What is the time complexity of math.pow(a,n) function in java? Because when we solve the same using recursion, the time complexity is O(n).
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Keith Stein
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2Does this answer your question? [Java Math.pow(a,b) time complexity](https://stackoverflow.com/questions/32418731/java-math-powa-b-time-complexity) – Nuwan Harshakumara Piyarathna May 03 '20 at 04:30
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Here is the link to the current implementation of pow. I don't see any loops or recursion, so the complexity has to be O(1):
/**
* Compute x**y
* n
* Method: Let x = 2 * (1+f)
* 1. Compute and return log2(x) in two pieces:
* log2(x) = w1 + w2,
* where w1 has 53 - 24 = 29 bit trailing zeros.
* 2. Perform y*log2(x) = n+y' by simulating multi-precision
* arithmetic, where |y'| <= 0.5.
* 3. Return x**y = 2**n*exp(y'*log2)
*
* Special cases:
* 1. (anything) ** 0 is 1
* 2. (anything) ** 1 is itself
* 3. (anything) ** NAN is NAN
* 4. NAN ** (anything except 0) is NAN
* 5. +-(|x| > 1) ** +INF is +INF
* 6. +-(|x| > 1) ** -INF is +0
* 7. +-(|x| < 1) ** +INF is +0
* 8. +-(|x| < 1) ** -INF is +INF
* 9. +-1 ** +-INF is NAN
* 10. +0 ** (+anything except 0, NAN) is +0
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
* 12. +0 ** (-anything except 0, NAN) is +INF
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
* 15. +INF ** (+anything except 0,NAN) is +INF
* 16. +INF ** (-anything except 0,NAN) is +0
* 17. -INF ** (anything) = -0 ** (-anything)
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
*
* Accuracy:
* pow(x,y) returns x**y nearly rounded. In particular
* pow(integer,integer)
* always returns the correct integer provided it is
* representable.
*/
public static class Pow {
private Pow() {
throw new UnsupportedOperationException();
}
public static strictfp double compute(final double x, final double y) {
double z;
double r, s, t, u, v, w;
int i, j, k, n;
// y == zero: x**0 = 1
if (y == 0.0)
return 1.0;
// +/-NaN return x + y to propagate NaN significands
if (Double.isNaN(x) || Double.isNaN(y))
return x + y;
final double y_abs = Math.abs(y);
double x_abs = Math.abs(x);
// Special values of y
if (y == 2.0) {
return x * x;
} else if (y == 0.5) {
if (x >= -Double.MAX_VALUE) // Handle x == -infinity later
return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0
} else if (y_abs == 1.0) { // y is +/-1
return (y == 1.0) ? x : 1.0 / x;
} else if (y_abs == INFINITY) { // y is +/-infinity
if (x_abs == 1.0)
return y - y; // inf**+/-1 is NaN
else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0
return (y >= 0) ? y : 0.0;
else // (|x| < 1)**-/+inf = inf, 0
return (y < 0) ? -y : 0.0;
}
final int hx = __HI(x);
int ix = hx & 0x7fffffff;
/*
* When x < 0, determine if y is an odd integer:
* y_is_int = 0 ... y is not an integer
* y_is_int = 1 ... y is an odd int
* y_is_int = 2 ... y is an even int
*/
int y_is_int = 0;
if (hx < 0) {
if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15
y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0
else if (y_abs >= 1.0) { // |y| >= 1.0
long y_abs_as_long = (long) y_abs;
if ( ((double) y_abs_as_long) == y_abs) {
y_is_int = 2 - (int)(y_abs_as_long & 0x1L);
}
}
}
// Special value of x
if (x_abs == 0.0 ||
x_abs == INFINITY ||
x_abs == 1.0) {
z = x_abs; // x is +/-0, +/-inf, +/-1
if (y < 0.0)
z = 1.0/z; // z = (1/|x|)
if (hx < 0) {
if (((ix - 0x3ff00000) | y_is_int) == 0) {
z = (z-z)/(z-z); // (-1)**non-int is NaN
} else if (y_is_int == 1)
z = -1.0 * z; // (x < 0)**odd = -(|x|**odd)
}
return z;
}
n = (hx >> 31) + 1;
// (x < 0)**(non-int) is NaN
if ((n | y_is_int) == 0)
return (x-x)/(x-x);
s = 1.0; // s (sign of result -ve**odd) = -1 else = 1
if ( (n | (y_is_int - 1)) == 0)
s = -1.0; // (-ve)**(odd int)
double p_h, p_l, t1, t2;
// |y| is huge
if (y_abs > 0x1.00000_ffff_ffffp31) { // if |y| > ~2**31
final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2
final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2
final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail
// Over/underflow if x is not close to one
if (x_abs < 0x1.fffff_0000_0000p-1) // |x| < ~0.9999995231628418
return (y < 0.0) ? s * INFINITY : s * 0.0;
if (x_abs > 0x1.00000_ffff_ffffp0) // |x| > ~1.0
return (y > 0.0) ? s * INFINITY : s * 0.0;
/*
* now |1-x| is tiny <= 2**-20, sufficient to compute
* log(x) by x - x^2/2 + x^3/3 - x^4/4
