Introduce an integer multiply-divide algorithm.

We have multiplies followed by divides all over the place and
most of them are difficult to handle regarding overflows. This
new algorithm handles this fine in all cases, as long as all
three operators and the overall result fits into 32 bits.

Makefile

@@ -80,7 +80,7 @@ PROGID = arduino
 
 PROGRAM = mendel
 
-SOURCES = $(PROGRAM).c dda.c gcode_parse.c gcode_process.c timer.c temp.c sermsg.c dda_queue.c watchdog.c debug.c sersendf.c heater.c analog.c intercom.c pinio.c clock.c home.c crc.c delay.c
+SOURCES = $(PROGRAM).c gcode_parse.c gcode_process.c dda.c dda_maths.c dda_queue.c timer.c temp.c sermsg.c watchdog.c debug.c sersendf.c heater.c analog.c intercom.c pinio.c clock.c home.c crc.c delay.c
 
 ARCH = avr-
 CC = $(ARCH)gcc

dda_maths.c

@@ -0,0 +1,60 @@
+
+/** \file
+	\brief Mathematic algorithms for the digital differential analyser (DDA).
+*/
+
+#include "dda_maths.h"
+
+#include <stdlib.h>
+#include <stdint.h>
+
+/*!
+  Integer multiply-divide algorithm.
+
+  \param multiplicand
+  \param multiplier
+  \param divisor
+  \return rounded result of multiplicand * multiplier / divisor
+
+  Calculate a * b / c, without overflowing and without using 64-bit integers.
+  Doing this the standard way, a * b could easily overflow, even if the correct
+  overall result fits into 32 bits. This algorithm avoids this intermediate
+  overflow and delivers valid results for all cases where each of the three
+  operators as well as the result fits into 32 bits.
+
+  Found on  http://stackoverflow.com/questions/4144232/
+  how-to-calculate-a-times-b-divided-by-c-only-using-32-bit-integer-types-even-i
+*/
+const uint32_t muldiv(uint32_t multiplicand, uint32_t multiplier,
+                      uint32_t divisor) {
+  uint32_t quotient = 0;
+  uint32_t remainder = 0;
+  uint32_t qn = multiplier / divisor;
+  uint32_t rn = multiplier % divisor;
+
+  while(multiplicand) {
+    if (multiplicand & 1) {
+      quotient += qn;
+      remainder += rn;
+      if (remainder >= divisor) {
+        quotient++;
+        remainder -= divisor;
+      }
+    }
+    multiplicand  >>= 1;
+    qn <<= 1;
+    rn <<= 1;
+    if (rn >= divisor) {
+      qn++; 
+      rn -= divisor;
+    }
+  }
+
+  // rounding
+  if (remainder > divisor / 2)
+    quotient++;
+
+  // remainder is valid here, but not returned
+  return quotient;
+}
+

dda_maths.h

@@ -0,0 +1,11 @@
+#ifndef	_DDA_MATHS_H
+#define	_DDA_MATHS_H
+
+#include	<stdint.h>
+
+
+// return rounded result of multiplicand * multiplier / divisor
+const uint32_t muldiv(uint32_t multiplicand, uint32_t multiplier,
+                      uint32_t divisor);
+
+#endif	/* _DDA_MATHS_H */