Spaces:
Sleeping
Sleeping
| from mpmath.libmp import * | |
| from mpmath import * | |
| import random | |
| import time | |
| import math | |
| import cmath | |
| def mpc_ae(a, b, eps=eps): | |
| res = True | |
| res = res and a.real.ae(b.real, eps) | |
| res = res and a.imag.ae(b.imag, eps) | |
| return res | |
| #---------------------------------------------------------------------------- | |
| # Constants and functions | |
| # | |
| tpi = "3.1415926535897932384626433832795028841971693993751058209749445923078\ | |
| 1640628620899862803482534211706798" | |
| te = "2.71828182845904523536028747135266249775724709369995957496696762772407\ | |
| 663035354759457138217852516642743" | |
| tdegree = "0.017453292519943295769236907684886127134428718885417254560971914\ | |
| 4017100911460344944368224156963450948221" | |
| teuler = "0.5772156649015328606065120900824024310421593359399235988057672348\ | |
| 84867726777664670936947063291746749516" | |
| tln2 = "0.693147180559945309417232121458176568075500134360255254120680009493\ | |
| 393621969694715605863326996418687542" | |
| tln10 = "2.30258509299404568401799145468436420760110148862877297603332790096\ | |
| 757260967735248023599720508959829834" | |
| tcatalan = "0.91596559417721901505460351493238411077414937428167213426649811\ | |
| 9621763019776254769479356512926115106249" | |
| tkhinchin = "2.6854520010653064453097148354817956938203822939944629530511523\ | |
| 4555721885953715200280114117493184769800" | |
| tglaisher = "1.2824271291006226368753425688697917277676889273250011920637400\ | |
| 2174040630885882646112973649195820237439420646" | |
| tapery = "1.2020569031595942853997381615114499907649862923404988817922715553\ | |
| 4183820578631309018645587360933525815" | |
| tphi = "1.618033988749894848204586834365638117720309179805762862135448622705\ | |
| 26046281890244970720720418939113748475" | |
| tmertens = "0.26149721284764278375542683860869585905156664826119920619206421\ | |
| 3924924510897368209714142631434246651052" | |
| ttwinprime = "0.660161815846869573927812110014555778432623360284733413319448\ | |
| 423335405642304495277143760031413839867912" | |
| def test_constants(): | |
| for prec in [3, 7, 10, 15, 20, 37, 80, 100, 29]: | |
| mp.dps = prec | |
| assert pi == mpf(tpi) | |
| assert e == mpf(te) | |
| assert degree == mpf(tdegree) | |
| assert euler == mpf(teuler) | |
| assert ln2 == mpf(tln2) | |
| assert ln10 == mpf(tln10) | |
| assert catalan == mpf(tcatalan) | |
| assert khinchin == mpf(tkhinchin) | |
| assert glaisher == mpf(tglaisher) | |
| assert phi == mpf(tphi) | |
| if prec < 50: | |
| assert mertens == mpf(tmertens) | |
| assert twinprime == mpf(ttwinprime) | |
| mp.dps = 15 | |
| assert pi >= -1 | |
| assert pi > 2 | |
