Introduction to Function Approximation with Andrew Bromage
References ▼
IEEE
754-2008 - IEEE Standard for Floating-Point Arithmetic
[1]2:42[2]10:36[4]27:28
[8]58:30
Float Toy
[3]10:36
Wikibooks
C Programming/​float.h
[5]30:22
MSDN
frexp
[6]39:07[7]53:58
Wikipedia
Horner's method
[9]1:06:29
Intel
Intel 64 and IA-32 Architectures Software Developer Manuals
[10]1:06:29[17]2:56:22
Wikipedia
Equioscillation theorem
[11]1:26:43
Wikipedia
Chebyshev nodes
[12]1:26:43
Wikipedia
Remez algorithm
[13]1:33:47
Wikipedia
Golden-section search
[14]1:42:24
Philip Wadler
As Natural as 0, 1, 2
[15]2:07:56
what-when-how
The Multiplane Downshooter (Non-Traditional Animation Techniques) Part 1
[16]2:37:32
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Casey Muratori
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Andrew Bromage
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Matt Mascarenhas
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0:00Welcome to a special episode with Andrew Bromage
0:00Welcome to a special episode with Andrew Bromage
0:00Welcome to a special episode with Andrew Bromage
2:42Andrew Bromage: How floating point numbers are represented by a computer1
2:42Andrew Bromage: How floating point numbers are represented by a computer1
2:42Andrew Bromage: How floating point numbers are represented by a computer1
4:36AB: Scientific notation, and how IEEE 754 represents numbers in binary
4:36AB: Scientific notation, and how IEEE 754 represents numbers in binary
4:36AB: Scientific notation, and how IEEE 754 represents numbers in binary
7:14Get Andrew back
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7:14Get Andrew back
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7:14Get Andrew back
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9:27AB: Return
9:27AB: Return
9:27AB: Return
10:36AB: Continuing IEEE 754's2 representation of floating point numbers3
10:36AB: Continuing IEEE 754's2 representation of floating point numbers3
10:36AB: Continuing IEEE 754's2 representation of floating point numbers3
16:20AB: Subnormal numbers, the special-case numbers infinity, quiet NaN and signaling NaN, and the quality of being "algebraically closed"
16:20AB: Subnormal numbers, the special-case numbers infinity, quiet NaN and signaling NaN, and the quality of being "algebraically closed"
16:20AB: Subnormal numbers, the special-case numbers infinity, quiet NaN and signaling NaN, and the quality of being "algebraically closed"
24:10AB: Any questions?
24:10AB: Any questions?
24:10AB: Any questions?
24:35Is it just a peculiarity of binary as a number system, that you can skip encoding the leading digit?
24:35Is it just a peculiarity of binary as a number system, that you can skip encoding the leading digit?
24:35Is it just a peculiarity of binary as a number system, that you can skip encoding the leading digit?
26:06desuused Q: Is there a representation for underflowing numbers?
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26:06desuused Q: Is there a representation for underflowing numbers?
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26:06desuused Q: Is there a representation for underflowing numbers?
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27:28AB: Note the binary and decimal representations of floating point numbers in the IEEE 754 standard4
27:28AB: Note the binary and decimal representations of floating point numbers in the IEEE 754 standard4
27:28AB: Note the binary and decimal representations of floating point numbers in the IEEE 754 standard4
27:51AB: Constants definitions in handmade_numerics.h
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27:51AB: Constants definitions in handmade_numerics.h
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27:51AB: Constants definitions in handmade_numerics.h
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30:22AB: Constants definitions in C's float.h5 as compared with those in handmade_numerics.h, with a special mention of machine epsilon
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30:22AB: Constants definitions in C's float.h5 as compared with those in handmade_numerics.h, with a special mention of machine epsilon
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30:22AB: Constants definitions in C's float.h5 as compared with those in handmade_numerics.h, with a special mention of machine epsilon
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33:32AB: Describe the IEEEBinary32 union, ieee754_number_category enum and the special-case number functions
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33:32AB: Describe the IEEEBinary32 union, ieee754_number_category enum and the special-case number functions
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33:32AB: Describe the IEEEBinary32 union, ieee754_number_category enum and the special-case number functions
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36:48AB: Describe Real32_Abs(), Real32_SetSign() and CategoryOfReal32()
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36:48AB: Describe Real32_Abs(), Real32_SetSign() and CategoryOfReal32()
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36:48AB: Describe Real32_Abs(), Real32_SetSign() and CategoryOfReal32()
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39:07AB: Describe ExtractExponent() as similar to the CRT's frexp()6 with an example of its use in a sqrt() function
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39:07AB: Describe ExtractExponent() as similar to the CRT's frexp()6 with an example of its use in a sqrt() function
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39:07AB: Describe ExtractExponent() as similar to the CRT's frexp()6 with an example of its use in a sqrt() function
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47:36When multiplying a subnormal number by a power of two, does the floating point unit first shift the numbers into the normal range before incrementing the exponent?
