1 // Copyright 2015 The Rust Project Developers. See the COPYRIGHT
2 // file at the top-level directory of this distribution and at
3 // http://rust-lang.org/COPYRIGHT.
5 // Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
6 // http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
7 // <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
8 // option. This file may not be copied, modified, or distributed
9 // except according to those terms.
11 //! Converting decimal strings into IEEE 754 binary floating point numbers.
13 //! # Problem statement
15 //! We are given a decimal string such as `12.34e56`. This string consists of integral (`12`),
16 //! fractional (`45`), and exponent (`56`) parts. All parts are optional and interpreted as zero
19 //! We seek the IEEE 754 floating point number that is closest to the exact value of the decimal
20 //! string. It is well-known that many decimal strings do not have terminating representations in
21 //! base two, so we round to 0.5 units in the last place (in other words, as well as possible).
22 //! Ties, decimal values exactly half-way between two consecutive floats, are resolved with the
23 //! half-to-even strategy, also known as banker's rounding.
25 //! Needless to say, this is quite hard, both in terms of implementation complexity and in terms
26 //! of CPU cycles taken.
30 //! First, we ignore signs. Or rather, we remove it at the very beginning of the conversion
31 //! process and re-apply it at the very end. This is correct in all edge cases since IEEE
32 //! floats are symmetric around zero, negating one simply flips the first bit.
34 //! Then we remove the decimal point by adjusting the exponent: Conceptually, `12.34e56` turns
35 //! into `1234e54`, which we describe with a positive integer `f = 1234` and an integer `e = 54`.
36 //! The `(f, e)` representation is used by almost all code past the parsing stage.
38 //! We then try a long chain of progressively more general and expensive special cases using
39 //! machine-sized integers and small, fixed-sized floating point numbers (first `f32`/`f64`, then
40 //! a type with 64 bit significand, `Fp`). When all these fail, we bite the bullet and resort to a
41 //! simple but very slow algorithm that involved computing `f * 10^e` fully and doing an iterative
42 //! search for the best approximation.
44 //! Primarily, this module and its children implement the algorithms described in:
45 //! "How to Read Floating Point Numbers Accurately" by William D. Clinger,
46 //! available online: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.4152
48 //! In addition, there are numerous helper functions that are used in the paper but not available
49 //! in Rust (or at least in core). Our version is additionally complicated by the need to handle
50 //! overflow and underflow and the desire to handle subnormal numbers. Bellerophon and
51 //! Algorithm R have trouble with overflow, subnormals, and underflow. We conservatively switch to
52 //! Algorithm M (with the modifications described in section 8 of the paper) well before the
53 //! inputs get into the critical region.
55 //! Another aspect that needs attention is the ``RawFloat`` trait by which almost all functions
56 //! are parametrized. One might think that it's enough to parse to `f64` and cast the result to
57 //! `f32`. Unfortunately this is not the world we live in, and this has nothing to do with using
58 //! base two or half-to-even rounding.
60 //! Consider for example two types `d2` and `d4` representing a decimal type with two decimal
61 //! digits and four decimal digits each and take "0.01499" as input. Let's use half-up rounding.
62 //! Going directly to two decimal digits gives `0.01`, but if we round to four digits first,
63 //! we get `0.0150`, which is then rounded up to `0.02`. The same principle applies to other
64 //! operations as well, if you want 0.5 ULP accuracy you need to do *everything* in full precision
65 //! and round *exactly once, at the end*, by considering all truncated bits at once.
67 //! FIXME Although some code duplication is necessary, perhaps parts of the code could be shuffled
68 //! around such that less code is duplicated. Large parts of the algorithms are independent of the
69 //! float type to output, or only needs access to a few constants, which could be passed in as
74 //! The conversion should *never* panic. There are assertions and explicit panics in the code,
75 //! but they should never be triggered and only serve as internal sanity checks. Any panics should
76 //! be considered a bug.
78 //! There are unit tests but they are woefully inadequate at ensuring correctness, they only cover
79 //! a small percentage of possible errors. Far more extensive tests are located in the directory
80 //! `src/etc/test-float-parse` as a Python script.
82 //! A note on integer overflow: Many parts of this file perform arithmetic with the decimal
83 //! exponent `e`. Primarily, we shift the decimal point around: Before the first decimal digit,
84 //! after the last decimal digit, and so on. This could overflow if done carelessly. We rely on
85 //! the parsing submodule to only hand out sufficiently small exponents, where "sufficient" means
86 //! "such that the exponent +/- the number of decimal digits fits into a 64 bit integer".