| assert pi > 3 | |
| assert pi < 4 | |
| def test_exact_sqrts(): | |
| for i in range(20000): | |
| assert sqrt(mpf(i*i)) == i | |
| random.seed(1) | |
| for prec in [100, 300, 1000, 10000]: | |
| mp.dps = prec | |
| for i in range(20): | |
| A = random.randint(10**(prec//2-2), 10**(prec//2-1)) | |
| assert sqrt(mpf(A*A)) == A | |
| mp.dps = 15 | |
| for i in range(100): | |
| for a in [1, 8, 25, 112307]: | |
| assert sqrt(mpf((a*a, 2*i))) == mpf((a, i)) | |
| assert sqrt(mpf((a*a, -2*i))) == mpf((a, -i)) | |
| def test_sqrt_rounding(): | |
| for i in [2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15]: | |
| i = from_int(i) | |
| for dps in [7, 15, 83, 106, 2000]: | |
| mp.dps = dps | |
| a = mpf_pow_int(mpf_sqrt(i, mp.prec, round_down), 2, mp.prec, round_down) | |
| b = mpf_pow_int(mpf_sqrt(i, mp.prec, round_up), 2, mp.prec, round_up) | |
| assert mpf_lt(a, i) | |
| assert mpf_gt(b, i) | |
| random.seed(1234) | |
| prec = 100 | |
| for rnd in [round_down, round_nearest, round_ceiling]: | |
| for i in range(100): | |
| a = mpf_rand(prec) | |
| b = mpf_mul(a, a) | |
| assert mpf_sqrt(b, prec, rnd) == a | |
| # Test some extreme cases | |
| mp.dps = 100 | |
| a = mpf(9) + 1e-90 | |
| b = mpf(9) - 1e-90 | |
| mp.dps = 15 | |
| assert sqrt(a, rounding='d') == 3 | |
| assert sqrt(a, rounding='n') == 3 | |
| assert sqrt(a, rounding='u') > 3 | |
| assert sqrt(b, rounding='d') < 3 | |
| assert sqrt(b, rounding='n') == 3 | |
| assert sqrt(b, rounding='u') == 3 | |
| # A worst case, from the MPFR test suite | |
| assert sqrt(mpf('7.0503726185518891')) == mpf('2.655253776675949') | |
| def test_float_sqrt(): | |
| mp.dps = 15 | |
| # These should round identically | |
| for x in [0, 1e-7, 0.1, 0.5, 1, 2, 3, 4, 5, 0.333, 76.19]: | |
| assert sqrt(mpf(x)) == float(x)**0.5 | |
| assert sqrt(-1) == 1j | |
| assert sqrt(-2).ae(cmath.sqrt(-2)) | |
| assert sqrt(-3).ae(cmath.sqrt(-3)) | |
| assert sqrt(-100).ae(cmath.sqrt(-100)) | |
| assert sqrt(1j).ae(cmath.sqrt(1j)) | |
| assert sqrt(-1j).ae(cmath.sqrt(-1j)) | |
| assert sqrt(math.pi + math.e*1j).ae(cmath.sqrt(math.pi + math.e*1j)) | |
| assert sqrt(math.pi - math.e*1j).ae(cmath.sqrt(math.pi - math.e*1j)) | |
| def test_hypot(): | |
| assert hypot(0, 0) == 0 | |
| assert hypot(0, 0.33) == mpf(0.33) | |
| assert hypot(0.33, 0) == mpf(0.33) | |
| assert hypot(-0.33, 0) == mpf(0.33) | |
| assert hypot(3, 4) == mpf(5) | |
| def test_exact_cbrt(): | |
| for i in range(0, 20000, 200): | |
| assert cbrt(mpf(i*i*i)) == i | |
| random.seed(1) | |
| for prec in [100, 300, 1000, 10000]: | |
| mp.dps = prec | |
| A = random.randint(10**(prec//2-2), 10**(prec//2-1)) | |
| assert cbrt(mpf(A*A*A)) == A | |
| mp.dps = 15 | |
| def test_exp(): | |
| assert exp(0) == 1 | |
