1 | /* Compute x^2 + y^2 - 1, without large cancellation error. |
---|---|

2 | Copyright (C) 2012 Free Software Foundation, Inc. |

3 | This file is part of the GNU C Library. |

4 | |

5 | The GNU C Library is free software; you can redistribute it and/or |

6 | modify it under the terms of the GNU Lesser General Public |

7 | License as published by the Free Software Foundation; either |

8 | version 2.1 of the License, or (at your option) any later version. |

9 | |

10 | The GNU C Library is distributed in the hope that it will be useful, |

11 | but WITHOUT ANY WARRANTY; without even the implied warranty of |

12 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |

13 | Lesser General Public License for more details. |

14 | |

15 | You should have received a copy of the GNU Lesser General Public |

16 | License along with the GNU C Library; if not, see |

17 | <http://www.gnu.org/licenses/>. */ |

18 | |

19 | #include "quadmath-imp.h" |

20 | #include <stdlib.h> |

21 | |

22 | /* Calculate X + Y exactly and store the result in *HI + *LO. It is |

23 | given that |X| >= |Y| and the values are small enough that no |

24 | overflow occurs. */ |

25 | |

26 | static inline void |

27 | add_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y) |

28 | { |

29 | /* Apply Dekker's algorithm. */ |

30 | *hi = x + y; |

31 | *lo = (x - *hi) + y; |

32 | } |

33 | |

34 | /* Calculate X * Y exactly and store the result in *HI + *LO. It is |

35 | given that the values are small enough that no overflow occurs and |

36 | large enough (or zero) that no underflow occurs. */ |

37 | |

38 | static inline void |

39 | mul_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y) |

40 | { |

41 | /* Fast built-in fused multiply-add. */ |

42 | *hi = x * y; |

43 | *lo = fmaq (x, y, -*hi); |

44 | } |

45 | |

46 | /* Compare absolute values of floating-point values pointed to by P |

47 | and Q for qsort. */ |

48 | |

49 | static int |

50 | compare (const void *p, const void *q) |

51 | { |

52 | __float128 pld = fabsq (*(const __float128 *) p); |

53 | __float128 qld = fabsq (*(const __float128 *) q); |

54 | if (pld < qld) |

55 | return -1; |

56 | else if (pld == qld) |

57 | return 0; |

58 | else |

59 | return 1; |

60 | } |

61 | |

62 | /* Return X^2 + Y^2 - 1, computed without large cancellation error. |

63 | It is given that 1 > X >= Y >= epsilon / 2, and that either X >= |

64 | 0.75 or Y >= 0.5. */ |

65 | |

66 | __float128 |

67 | __quadmath_x2y2m1q (__float128 x, __float128 y) |

68 | { |

69 | __float128 vals[4]; |

70 | size_t i; |

71 | |

72 | /* FIXME: SET_RESTORE_ROUNDL (FE_TONEAREST); */ |

73 | mul_split (&vals[1], &vals[0], x, x); |

74 | mul_split (&vals[3], &vals[2], y, y); |

75 | if (x >= 0.75Q) |

76 | vals[1] -= 1.0Q; |

77 | else |

78 | { |

79 | vals[1] -= 0.5Q; |

80 | vals[3] -= 0.5Q; |

81 | } |

82 | qsort (vals, 4, sizeof (__float128), compare); |

83 | /* Add up the values so that each element of VALS has absolute value |

84 | at most equal to the last set bit of the next nonzero |

85 | element. */ |

86 | for (i = 0; i <= 2; i++) |

87 | { |

88 | add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]); |

89 | qsort (vals + i + 1, 3 - i, sizeof (__float128), compare); |

90 | } |

91 | /* Now any error from this addition will be small. */ |

92 | return vals[3] + vals[2] + vals[1] + vals[0]; |

93 | } |

94 |