87 //! Larger exponents are accepted, but we don't do arithmetic with them, they are immediately
88 //! turned into {positive,negative} {zero,infinity}.
91 #![unstable(feature = "dec2flt",
92 reason = "internal routines only exposed for testing",
98 use self::parse::{parse_decimal, Decimal, Sign, ParseResult};
99 use self::num::digits_to_big;
100 use self::rawfp::RawFloat;
105 // These two have their own tests.
109 macro_rules! from_str_float_impl {
111 #[stable(feature = "rust1", since = "1.0.0")]
112 impl FromStr for $t {
113 type Err = ParseFloatError;
115 /// Converts a string in base 10 to a float.
116 /// Accepts an optional decimal exponent.
118 /// This function accepts strings such as
122 /// * '2.5E10', or equivalently, '2.5e10'
125 /// * '.5', or, equivalently, '0.5'
126 /// * 'inf', '-inf', 'NaN'
128 /// Leading and trailing whitespace represent an error.
136 /// `Err(ParseFloatError)` if the string did not represent a valid
137 /// number. Otherwise, `Ok(n)` where `n` is the floating-point
138 /// number represented by `src`.
140 fn from_str(src: &str) -> Result<Self, ParseFloatError> {
146 from_str_float_impl!(f32);
147 from_str_float_impl!(f64);
149 /// An error which can be returned when parsing a float.
151 /// This error is used as the error type for the [`FromStr`] implementation
152 /// for [`f32`] and [`f64`].
154 /// [`FromStr`]: ../str/trait.FromStr.html
155 /// [`f32`]: ../../std/primitive.f32.html
156 /// [`f64`]: ../../std/primitive.f64.html
157 #[derive(Debug, Clone, PartialEq, Eq)]
158 #[stable(feature = "rust1", since = "1.0.0")]
159 pub struct ParseFloatError {
163 #[derive(Debug, Clone, PartialEq, Eq)]
164 enum FloatErrorKind {
169 impl ParseFloatError {
170 #[unstable(feature = "int_error_internals",
171 reason = "available through Error trait and this method should \
172 not be exposed publicly",
175 pub fn __description(&self) -> &str {
177 FloatErrorKind::Empty => "cannot parse float from empty string",
178 FloatErrorKind::Invalid => "invalid float literal",
183 #[stable(feature = "rust1", since = "1.0.0")]
184 impl fmt::Display for ParseFloatError {
185 fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
186 self.__description().fmt(f)
190 fn pfe_empty() -> ParseFloatError {
191 ParseFloatError { kind: FloatErrorKind::Empty }
194 fn pfe_invalid() -> ParseFloatError {
195 ParseFloatError { kind: FloatErrorKind::Invalid }
198 /// Split decimal string into sign and the rest, without inspecting or validating the rest.
199 fn extract_sign(s: &str) -> (Sign, &str) {
200 match s.as_bytes()[0] {
201 b'+' => (Sign::Positive, &s[1..]),
202 b'-' => (Sign::Negative, &s[1..]),
203 // If the string is invalid, we never use the sign, so we don't need to validate here.
204 _ => (Sign::Positive, s),
208 /// Convert a decimal string into a floating point number.
209 fn dec2flt<T: RawFloat>(s: &str) -> Result<T, ParseFloatError> {
211 return Err(pfe_empty())
213 let (sign, s) = extract_sign(s);
214 let flt = match parse_decimal(s) {
215 ParseResult::Valid(decimal) => convert(decimal)?,
216 ParseResult::ShortcutToInf => T::INFINITY,
217 ParseResult::ShortcutToZero => T::ZERO,
218 ParseResult::Invalid => match s {
219 "inf" => T::INFINITY,
221 _ => { return Err(pfe_invalid()); }
226 Sign::Positive => Ok(flt),
227 Sign::Negative => Ok(-flt),
231 /// The main workhorse for the decimal-to-float conversion: Orchestrate all the preprocessing
232 /// and figure out which algorithm should do the actual conversion.
233 fn convert<T: RawFloat>(mut decimal: Decimal) -> Result<T, ParseFloatError> {
234 simplify(&mut decimal);
235 if let Some(x) = trivial_cases(&decimal) {
238 // Remove/shift out the decimal point.
239 let e = decimal.exp - decimal.fractional.len() as i64;
240 if let Some(x) = algorithm::fast_path(decimal.integral, decimal.fractional, e) {
243 // Big32x40 is limited to 1280 bits, which translates to about 385 decimal digits.