| assert exp(10000).ae(mpf('8.8068182256629215873e4342')) | |
| assert exp(-10000).ae(mpf('1.1354838653147360985e-4343')) | |
| a = exp(mpf((1, 8198646019315405, -53, 53))) | |
| assert(a.bc == bitcount(a.man)) | |
| mp.prec = 67 | |
| a = exp(mpf((1, 1781864658064754565, -60, 61))) | |
| assert(a.bc == bitcount(a.man)) | |
| mp.prec = 53 | |
| assert exp(ln2 * 10).ae(1024) | |
| assert exp(2+2j).ae(cmath.exp(2+2j)) | |
| def test_issue_73(): | |
| mp.dps = 512 | |
| a = exp(-1) | |
| b = exp(1) | |
| mp.dps = 15 | |
| assert (+a).ae(0.36787944117144233) | |
| assert (+b).ae(2.7182818284590451) | |
| def test_log(): | |
| mp.dps = 15 | |
| assert log(1) == 0 | |
| for x in [0.5, 1.5, 2.0, 3.0, 100, 10**50, 1e-50]: | |
| assert log(x).ae(math.log(x)) | |
| assert log(x, x) == 1 | |
| assert log(1024, 2) == 10 | |
| assert log(10**1234, 10) == 1234 | |
| assert log(2+2j).ae(cmath.log(2+2j)) | |
| # Accuracy near 1 | |
| assert (log(0.6+0.8j).real*10**17).ae(2.2204460492503131) | |
| assert (log(0.6-0.8j).real*10**17).ae(2.2204460492503131) | |
| assert (log(0.8-0.6j).real*10**17).ae(2.2204460492503131) | |
| assert (log(1+1e-8j).real*10**16).ae(0.5) | |
| assert (log(1-1e-8j).real*10**16).ae(0.5) | |
| assert (log(-1+1e-8j).real*10**16).ae(0.5) | |
| assert (log(-1-1e-8j).real*10**16).ae(0.5) | |
| assert (log(1j+1e-8).real*10**16).ae(0.5) | |
| assert (log(1j-1e-8).real*10**16).ae(0.5) | |
| assert (log(-1j+1e-8).real*10**16).ae(0.5) | |
| assert (log(-1j-1e-8).real*10**16).ae(0.5) | |
| assert (log(1+1e-40j).real*10**80).ae(0.5) | |
| assert (log(1j+1e-40).real*10**80).ae(0.5) | |
| # Huge | |
| assert log(ldexp(1.234,10**20)).ae(log(2)*1e20) | |
| assert log(ldexp(1.234,10**200)).ae(log(2)*1e200) | |
| # Some special values | |
| assert log(mpc(0,0)) == mpc(-inf,0) | |
| assert isnan(log(mpc(nan,0)).real) | |
| assert isnan(log(mpc(nan,0)).imag) | |
| assert isnan(log(mpc(0,nan)).real) | |
| assert isnan(log(mpc(0,nan)).imag) | |
| assert isnan(log(mpc(nan,1)).real) | |
| assert isnan(log(mpc(nan,1)).imag) | |
| assert isnan(log(mpc(1,nan)).real) | |
| assert isnan(log(mpc(1,nan)).imag) | |
| def test_trig_hyperb_basic(): | |
| for x in (list(range(100)) + list(range(-100,0))): | |
| t = x / 4.1 | |
| assert cos(mpf(t)).ae(math.cos(t)) | |
| assert sin(mpf(t)).ae(math.sin(t)) | |
| assert tan(mpf(t)).ae(math.tan(t)) | |
| assert cosh(mpf(t)).ae(math.cosh(t)) | |
| assert sinh(mpf(t)).ae(math.sinh(t)) | |
| assert tanh(mpf(t)).ae(math.tanh(t)) | |
| assert sin(1+1j).ae(cmath.sin(1+1j)) | |
| assert sin(-4-3.6j).ae(cmath.sin(-4-3.6j)) | |
| assert cos(1+1j).ae(cmath.cos(1+1j)) | |
| assert cos(-4-3.6j).ae(cmath.cos(-4-3.6j)) | |
| def test_degrees(): | |
| assert cos(0*degree) == 1 | |