47:36When multiplying a subnormal number by a power of two, does the floating point unit first shift the numbers into the normal range before incrementing the exponent?
47:36When multiplying a subnormal number by a power of two, does the floating point unit first shift the numbers into the normal range before incrementing the exponent?
50:38AB: Describe ScaleByExponent()
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50:38AB: Describe ScaleByExponent()
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50:38AB: Describe ScaleByExponent()
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53:58AB: Note the differing range of absolute values of the mantissa in text books (as used in handmade_numerics.h) and the CRT's frexp()7
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53:58AB: Note the differing range of absolute values of the mantissa in text books (as used in handmade_numerics.h) and the CRT's frexp()7
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53:58AB: Note the differing range of absolute values of the mantissa in text books (as used in handmade_numerics.h) and the CRT's frexp()7
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57:01AB: Quote James H. Wilkinson on the state of computer arithmetic in 1971
57:01AB: Quote James H. Wilkinson on the state of computer arithmetic in 1971
57:01AB: Quote James H. Wilkinson on the state of computer arithmetic in 1971
58:30AB: Describe SlowDivision(), with emphasis on the sheer amount of specification compliance it contains8
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58:30AB: Describe SlowDivision(), with emphasis on the sheer amount of specification compliance it contains8
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58:30AB: Describe SlowDivision(), with emphasis on the sheer amount of specification compliance it contains8
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1:06:29AB: Walk through an example of SlowDivision(), noting why it uses 11 bits of precision in its application of Horner's rule9 (IA-32's RCPSS instruction10)
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1:06:29AB: Walk through an example of SlowDivision(), noting why it uses 11 bits of precision in its application of Horner's rule9 (IA-32's RCPSS instruction10)
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1:06:29AB: Walk through an example of SlowDivision(), noting why it uses 11 bits of precision in its application of Horner's rule9 (IA-32's RCPSS instruction10)
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1:14:13AB: How SlowDivision() finishes up its computation to the highest precision it can
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1:14:13AB: How SlowDivision() finishes up its computation to the highest precision it can
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1:14:13AB: How SlowDivision() finishes up its computation to the highest precision it can
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1:16:54AB: Note the difference between SlowDivision() and how our FPU performs division
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1:16:54AB: Note the difference between SlowDivision() and how our FPU performs division
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1:16:54AB: Note the difference between SlowDivision() and how our FPU performs division
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1:19:13AB: Calculating those polynomial approximations from SlowDivision(), with range illustrations in Mathematica
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1:19:13AB: Calculating those polynomial approximations from SlowDivision(), with range illustrations in Mathematica
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1:19:13AB: Calculating those polynomial approximations from SlowDivision(), with range illustrations in Mathematica
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1:25:12AB: Relative vs absolute error
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1:25:12AB: Relative vs absolute error
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1:25:12AB: Relative vs absolute error
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1:26:43AB: Plot the error function in Mathematica, with a mention of Chebyshev's Equioscillation theorem11 and Chebyshev nodes12
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1:26:43AB: Plot the error function in Mathematica, with a mention of Chebyshev's Equioscillation theorem11 and Chebyshev nodes12
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1:26:43AB: Plot the error function in Mathematica, with a mention of Chebyshev's Equioscillation theorem11 and Chebyshev nodes12
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1:31:35AB: Plot the eighth order Chebyshev polynomial in the range -1 to 1 in Mathematica
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1:31:35AB: Plot the eighth order Chebyshev polynomial in the range -1 to 1 in Mathematica
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1:31:35AB: Plot the eighth order Chebyshev polynomial in the range -1 to 1 in Mathematica
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1:33:47AB: Using the Remez exchange algorithm13 to find approximations to Chebyshev's set of polynomials
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1:33:47AB: Using the Remez exchange algorithm13 to find approximations to Chebyshev's set of polynomials
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1:33:47AB: Using the Remez exchange algorithm13 to find approximations to Chebyshev's set of polynomials
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1:38:21On the initial guesses in the Remez exchange algorithm
1:38:21On the initial guesses in the Remez exchange algorithm
1:38:21On the initial guesses in the Remez exchange algorithm
1:39:15AB: Plot the ninth order Chebyshev polynomial, for comparison with the eighth order, to explain extrema
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1:39:15AB: Plot the ninth order Chebyshev polynomial, for comparison with the eighth order, to explain extrema
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1:39:15AB: Plot the ninth order Chebyshev polynomial, for comparison with the eighth order, to explain extrema
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1:40:20Struggle with the communication link
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1:40:20Struggle with the communication link
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1:40:20Struggle with the communication link
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1:41:49As the Remez exchange algorithm proceeds, what values does it use as its new guesses?