*/
t = x_abs - 1.0; // t has 20 trailing zeros
w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits
v = t * INV_LN2_L - w * INV_LN2;
t1 = u + v;
t1 =__LO(t1, 0);
t2 = v - (t1 - u);
} else {
final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2)
final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp
final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H
double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l;
n = 0;
// Take care of subnormal numbers
if (ix < 0x00100000) {
x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0
n -= 53;
ix = __HI(x_abs);
}
n += ((ix) >> 20) - 0x3ff;
j = ix & 0x000fffff;
// Determine interval
ix = j | 0x3ff00000; // Normalize ix
if (j <= 0x3988E)
k = 0; // |x| <sqrt(3/2)
else if (j < 0xBB67A)
k = 1; // |x| <sqrt(3)
else {
k = 0;
n += 1;
ix -= 0x00100000;
}
x_abs = __HI(x_abs, ix);
// Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5)
final double BP[] = {1.0,
1.5};
final double DP_H[] = {0.0,
0x1.2b80_34p-1}; // 5.84962487220764160156e-01
final double DP_L[] = {0.0,
0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08
// Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
final double L1 = 0x1.3333_3333_33303p-1; // 5.99999999999994648725e-01
final double L2 = 0x1.b6db_6db6_fabffp-2; // 4.28571428578550184252e-01
final double L3 = 0x1.5555_5518_f264dp-2; // 3.33333329818377432918e-01
final double L4 = 0x1.1746_0a91_d4101p-2; // 2.72728123808534006489e-01
final double L5 = 0x1.d864_a93c_9db65p-3; // 2.30660745775561754067e-01
final double L6 = 0x1.a7e2_84a4_54eefp-3; // 2.06975017800338417784e-01
u = x_abs - BP[k]; // BP[0]=1.0, BP[1]=1.5
v = 1.0 / (x_abs + BP[k]);
ss = u * v;
s_h = ss;
s_h = __LO(s_h, 0);
// t_h=x_abs + BP[k] High
t_h = 0.0;
t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) );
t_l = x_abs - (t_h - BP[k]);
s_l = v * ((u - s_h * t_h) - s_h * t_l);
// Compute log(x_abs)
s2 = ss * ss;
r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
r += s_l * (s_h + ss);
s2 = s_h * s_h;
t_h = 3.0 + s2 + r;
t_h = __LO(t_h, 0);
t_l = r - ((t_h - 3.0) - s2);
// u+v = ss*(1+...)
u = s_h * t_h;
v = s_l * t_h + t_l * ss;
// 2/(3log2)*(ss + ...)
p_h = u + v;
p_h = __LO(p_h, 0);
p_l = v - (p_h - u);
z_h = CP_H * p_h; // CP_H + CP_L = 2/(3*log2)
z_l = CP_L * p_h + p_l * CP + DP_L[k];
// log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l
t = (double)n;
t1 = (((z_h + z_l) + DP_H[k]) + t);
t1 = __LO(t1, 0);
t2 = z_l - (((t1 - t) - DP_H[k]) - z_h);
}
// Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2)
double y1 = y;
y1 = __LO(y1, 0);
p_l = (y - y1) * t1 + y * t2;
p_h = y1 * t1;
z = p_l + p_h;
j = __HI(z);
i = __LO(z);
if (j >= 0x40900000) { // z >= 1024
if (((j - 0x40900000) | i)!=0) // if z > 1024
return s * INFINITY; // Overflow
else {
final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp))
if (p_l + OVT > z - p_h)
return s * INFINITY; // Overflow
}
} else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075
if (((j - 0xc090cc00) | i)!=0) // z < -1075
return s * 0.0; // Underflow
else {
if (p_l <= z - p_h)
return s * 0.0; // Underflow
}
}
/*
* Compute 2**(p_h+p_l)
*/
// Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01
final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03
final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05
final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06
final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08
final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01
final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01
final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09
i = j & 0x7fffffff;
k = (i >> 20) - 0x3ff;
n = 0;
if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z + 0.5]
n = j + (0x00100000 >> (k + 1));
k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n
t = 0.0;
t = __HI(t, (n & ~(0x000fffff >> k)) );
n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
if (j < 0)
n = -n;
p_h -= t;
}
t = p_l + p_h;
t = __LO(t, 0);
u = t * LG2_H;
v = (p_l - (t - p_h)) * LG2 + t * LG2_L;
z = u + v;
w = v - (z - u);
t = z * z;
t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
r = (z * t1)/(t1 - 2.0) - (w + z * w);
z = 1.0 - (r - z);
j = __HI(z);
j += (n << 20);
if ((j >> 20) <= 0)
z = Math.scalb(z, n); // subnormal output
else {
int z_hi = __HI(z);
z_hi += (n << 20);
z = __HI(z, z_hi);
}
return s * z;
}
}
Philipp Claßen
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