| assert cos(90*degree).ae(0) | |
| assert cos(180*degree).ae(-1) | |
| assert cos(270*degree).ae(0) | |
| assert cos(360*degree).ae(1) | |
| assert sin(0*degree) == 0 | |
| assert sin(90*degree).ae(1) | |
| assert sin(180*degree).ae(0) | |
| assert sin(270*degree).ae(-1) | |
| assert sin(360*degree).ae(0) | |
| def random_complexes(N): | |
| random.seed(1) | |
| a = [] | |
| for i in range(N): | |
| x1 = random.uniform(-10, 10) | |
| y1 = random.uniform(-10, 10) | |
| x2 = random.uniform(-10, 10) | |
| y2 = random.uniform(-10, 10) | |
| z1 = complex(x1, y1) | |
| z2 = complex(x2, y2) | |
| a.append((z1, z2)) | |
| return a | |
| def test_complex_powers(): | |
| for dps in [15, 30, 100]: | |
| # Check accuracy for complex square root | |
| mp.dps = dps | |
| a = mpc(1j)**0.5 | |
| assert a.real == a.imag == mpf(2)**0.5 / 2 | |
| mp.dps = 15 | |
| random.seed(1) | |
| for (z1, z2) in random_complexes(100): | |
| assert (mpc(z1)**mpc(z2)).ae(z1**z2, 1e-12) | |
| assert (e**(-pi*1j)).ae(-1) | |
| mp.dps = 50 | |
| assert (e**(-pi*1j)).ae(-1) | |
| mp.dps = 15 | |
| def test_complex_sqrt_accuracy(): | |
| def test_mpc_sqrt(lst): | |
| for a, b in lst: | |
| z = mpc(a + j*b) | |
| assert mpc_ae(sqrt(z*z), z) | |
| z = mpc(-a + j*b) | |
| assert mpc_ae(sqrt(z*z), -z) | |
| z = mpc(a - j*b) | |
| assert mpc_ae(sqrt(z*z), z) | |
| z = mpc(-a - j*b) | |
| assert mpc_ae(sqrt(z*z), -z) | |
| random.seed(2) | |
| N = 10 | |
| mp.dps = 30 | |
| dps = mp.dps | |
| test_mpc_sqrt([(random.uniform(0, 10),random.uniform(0, 10)) for i in range(N)]) | |
| test_mpc_sqrt([(i + 0.1, (i + 0.2)*10**i) for i in range(N)]) | |
| mp.dps = 15 | |
| def test_atan(): | |
| mp.dps = 15 | |
| assert atan(-2.3).ae(math.atan(-2.3)) | |
| assert atan(1e-50) == 1e-50 | |
| assert atan(1e50).ae(pi/2) | |
| assert atan(-1e-50) == -1e-50 | |
| assert atan(-1e50).ae(-pi/2) | |
| assert atan(10**1000).ae(pi/2) | |
| for dps in [25, 70, 100, 300, 1000]: | |
| mp.dps = dps | |
| assert (4*atan(1)).ae(pi) | |
| mp.dps = 15 | |
| pi2 = pi/2 | |
| assert atan(mpc(inf,-1)).ae(pi2) | |
| assert atan(mpc(inf,0)).ae(pi2) | |
| assert atan(mpc(inf,1)).ae(pi2) | |
| assert atan(mpc(1,inf)).ae(pi2) | |
| assert atan(mpc(0,inf)).ae(pi2) | |
| assert atan(mpc(-1,inf)).ae(-pi2) | |
| assert atan(mpc(-inf,1)).ae(-pi2) | |
| assert atan(mpc(-inf,0)).ae(-pi2) | |
| assert atan(mpc(-inf,-1)).ae(-pi2) | |
| assert atan(mpc(-1,-inf)).ae(-pi2) | |
| assert atan(mpc(0,-inf)).ae(-pi2) | |
| assert atan(mpc(1,-inf)).ae(pi2) | |
| def test_atan2(): | |
| mp.dps = 15 | |
| assert atan2(1,1).ae(pi/4) | |
| assert atan2(1,-1).ae(3*pi/4) | |
| assert atan2(-1,-1).ae(-3*pi/4) | |
| assert atan2(-1,1).ae(-pi/4) | |
| assert atan2(-1,0).ae(-pi/2) | |
