You may be right, that in the general case the output may change (most probably very slightly and unnoticable, one really needs some special cases to see real changes), but I would regard such an algorithm close to numerical unstable. And believe me, Milkyway isn't at that point. Hell, the bug with the number of the integration points didn't change the (decimal) output!

This would be true (if it were not more a problem of truncation rather than 5/5 rounding) for addition, but this is no more true with multiplication.

The error goes both ways for multiplications too. As I said before, given that the finite math of floating-point calculations on computers implies in an error of 0.5 bit of the mantissa, by your rationale all calculations done on a computer would be increasingly wrong, which is patently incorrect.

We are not speaking about benchmarking here, but about science application where accuracy does matter.

I mentioned SPEC because it's a benchmark that REQUIRES AND VERIFIES correct results. So, if -ffast-math doesn't affect the results of over 20 scientific applications from SPEC CPU2006, I doubt that it would affect Milkyway.

But if you prefer to live with your preconceptions and ignore the facts, fine.

Enough said.

An error is an error in both directions, sometimes up, sometimes down. So it stays in the 15th digit.

-ffast-math is used in SPEC benchmarks, which validate the results for acceptance, without any issue.

We're talking about a difference smaller than 15 decimal digits! If the output of the application is truncated to the default 5 digits, it'll never even show up.

And here's what they mean:

• -fno-math-errno: don't bother to set the global variable errno in case of a math error (diagnostic purposes)
  
• -funsafe-math-optimizations: IEEE754 calls for following the order in the source code, ruling out commutative operations, such as two multiplies, or associative operations, such as multiplication of a sum by a factor; this option does away with this restriction.
  
• -fno-trapping-math: some math operations can result in math errors, such as log (-1), so compilers tip-toe around such operations at the expense of performance; this option assumes that such invalid operation won't occur (diagnostic purposes).
  
• -ffinite-math-only: assumes that intermediary results will always be normal numbers (e.g., never infinite).
  
• -fno-rounding-math: this is the default anyways.
  
• -fno-signaling-nans: assumes that invalid results will not raise an exception (diagnostic purposes).
  

As you can see, most options are for error reporting purposes and the only option that causes different results is -funsafe-math-optimizations

-funsafe-math-optimizations
           Allow optimizations for floating-point arithmetic that (a) assume that arguments and results are valid and (b) may violate IEEE or ANSI standards.  When used at link-time, it
           may include libraries or startup files that change the default FPU control word or other similar optimizations.

           This option is not turned on by any -O option since it can result in incorrect output for programs which depend on an exact implementation of IEEE or ISO rules/specifications
           for math functions. It may, however, yield faster code for programs that do not require the guarantees of these specifications.  Enables -fno-signed-zeros, -fno-trapping-math,
           -fassociative-math and -freciprocal-math.

           The default is -fno-unsafe-math-optimizations.

but different by one or two bits in the mantissa, which never shows up when outputting the values in decimal format.

Could just be that there isn't much work at the moment.

#-*- Makefile -*-
#
# usage:
#
#  make -f make.linux app_x86_64
#
#  make -f make.linux app_i686
#
#
# change BOINC_DIR locally or define it on the make command line
#
APP_VERSION = 0.3

BOINC_DIR = /usr/src/rpm/BUILD/boinc
BOINC_API_DIR = $(BOINC_DIR)/api
BOINC_LIB_DIR = $(BOINC_DIR)/lib
BOINC_LINK_DIR = /usr/src/rpm/BUILD/boinc
BOINC_LIB_LINK_DIR = $(BOINC_LINK_DIR)/lib
BOINC_API_LINK_DIR = $(BOINC_LINK_DIR)/api

VARIANTFLAGS = -DGMLE_BOINC

CXXFLAGS_ALL = $(VARIANTFLAGS) \
	-O2 -fomit-frame-pointer -frename-registers -fweb -fexpensive-optimizations -fno-strict-aliasing -march=i686 -msse3 -mfpmath=sse -pipe \
	-I$(BOINC_DIR) -I$(BOINC_LIB_DIR) -I$(BOINC_API_DIR)

CXX_i686 = g++
CXXFLAGS_i686 = -m32 $(CXXFLAGS_ALL)
LDFLAGS_i686 = -L/usr/X11R6/lib -L$(BOINC_LIB_LINK_DIR) -L$(BOINC_API_LINK_DIR)