1:41:49As the Remez exchange algorithm proceeds, what values does it use as its new guesses?
1:41:49As the Remez exchange algorithm proceeds, what values does it use as its new guesses?
1:42:24AB: Searching for the extremum, perhaps using golden-section search,14 as the Remez exchange algorithm proceeds
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1:42:24AB: Searching for the extremum, perhaps using golden-section search,14 as the Remez exchange algorithm proceeds
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1:42:24AB: Searching for the extremum, perhaps using golden-section search,14 as the Remez exchange algorithm proceeds
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1:46:50AB: Plot sine in Mathematica, and introduce the computation of sine in the range 0 to 2π
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1:46:50AB: Plot sine in Mathematica, and introduce the computation of sine in the range 0 to 2π
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1:46:50AB: Plot sine in Mathematica, and introduce the computation of sine in the range 0 to 2π
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1:56:43gg_nate Can you set video / audio bit rate on hangouts? If he could lower the bit rate of the video the audio might work better
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1:56:43gg_nate Can you set video / audio bit rate on hangouts? If he could lower the bit rate of the video the audio might work better
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1:56:43gg_nate Can you set video / audio bit rate on hangouts? If he could lower the bit rate of the video the audio might work better
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1:56:57AB: Describe SinCos_TableVersion()
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1:56:57AB: Describe SinCos_TableVersion()
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1:56:57AB: Describe SinCos_TableVersion()
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1:58:31AB: Deriving trigonometric identities
1:58:31AB: Deriving trigonometric identities
1:58:31AB: Deriving trigonometric identities
2:02:28AB: Calculating cosine around "a", branch-free
2:02:28AB: Calculating cosine around "a", branch-free
2:02:28AB: Calculating cosine around "a", branch-free
2:04:00AB: Point out the experimental SinCos_QuadrantVersion() for SIMD sines and cosines
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2:04:00AB: Point out the experimental SinCos_QuadrantVersion() for SIMD sines and cosines
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2:04:00AB: Point out the experimental SinCos_QuadrantVersion() for SIMD sines and cosines
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2:05:54AB: Describe the XSinCosX table look up
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2:05:54AB: Describe the XSinCosX table look up
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2:05:54AB: Describe the XSinCosX table look up
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2:07:56AB: Counting has always started at zero15
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2:07:56AB: Counting has always started at zero15
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2:07:56AB: Counting has always started at zero15
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2:08:47AB: Continued description of the XSinCosX table look up
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2:08:47AB: Continued description of the XSinCosX table look up
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2:08:47AB: Continued description of the XSinCosX table look up
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2:10:38AB: Run calculate_sincos_tables and explain the result for .5
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2:10:38AB: Run calculate_sincos_tables and explain the result for .5
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2:10:38AB: Run calculate_sincos_tables and explain the result for .5
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2:12:10AB: Describe how FindSinCosAround() searches adjacent floating point numbers
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2:12:10AB: Describe how FindSinCosAround() searches adjacent floating point numbers
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2:12:10AB: Describe how FindSinCosAround() searches adjacent floating point numbers
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2:15:55Could you explain why part of the table version looks up into XSinCosX while the polynomial part is the same no matter where you are in the table?
2:15:55Could you explain why part of the table version looks up into XSinCosX while the polynomial part is the same no matter where you are in the table?
2:15:55Could you explain why part of the table version looks up into XSinCosX while the polynomial part is the same no matter where you are in the table?
2:17:10AB: Further explain the table lookups of cos(a) and sin(a) and the sin(e) and cos(e) polynomial approximations
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2:17:10AB: Further explain the table lookups of cos(a) and sin(a) and the sin(e) and cos(e) polynomial approximations
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2:17:10AB: Further explain the table lookups of cos(a) and sin(a) and the sin(e) and cos(e) polynomial approximations
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2:19:46Since table lookups are hard in SIMD, what sort of stuff would you end up doing if you couldn't use a table?
2:19:46Since table lookups are hard in SIMD, what sort of stuff would you end up doing if you couldn't use a table?
2:19:46Since table lookups are hard in SIMD, what sort of stuff would you end up doing if you couldn't use a table?
2:20:09AB: Describe the experimental SinCos_QuadrantVersion()
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2:20:09AB: Describe the experimental SinCos_QuadrantVersion()
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2:20:09AB: Describe the experimental SinCos_QuadrantVersion()
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2:28:25x13pixels Q: Why do C0, C2, C4, C6, etc. have more precision than can fit in a float?
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2:28:25x13pixels Q: Why do C0, C2, C4, C6, etc. have more precision than can fit in a float?
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2:28:25x13pixels Q: Why do C0, C2, C4, C6, etc. have more precision than can fit in a float?