| assert atan2(1,0).ae(pi/2) | |
| assert atan2(0,0) == 0 | |
| assert atan2(inf,0).ae(pi/2) | |
| assert atan2(-inf,0).ae(-pi/2) | |
| assert isnan(atan2(inf,inf)) | |
| assert isnan(atan2(-inf,inf)) | |
| assert isnan(atan2(inf,-inf)) | |
| assert isnan(atan2(3,nan)) | |
| assert isnan(atan2(nan,3)) | |
| assert isnan(atan2(0,nan)) | |
| assert isnan(atan2(nan,0)) | |
| assert atan2(0,inf) == 0 | |
| assert atan2(0,-inf).ae(pi) | |
| assert atan2(10,inf) == 0 | |
| assert atan2(-10,inf) == 0 | |
| assert atan2(-10,-inf).ae(-pi) | |
| assert atan2(10,-inf).ae(pi) | |
| assert atan2(inf,10).ae(pi/2) | |
| assert atan2(inf,-10).ae(pi/2) | |
| assert atan2(-inf,10).ae(-pi/2) | |
| assert atan2(-inf,-10).ae(-pi/2) | |
| def test_areal_inverses(): | |
| assert asin(mpf(0)) == 0 | |
| assert asinh(mpf(0)) == 0 | |
| assert acosh(mpf(1)) == 0 | |
| assert isinstance(asin(mpf(0.5)), mpf) | |
| assert isinstance(asin(mpf(2.0)), mpc) | |
| assert isinstance(acos(mpf(0.5)), mpf) | |
| assert isinstance(acos(mpf(2.0)), mpc) | |
| assert isinstance(atanh(mpf(0.1)), mpf) | |
| assert isinstance(atanh(mpf(1.1)), mpc) | |
| random.seed(1) | |
| for i in range(50): | |
| x = random.uniform(0, 1) | |
| assert asin(mpf(x)).ae(math.asin(x)) | |
| assert acos(mpf(x)).ae(math.acos(x)) | |
| x = random.uniform(-10, 10) | |
| assert asinh(mpf(x)).ae(cmath.asinh(x).real) | |
| assert isinstance(asinh(mpf(x)), mpf) | |
| x = random.uniform(1, 10) | |
| assert acosh(mpf(x)).ae(cmath.acosh(x).real) | |
| assert isinstance(acosh(mpf(x)), mpf) | |
| x = random.uniform(-10, 0.999) | |
| assert isinstance(acosh(mpf(x)), mpc) | |
| x = random.uniform(-1, 1) | |
| assert atanh(mpf(x)).ae(cmath.atanh(x).real) | |
| assert isinstance(atanh(mpf(x)), mpf) | |
| dps = mp.dps | |
| mp.dps = 300 | |
| assert isinstance(asin(0.5), mpf) | |
| mp.dps = 1000 | |
| assert asin(1).ae(pi/2) | |
| assert asin(-1).ae(-pi/2) | |
| mp.dps = dps | |
| def test_invhyperb_inaccuracy(): | |
| mp.dps = 15 | |
| assert (asinh(1e-5)*10**5).ae(0.99999999998333333) | |
| assert (asinh(1e-10)*10**10).ae(1) | |
| assert (asinh(1e-50)*10**50).ae(1) | |
| assert (asinh(-1e-5)*10**5).ae(-0.99999999998333333) | |
| assert (asinh(-1e-10)*10**10).ae(-1) | |
| assert (asinh(-1e-50)*10**50).ae(-1) | |
| assert asinh(10**20).ae(46.744849040440862) | |
| assert asinh(-10**20).ae(-46.744849040440862) | |
| assert (tanh(1e-10)*10**10).ae(1) | |
| assert (tanh(-1e-10)*10**10).ae(-1) | |
| assert (atanh(1e-10)*10**10).ae(1) | |
| assert (atanh(-1e-10)*10**10).ae(-1) | |
| def test_complex_functions(): | |
| for x in (list(range(10)) + list(range(-10,0))): | |
| for y in (list(range(10)) + list(range(-10,0))): | |
| z = complex(x, y)/4.3 + 0.01j | |
