println("%");
+  Serial.println();
+  Serial.println();
+  delay(10);
+}
+
+void test_itan_error_1(bool show)
+{
+  Serial.println(__FUNCTION__);
+  Serial.println("ITAN 0-3600 calls: \t");
+  delay(10);
+
+  float mx = 0;
+  float mxr = 0;
+  float z = 0;
+  float zz = 0;
+  for (int i = 0; i < 3600; i++)
+  {
+    if (((i + 900 ) % 1800) == 0) continue;
+    float a = tan(i * 0.1 * PI / 180);
+    float b = itan(i * 0.1);
+    float y = abs(a - b);  // abs error - rel error ~ 1%
+    z += y;
+    if (a > 0) zz += y / a; // not 100% correct but almost.
+    if (mx < y) mx = y;
+    if (show)
+    {
+      Serial.print(i);
+      Serial.print("\t");
+      Serial.print(a, 6);
+      Serial.print("\t");
+      Serial.print(b, 6);
+      Serial.print("\t");
+      Serial.print(a - b, 6);
+      Serial.print("\t");
+      Serial.println((a - b) / a, 6);
+    }
+    if (abs(a) > 0.00001 && mxr < y / a)
+    {
+      //      Serial.print(i);
+      //      Serial.print('\t');
+      //      Serial.print(a, 6);
+      //      Serial.print('\t');
+      //      Serial.print(b, 6);
+      //      Serial.print('\t');
+      //      Serial.print(y/a, 6);
+      //      Serial.println();
+      mxr = y / a;
+    }
+  }
+  Serial.print("max abs error: ");
+  Serial.println(mx, 8);
+  Serial.print("avg abs error: ");
+  Serial.println(z / 3600, 8);
+  Serial.print("max rel error: ");
+  Serial.println(mxr, 8);
+  Serial.print("avg rel error: ");
+  Serial.println(zz / 3600, 8);
+  Serial.println();
+  Serial.println();
+  delay(10);
+}
+
+
+void loop()
+{
+}
+
+// -- END OF FILE --

libraries/FastTrig/keywords.txt

@@ -6,6 +6,7 @@
 isin	KEYWORD2
 icos	KEYWORD2
 itan	KEYWORD2
+icot	KEYWORD2
 iasin	KEYWORD2
 iacos	KEYWORD2
 

libraries/FastTrig/library.json

@@ -1,7 +1,7 @@
 {
   "name": "FastTrig",
-  "keywords": "sin, cos, tan, isin, icos, itan, fast",
-  "description": "Arduino library with interpolated lookup for sin() and cos(). Trades speed for accuracy. Check readme for details.",
+  "keywords": "sin, cos, tan, isin, icos, itan, icot, fast",
+  "description": "Arduino library with interpolated lookup for sin() cos() tan(). Trades speed for accuracy. Check readme for details.",
   "authors":
   [
     {
@@ -15,7 +15,7 @@
     "type": "git",
     "url": "https://github.com/RobTillaart/FastTrig"
   },
-  "version":"0.1.7",
+  "version": "0.1.8",
   "license": "MIT",
   "frameworks": "*",
   "platforms": "*"