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2:29:32AB: Describe ATan() and ATan2(), noting the current use of atan2 in Handmade Hero
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2:29:32AB: Describe ATan() and ATan2(), noting the current use of atan2 in Handmade Hero
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2:29:32AB: Describe ATan() and ATan2(), noting the current use of atan2 in Handmade Hero
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2:35:01Determine to remove atan2 from Handmade Hero with thanks to Andrew
2:35:01Determine to remove atan2 from Handmade Hero with thanks to Andrew
2:35:01Determine to remove atan2 from Handmade Hero with thanks to Andrew
2:35:31Q&A
2:35:31Q&A
2:35:31Q&A
2:36:240b0000000000000 If the values are denormal, they will run way slower
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2:36:240b0000000000000 If the values are denormal, they will run way slower
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2:36:240b0000000000000 If the values are denormal, they will run way slower
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2:36:46filiadelski Q: What was the reason we couldn't do two's complement in the exponent?
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2:36:46filiadelski Q: What was the reason we couldn't do two's complement in the exponent?
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2:36:46filiadelski Q: What was the reason we couldn't do two's complement in the exponent?
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2:37:320b0000000000000 sin and cos16
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2:37:320b0000000000000 sin and cos16
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2:37:320b0000000000000 sin and cos16
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2:41:320b0000000000000 Unless you explicitly flush them to zero, they will run super slow
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2:41:320b0000000000000 Unless you explicitly flush them to zero, they will run super slow
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2:41:320b0000000000000 Unless you explicitly flush them to zero, they will run super slow
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2:42:44spacealiens Q: Is there a version of this maths source code anywhere, or will it be included in the Handmade Hero project eventually?
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2:42:44spacealiens Q: Is there a version of this maths source code anywhere, or will it be included in the Handmade Hero project eventually?
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2:42:44spacealiens Q: Is there a version of this maths source code anywhere, or will it be included in the Handmade Hero project eventually?
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2:43:17AB: Note the educational nature of this sine and cosine implementation, with a mention of Cody-Waite reduction
2:43:17AB: Note the educational nature of this sine and cosine implementation, with a mention of Cody-Waite reduction
2:43:17AB: Note the educational nature of this sine and cosine implementation, with a mention of Cody-Waite reduction
2:45:43vateferfout Q: Will you go through acos as well in a later stream?
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2:45:43vateferfout Q: Will you go through acos as well in a later stream?
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2:45:43vateferfout Q: Will you go through acos as well in a later stream?
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2:46:100b0000000000000 What is the guy speaking's username?
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2:46:100b0000000000000 What is the guy speaking's username?
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2:46:100b0000000000000 What is the guy speaking's username?
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2:46:51SSE denormal flushing
2:46:51SSE denormal flushing
2:46:51SSE denormal flushing
2:48:130b0000000000000 I can show you some versions that are completely branchless without tables if you are interested
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2:48:130b0000000000000 I can show you some versions that are completely branchless without tables if you are interested
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2:48:130b0000000000000 I can show you some versions that are completely branchless without tables if you are interested
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2:49:05sneakybob_wot Q: Have you done a speed comparison vs the C versions?
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2:49:05sneakybob_wot Q: Have you done a speed comparison vs the C versions?
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2:49:05sneakybob_wot Q: Have you done a speed comparison vs the C versions?
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2:50:34AB: Re-emphasise the slowness of SlowDivision()
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2:50:34AB: Re-emphasise the slowness of SlowDivision()
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2:50:34AB: Re-emphasise the slowness of SlowDivision()
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2:52:55staythirsty90 Not industrial strength?! Why am I here??
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2:52:55staythirsty90 Not industrial strength?! Why am I here??
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2:52:55staythirsty90 Not industrial strength?! Why am I here??
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2:53:15vateferfout Q: To be sure, it means the SIMD intrinsics are not standards-compliant?
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2:53:15vateferfout Q: To be sure, it means the SIMD intrinsics are not standards-compliant?
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2:53:15vateferfout Q: To be sure, it means the SIMD intrinsics are not standards-compliant?
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2:56:22AB: Quote the Intel 64 and IA-32 Architectures Optimization Reference Manual:17 "Although x87 supports transcendental instructions, software library implementation of transcendental function can be faster in many cases"
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2:56:22AB: Quote the Intel 64 and IA-32 Architectures Optimization Reference Manual:17 "Although x87 supports transcendental instructions, software library implementation of transcendental function can be faster in many cases"
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2:56:22AB: Quote the Intel 64 and IA-32 Architectures Optimization Reference Manual:17 "Although x87 supports transcendental instructions, software library implementation of transcendental function can be faster in many cases"
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2:57:04Thank you to Andrew for walking us through sine and cosine, with closing thoughts on numerical approximations
2:57:04Thank you to Andrew for walking us through sine and cosine, with closing thoughts on numerical approximations
2:57:04Thank you to Andrew for walking us through sine and cosine, with closing thoughts on numerical approximations