| assert exp(mpc(z)).ae(cmath.exp(z)) | |
| assert log(mpc(z)).ae(cmath.log(z)) | |
| assert cos(mpc(z)).ae(cmath.cos(z)) | |
| assert sin(mpc(z)).ae(cmath.sin(z)) | |
| assert tan(mpc(z)).ae(cmath.tan(z)) | |
| assert sinh(mpc(z)).ae(cmath.sinh(z)) | |
| assert cosh(mpc(z)).ae(cmath.cosh(z)) | |
| assert tanh(mpc(z)).ae(cmath.tanh(z)) | |
| def test_complex_inverse_functions(): | |
| mp.dps = 15 | |
| iv.dps = 15 | |
| for (z1, z2) in random_complexes(30): | |
| # apparently cmath uses a different branch, so we | |
| # can't use it for comparison | |
| assert sinh(asinh(z1)).ae(z1) | |
| # | |
| assert acosh(z1).ae(cmath.acosh(z1)) | |
| assert atanh(z1).ae(cmath.atanh(z1)) | |
| assert atan(z1).ae(cmath.atan(z1)) | |
| # the reason we set a big eps here is that the cmath | |
| # functions are inaccurate | |
| assert asin(z1).ae(cmath.asin(z1), rel_eps=1e-12) | |
| assert acos(z1).ae(cmath.acos(z1), rel_eps=1e-12) | |
| one = mpf(1) | |
| for i in range(-9, 10, 3): | |
| for k in range(-9, 10, 3): | |
| a = 0.9*j*10**k + 0.8*one*10**i | |
| b = cos(acos(a)) | |
| assert b.ae(a) | |
| b = sin(asin(a)) | |
| assert b.ae(a) | |
| one = mpf(1) | |
| err = 2*10**-15 | |
| for i in range(-9, 9, 3): | |
| for k in range(-9, 9, 3): | |
| a = -0.9*10**k + j*0.8*one*10**i | |
| b = cosh(acosh(a)) | |
| assert b.ae(a, err) | |
| b = sinh(asinh(a)) | |
| assert b.ae(a, err) | |
| def test_reciprocal_functions(): | |
| assert sec(3).ae(-1.01010866590799375) | |
| assert csc(3).ae(7.08616739573718592) | |
| assert cot(3).ae(-7.01525255143453347) | |
| assert sech(3).ae(0.0993279274194332078) | |
| assert csch(3).ae(0.0998215696688227329) | |
| assert coth(3).ae(1.00496982331368917) | |
| assert asec(3).ae(1.23095941734077468) | |
| assert acsc(3).ae(0.339836909454121937) | |
| assert acot(3).ae(0.321750554396642193) | |
| assert asech(0.5).ae(1.31695789692481671) | |
| assert acsch(3).ae(0.327450150237258443) | |
| assert acoth(3).ae(0.346573590279972655) | |
| assert acot(0).ae(1.5707963267948966192) | |
| assert acoth(0).ae(1.5707963267948966192j) | |
| def test_ldexp(): | |
| mp.dps = 15 | |
| assert ldexp(mpf(2.5), 0) == 2.5 | |
| assert ldexp(mpf(2.5), -1) == 1.25 | |
| assert ldexp(mpf(2.5), 2) == 10 | |
| assert ldexp(mpf('inf'), 3) == mpf('inf') | |
| def test_frexp(): | |
| mp.dps = 15 | |
| assert frexp(0) == (0.0, 0) | |
| assert frexp(9) == (0.5625, 4) | |
| assert frexp(1) == (0.5, 1) | |
| assert frexp(0.2) == (0.8, -2) | |
| assert frexp(1000) == (0.9765625, 10) | |
| def test_aliases(): | |
| assert ln(7) == log(7) | |
| assert log10(3.75) == log(3.75,10) | |
| assert degrees(5.6) == 5.6 / degree | |
| assert radians(5.6) == 5.6 * degree | |
| assert power(-1,0.5) == j | |
