pow.go

     1  // Copyright 2009 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  package math
     6  
     7  func isOddInt(x float64) bool {
     8  	if Abs(x) >= (1 << 53) {
     9  		// 1 << 53 is the largest exact integer in the float64 format.
    10  		// Any number outside this range will be truncated before the decimal point and therefore will always be
    11  		// an even integer.
    12  		// Without this check and if x overflows int64 the int64(xi) conversion below may produce incorrect results
    13  		// on some architectures (and does so on arm64). See issue #57465.
    14  		return false
    15  	}
    16  
    17  	xi, xf := Modf(x)
    18  	return xf == 0 && int64(xi)&1 == 1
    19  }
    20  
    21  // Special cases taken from FreeBSD's /usr/src/lib/msun/src/e_pow.c
    22  // updated by IEEE Std. 754-2008 "Section 9.2.1 Special values".
    23  
    24  // Pow returns x**y, the base-x exponential of y.
    25  //
    26  // Special cases are (in order):
    27  //
    28  //	Pow(x, ±0) = 1 for any x
    29  //	Pow(1, y) = 1 for any y
    30  //	Pow(x, 1) = x for any x
    31  //	Pow(NaN, y) = NaN
    32  //	Pow(x, NaN) = NaN
    33  //	Pow(±0, y) = ±Inf for y an odd integer < 0
    34  //	Pow(±0, -Inf) = +Inf
    35  //	Pow(±0, +Inf) = +0
    36  //	Pow(±0, y) = +Inf for finite y < 0 and not an odd integer
    37  //	Pow(±0, y) = ±0 for y an odd integer > 0
    38  //	Pow(±0, y) = +0 for finite y > 0 and not an odd integer
    39  //	Pow(-1, ±Inf) = 1
    40  //	Pow(x, +Inf) = +Inf for |x| > 1
    41  //	Pow(x, -Inf) = +0 for |x| > 1
    42  //	Pow(x, +Inf) = +0 for |x| < 1
    43  //	Pow(x, -Inf) = +Inf for |x| < 1
    44  //	Pow(+Inf, y) = +Inf for y > 0
    45  //	Pow(+Inf, y) = +0 for y < 0
    46  //	Pow(-Inf, y) = Pow(-0, -y)
    47  //	Pow(x, y) = NaN for finite x < 0 and finite non-integer y
    48  func Pow(x, y float64) float64 {
    49  	if haveArchPow {
    50  		return archPow(x, y)
    51  	}
    52  	return pow(x, y)
    53  }
    54  
    55  func pow(x, y float64) float64 {
    56  	switch {
    57  	case y == 0 || x == 1:
    58  		return 1
    59  	case y == 1:
    60  		return x
    61  	case IsNaN(x) || IsNaN(y):
    62  		return NaN()
    63  	case x == 0:
    64  		switch {
    65  		case y < 0:
    66  			if Signbit(x) && isOddInt(y) {
    67  				return Inf(-1)
    68  			}
    69  			return Inf(1)
    70  		case y > 0:
    71  			if Signbit(x) && isOddInt(y) {
    72  				return x
    73  			}
    74  			return 0
    75  		}
    76  	case IsInf(y, 0):
    77  		switch {
    78  		case x == -1:
    79  			return 1
    80  		case (Abs(x) < 1) == IsInf(y, 1):
    81  			return 0
    82  		default:
    83  			return Inf(1)
    84  		}
    85  	case IsInf(x, 0):
    86  		if IsInf(x, -1) {
    87  			return Pow(1/x, -y) // Pow(-0, -y)
    88  		}
    89  		switch {
    90  		case y < 0:
    91  			return 0
    92  		case y > 0:
    93  			return Inf(1)
    94  		}
    95  	case y == 0.5:
    96  		return Sqrt(x)
    97  	case y == -0.5:
    98  		return 1 / Sqrt(x)
    99  	}
   100  
   101  	yi, yf := Modf(Abs(y))
   102  	if yf != 0 && x < 0 {
   103  		return NaN()
   104  	}
   105  	if yi >= 1<<63 {
   106  		// yi is a large even int that will lead to overflow (or underflow to 0)
   107  		// for all x except -1 (x == 1 was handled earlier)
   108  		switch {
   109  		case x == -1:
   110  			return 1
   111  		case (Abs(x) < 1) == (y > 0):
   112  			return 0
   113  		default:
   114  			return Inf(1)
   115  		}
   116  	}
   117  
   118  	// ans = a1 * 2**ae (= 1 for now).
   119  	a1 := 1.0
   120  	ae := 0
   121  
   122  	// ans *= x**yf
   123  	if yf != 0 {
   124  		if yf > 0.5 {
   125  			yf--
   126  			yi++
   127  		}
   128  		a1 = Exp(yf * Log(x))
   129  	}
   130  
   131  	// ans *= x**yi
   132  	// by multiplying in successive squarings
   133  	// of x according to bits of yi.
   134  	// accumulate powers of two into exp.
   135  	x1, xe := Frexp(x)
   136  	for i := int64(yi); i != 0; i >>= 1 {
   137  		if xe < -1<<12 || 1<<12 < xe {
   138  			// catch xe before it overflows the left shift below
   139  			// Since i !=0 it has at least one bit still set, so ae will accumulate xe
   140  			// on at least one more iteration, ae += xe is a lower bound on ae
   141  			// the lower bound on ae exceeds the size of a float64 exp
   142  			// so the final call to Ldexp will produce under/overflow (0/Inf)
   143  			ae += xe
   144  			break
   145  		}
   146  		if i&1 == 1 {
   147  			a1 *= x1
   148  			ae += xe
   149  		}
   150  		x1 *= x1
   151  		xe <<= 1
   152  		if x1 < .5 {
   153  			x1 += x1
   154  			xe--
   155  		}
   156  	}
   157  
   158  	// ans = a1*2**ae
   159  	// if y < 0 { ans = 1 / ans }
   160  	// but in the opposite order
   161  	if y < 0 {
   162  		a1 = 1 / a1
   163  		ae = -ae
   164  	}
   165  	return Ldexp(a1, ae)
   166  }
   167
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