Exact computations with approximate arithmetic Jean-Michel Muller CNRS - Laboratoire LIP (CNRS-INRIA-Université de Lyon) october 2007 http://perso.ens-lyon.fr/jean-michel.muller/ -1- Better approach ? Floating-Point Arithmetic ? used everywhere in scientific calculation; x = mx ×βex; “fuzzy” approach: computed value of x +y = (x +y)(1+(cid:15)). -2- Floating-Point Arithmetic ? used everywhere in scientific calculation; x = mx ×βex; “fuzzy” approach: computed value of x +y = (x +y)(1+(cid:15)). Better approach ? -2- accuracy: some predictions of general relativity or quantum mechanics verified within relative accuracy 10−14 intermediate calculations: quad precision and smart tricks required for very-long term stability of the Solar system (J. Laskar, Paris Observatory). Good news: we seem to be safe for the next 40 million years; Needs ? A few figures... span: Estimated diameter of observable universe ≈1062 Planck’s length -3- intermediate calculations: quad precision and smart tricks required for very-long term stability of the Solar system (J. Laskar, Paris Observatory). Good news: we seem to be safe for the next 40 million years; Needs ? A few figures... span: Estimated diameter of observable universe ≈1062 Planck’s length accuracy: some predictions of general relativity or quantum mechanics verified within relative accuracy 10−14 -3- Needs ? A few figures... span: Estimated diameter of observable universe ≈1062 Planck’s length accuracy: some predictions of general relativity or quantum mechanics verified within relative accuracy 10−14 intermediate calculations: quad precision and smart tricks required for very-long term stability of the Solar system (J. Laskar, Paris Observatory). Good news: we seem to be safe for the next 40 million years; -3- Note that 2147483648 = 231. Excel’2007, compute 65535−2−37, you get 100000; On some Cray computers, it was possible to get an overflow when multiplying by 1; Maple version 7.0, enter 5001! and you get 1 instead of 5001; 5000! Version 6.0 was even worse. Enter 214748364810, you get 10. We can do a very poor job... Pentium 1 division bug: 8391667/12582905 gave 0.666869··· instead of 0.666910···; -4- Maple version 7.0, enter 5001! and you get 1 instead of 5001; 5000! Version 6.0 was even worse. Enter 214748364810, you get 10. Note that 2147483648 = 231. Excel’2007, compute 65535−2−37, you get 100000; We can do a very poor job... Pentium 1 division bug: 8391667/12582905 gave 0.666869··· instead of 0.666910···; On some Cray computers, it was possible to get an overflow when multiplying by 1; -4- Version 6.0 was even worse. Enter 214748364810, you get 10. Note that 2147483648 = 231. Excel’2007, compute 65535−2−37, you get 100000; We can do a very poor job... Pentium 1 division bug: 8391667/12582905 gave 0.666869··· instead of 0.666910···; On some Cray computers, it was possible to get an overflow when multiplying by 1; Maple version 7.0, enter 5001! and you get 1 instead of 5001; 5000! -4- Excel’2007, compute 65535−2−37, you get 100000; We can do a very poor job... Pentium 1 division bug: 8391667/12582905 gave 0.666869··· instead of 0.666910···; On some Cray computers, it was possible to get an overflow when multiplying by 1; Maple version 7.0, enter 5001! and you get 1 instead of 5001; 5000! Version 6.0 was even worse. Enter 214748364810, you get 10. Note that 2147483648 = 231. -4-