| assert fmod(25,7) == 4.0 and isinstance(fmod(25,7), mpf) | |
| def test_arg_sign(): | |
| assert arg(3) == 0 | |
| assert arg(-3).ae(pi) | |
| assert arg(j).ae(pi/2) | |
| assert arg(-j).ae(-pi/2) | |
| assert arg(0) == 0 | |
| assert isnan(atan2(3,nan)) | |
| assert isnan(atan2(nan,3)) | |
| assert isnan(atan2(0,nan)) | |
| assert isnan(atan2(nan,0)) | |
| assert isnan(atan2(nan,nan)) | |
| assert arg(inf) == 0 | |
| assert arg(-inf).ae(pi) | |
| assert isnan(arg(nan)) | |
| #assert arg(inf*j).ae(pi/2) | |
| assert sign(0) == 0 | |
| assert sign(3) == 1 | |
| assert sign(-3) == -1 | |
| assert sign(inf) == 1 | |
| assert sign(-inf) == -1 | |
| assert isnan(sign(nan)) | |
| assert sign(j) == j | |
| assert sign(-3*j) == -j | |
| assert sign(1+j).ae((1+j)/sqrt(2)) | |
| def test_misc_bugs(): | |
| # test that this doesn't raise an exception | |
| mp.dps = 1000 | |
| log(1302) | |
| mp.dps = 15 | |
| def test_arange(): | |
| assert arange(10) == [mpf('0.0'), mpf('1.0'), mpf('2.0'), mpf('3.0'), | |
| mpf('4.0'), mpf('5.0'), mpf('6.0'), mpf('7.0'), | |
| mpf('8.0'), mpf('9.0')] | |
| assert arange(-5, 5) == [mpf('-5.0'), mpf('-4.0'), mpf('-3.0'), | |
| mpf('-2.0'), mpf('-1.0'), mpf('0.0'), | |
| mpf('1.0'), mpf('2.0'), mpf('3.0'), mpf('4.0')] | |
| assert arange(0, 1, 0.1) == [mpf('0.0'), mpf('0.10000000000000001'), | |
| mpf('0.20000000000000001'), | |
| mpf('0.30000000000000004'), | |
| mpf('0.40000000000000002'), | |
| mpf('0.5'), mpf('0.60000000000000009'), | |
| mpf('0.70000000000000007'), | |
| mpf('0.80000000000000004'), | |
| mpf('0.90000000000000002')] | |
| assert arange(17, -9, -3) == [mpf('17.0'), mpf('14.0'), mpf('11.0'), | |
| mpf('8.0'), mpf('5.0'), mpf('2.0'), | |
| mpf('-1.0'), mpf('-4.0'), mpf('-7.0')] | |
| assert arange(0.2, 0.1, -0.1) == [mpf('0.20000000000000001')] | |
| assert arange(0) == [] | |
| assert arange(1000, -1) == [] | |
| assert arange(-1.23, 3.21, -0.0000001) == [] | |
| def test_linspace(): | |
| assert linspace(2, 9, 7) == [mpf('2.0'), mpf('3.166666666666667'), | |
| mpf('4.3333333333333339'), mpf('5.5'), mpf('6.666666666666667'), | |
| mpf('7.8333333333333339'), mpf('9.0')] | |
| assert linspace(2, 9, 7, endpoint=0) == [mpf('2.0'), mpf('3.0'), mpf('4.0'), | |
| mpf('5.0'), mpf('6.0'), mpf('7.0'), mpf('8.0')] | |
| assert linspace(2, 7, 1) == [mpf(2)] | |
| def test_float_cbrt(): | |
| mp.dps = 30 | |
| for a in arange(0,10,0.1): | |
| assert cbrt(a*a*a).ae(a, eps) | |
| assert cbrt(-1).ae(0.5 + j*sqrt(3)/2) | |
| one_third = mpf(1)/3 | |
| for a in arange(0,10,2.7) + [0.1 + 10**5]: | |
| a = mpc(a + 1.1j) | |
| r1 = cbrt(a) | |
| mp.dps += 10 | |
| r2 = pow(a, one_third) | |
| mp.dps -= 10 | |
| assert r1.ae(r2, eps) | |
| mp.dps = 100 | |
| for n in range(100, 301, 100): | |