244 // If we exceed this, we'll crash, so we error out before getting too close (within 10^10).
245 let upper_bound = bound_intermediate_digits(&decimal, e);
246 if upper_bound > 375 {
247 return Err(pfe_invalid());
249 let f = digits_to_big(decimal.integral, decimal.fractional);
251 // Now the exponent certainly fits in 16 bit, which is used throughout the main algorithms.
253 // FIXME These bounds are rather conservative. A more careful analysis of the failure modes
254 // of Bellerophon could allow using it in more cases for a massive speed up.
255 let exponent_in_range = table::MIN_E <= e && e <= table::MAX_E;
256 let value_in_range = upper_bound <= T::MAX_NORMAL_DIGITS as u64;
257 if exponent_in_range && value_in_range {
258 Ok(algorithm::bellerophon(&f, e))
260 Ok(algorithm::algorithm_m(&f, e))
264 // As written, this optimizes badly (see #27130, though it refers to an old version of the code).
265 // `inline(always)` is a workaround for that. There are only two call sites overall and it doesn't
266 // make code size worse.
268 /// Strip zeros where possible, even when this requires changing the exponent
270 fn simplify(decimal: &mut Decimal) {
271 let is_zero = &|&&d: &&u8| -> bool { d == b'0' };
272 // Trimming these zeros does not change anything but may enable the fast path (< 15 digits).
273 let leading_zeros = decimal.integral.iter().take_while(is_zero).count();
274 decimal.integral = &decimal.integral[leading_zeros..];
275 let trailing_zeros = decimal.fractional.iter().rev().take_while(is_zero).count();
276 let end = decimal.fractional.len() - trailing_zeros;
277 decimal.fractional = &decimal.fractional[..end];
278 // Simplify numbers of the form 0.0...x and x...0.0, adjusting the exponent accordingly.
279 // This may not always be a win (possibly pushes some numbers out of the fast path), but it
280 // simplifies other parts significantly (notably, approximating the magnitude of the value).
281 if decimal.integral.is_empty() {
282 let leading_zeros = decimal.fractional.iter().take_while(is_zero).count();
283 decimal.fractional = &decimal.fractional[leading_zeros..];
284 decimal.exp -= leading_zeros as i64;
285 } else if decimal.fractional.is_empty() {
286 let trailing_zeros = decimal.integral.iter().rev().take_while(is_zero).count();
287 let end = decimal.integral.len() - trailing_zeros;
288 decimal.integral = &decimal.integral[..end];
289 decimal.exp += trailing_zeros as i64;
293 /// Quick and dirty upper bound on the size (log10) of the largest value that Algorithm R and
294 /// Algorithm M will compute while working on the given decimal.
295 fn bound_intermediate_digits(decimal: &Decimal, e: i64) -> u64 {
296 // We don't need to worry too much about overflow here thanks to trivial_cases() and the
297 // parser, which filter out the most extreme inputs for us.
298 let f_len: u64 = decimal.integral.len() as u64 + decimal.fractional.len() as u64;
300 // In the case e >= 0, both algorithms compute about `f * 10^e`. Algorithm R proceeds to
301 // do some complicated calculations with this but we can ignore that for the upper bound
302 // because it also reduces the fraction beforehand, so we have plenty of buffer there.
305 // If e < 0, Algorithm R does roughly the same thing, but Algorithm M differs:
306 // It tries to find a positive number k such that `f << k / 10^e` is an in-range
307 // significand. This will result in about `2^53 * f * 10^e` < `10^17 * f * 10^e`.
308 // One input that triggers this is 0.33...33 (375 x 3).
309 f_len + (e.abs() as u64) + 17
313 /// Detect obvious overflows and underflows without even looking at the decimal digits.
314 fn trivial_cases<T: RawFloat>(decimal: &Decimal) -> Option<T> {
315 // There were zeros but they were stripped by simplify()
316 if decimal.integral.is_empty() && decimal.fractional.is_empty() {
317 return Some(T::ZERO);
319 // This is a crude approximation of ceil(log10(the real value)). We don't need to worry too
320 // much about overflow here because the input length is tiny (at least compared to 2^64) and
321 // the parser already handles exponents whose absolute value is greater than 10^18
322 // (which is still 10^19 short of 2^64).
323 let max_place = decimal.exp + decimal.integral.len() as i64;
324 if max_place > T::INF_CUTOFF {
325 return Some(T::INFINITY);
326 } else if max_place < T::ZERO_CUTOFF {
327 return Some(T::ZERO);