CXX_x86_64 = g++
CXXFLAGS_x86_64 = $(CXXFLAGS_ALL)
LDFLAGS_x86_64 = -L/usr/local/lib
#LDFLAGS_x86_64 = -L/usr/local/lib -L/usr/X11R6/lib
#LDFLAGS_x86_64 = -L$(BOINC_LIB_LINK_DIR) -L$(BOINC_API_LINK_DIR)


APP_DIR = ../astronomy
FGDO_DIR = ..
SCHED_DIR = $(FGDO_DIR)/evaluation
SEARCH_DIR = $(FGDO_DIR)/searches
UTIL_DIR = $(FGDO_DIR)/util

APP_OBJS = \
	$(APP_DIR)/boinc_astronomy.o \
	$(APP_DIR)/atSurveyGeometry.o \
	$(APP_DIR)/numericalIntegration.o \
	$(APP_DIR)/parameters.o \
	$(APP_DIR)/probability.o \
	$(APP_DIR)/stCoords.o \
	$(APP_DIR)/stCnum.o \
	$(APP_DIR)/stMath.o \
	$(APP_DIR)/stVector.o \
	$(APP_DIR)/star_points.o \
	$(APP_DIR)/evaluation_optimized.o \
	$(APP_DIR)/evaluation_state.o
SEARCH_OBJS = \
	$(SEARCH_DIR)/search_parameters.o
UTIL_OBJS = \
	$(UTIL_DIR)/io_util.o \
	$(UTIL_DIR)/settings.o

PROGS = milkyway_$(APP_VERSION)_i686-pc-linux-gnu milkyway_$(APP_VERSION)_x86_64-pc-linux-gnu

all: $(PROGS)

app_i686: OBJ_CXX = $(CXX_i686)
app_i686: OBJ_CXXFLAGS = $(CXXFLAGS_i686)
app_i686: $(APP_OBJS) $(SEARCH_OBJS) $(UTIL_OBJS) $(BOINC_API_LINK_DIR)/libboinc_api.a $(BOINC_LIB_LINK_DIR)/libboinc.a
	$(CXX_i686) $(LDFLAGS_i686) $(CXXFLAGS_i686) -Wl --export_dynamic -o milkyway_$(APP_VERSION)_i686-pc-linux-gnu $(APP_OBJS) $(SEARCH_OBJS) $(UTIL_OBJS) -lm -lboinc_api -lboinc -pthread

app_x86_64: OBJ_CXX = $(CXX_x86_64)
app_x86_64: OBJ_CXXFLAGS = $(CXXFLAGS_x86_64)
app_x86_64: $(APP_OBJS) $(SEARCH_OBJS) $(UTIL_OBJS) $(BOINC_API_LINK_DIR)/libboinc_api.a $(BOINC_LIB_LINK_DIR)/libboinc.a
	$(CXX_x86_64) $(LDFLAGS_x86_64) $(CXXFLAGS_x86_64) -Wl --export_dynamic -o milkyway_$(APP_VERSION)_x86_64-pc-linux-gnu $(APP_OBJS) $(SEARCH_OBJS) $(UTIL_OBJS) -lm -lboinc_api -lboinc -pthread

.C.o:
	$(OBJ_CXX) $(OBJ_CXXFLAGS) $(INC) -Wall -x c++ -c $< -o $@
.c.o:
	$(OBJ_CXX) $(OBJ_CXXFLAGS) $(INC) -Wall -x c++ -c $< -o $@

clean:
	rm -f $(APP_OBJS) $(UTIL_OBJS) $(SEARCH_OBJS);

clean_all:
	rm -f $(PROGS) $(APP_OBJS) $(UTIL_OBJS) $(SEARCH_OBJS);

<app_info>
  <app>
    <name>milkyway</name>
  </app>
<file_info>
  <name>milkyway_0.3_i686-pc-linux-gnu</name>
  <executable/>
</file_info>
<app_version>
  <app_name>milkyway</app_name>
  <version_num>0.3</version_num>
  <file_ref>
    <file_name>milkyway_0.3_i686-pc-linux-gnu</file_name>
    <main_program/>
  </file_ref>
</app_version>
<app_info>