| w = 10**n + j*10**-3 | |
| z = w*w*w | |
| r = cbrt(z) | |
| assert mpc_ae(r, w, eps) | |
| mp.dps = 15 | |
| def test_root(): | |
| mp.dps = 30 | |
| random.seed(1) | |
| a = random.randint(0, 10000) | |
| p = a*a*a | |
| r = nthroot(mpf(p), 3) | |
| assert r == a | |
| for n in range(4, 10): | |
| p = p*a | |
| assert nthroot(mpf(p), n) == a | |
| mp.dps = 40 | |
| for n in range(10, 5000, 100): | |
| for a in [random.random()*10000, random.random()*10**100]: | |
| r = nthroot(a, n) | |
| r1 = pow(a, mpf(1)/n) | |
| assert r.ae(r1) | |
| r = nthroot(a, -n) | |
| r1 = pow(a, -mpf(1)/n) | |
| assert r.ae(r1) | |
| # XXX: this is broken right now | |
| # tests for nthroot rounding | |
| for rnd in ['nearest', 'up', 'down']: | |
| mp.rounding = rnd | |
| for n in [-5, -3, 3, 5]: | |
| prec = 50 | |
| for i in range(10): | |
| mp.prec = prec | |
| a = rand() | |
| mp.prec = 2*prec | |
| b = a**n | |
| mp.prec = prec | |
| r = nthroot(b, n) | |
| assert r == a | |
| mp.dps = 30 | |
| for n in range(3, 21): | |
| a = (random.random() + j*random.random()) | |
| assert nthroot(a, n).ae(pow(a, mpf(1)/n)) | |
| assert mpc_ae(nthroot(a, n), pow(a, mpf(1)/n)) | |
| a = (random.random()*10**100 + j*random.random()) | |
| r = nthroot(a, n) | |
| mp.dps += 4 | |
| r1 = pow(a, mpf(1)/n) | |
| mp.dps -= 4 | |
| assert r.ae(r1) | |
| assert mpc_ae(r, r1, eps) | |
| r = nthroot(a, -n) | |
| mp.dps += 4 | |
| r1 = pow(a, -mpf(1)/n) | |
| mp.dps -= 4 | |
| assert r.ae(r1) | |
| assert mpc_ae(r, r1, eps) | |
| mp.dps = 15 | |
| assert nthroot(4, 1) == 4 | |
| assert nthroot(4, 0) == 1 | |
| assert nthroot(4, -1) == 0.25 | |
| assert nthroot(inf, 1) == inf | |
| assert nthroot(inf, 2) == inf | |
| assert nthroot(inf, 3) == inf | |
| assert nthroot(inf, -1) == 0 | |
| assert nthroot(inf, -2) == 0 | |
| assert nthroot(inf, -3) == 0 | |
| assert nthroot(j, 1) == j | |
| assert nthroot(j, 0) == 1 | |
| assert nthroot(j, -1) == -j | |
| assert isnan(nthroot(nan, 1)) | |
| assert isnan(nthroot(nan, 0)) | |
| assert isnan(nthroot(nan, -1)) | |
| assert isnan(nthroot(inf, 0)) | |
| assert root(2,3) == nthroot(2,3) | |
| assert root(16,4,0) == 2 | |
| assert root(16,4,1) == 2j | |
| assert root(16,4,2) == -2 | |
| assert root(16,4,3) == -2j | |
| assert root(16,4,4) == 2 | |
| assert root(-125,3,1) == -5 | |
| def test_issue_136(): | |
| for dps in [20, 80]: | |
| mp.dps = dps | |
| r = nthroot(mpf('-1e-20'), 4) | |
| assert r.ae(mpf(10)**(-5) * (1 + j) * mpf(2)**(-0.5)) | |
| mp.dps = 80 | |
| assert nthroot('-1e-3', 4).ae(mpf(10)**(-3./4) * (1 + j)/sqrt(2)) | |
| assert nthroot('-1e-6', 4).ae((1 + j)/(10 * sqrt(20))) | |
| # Check that this doesn't take eternity to compute | |
| mp.dps = 20 | |
| assert nthroot('-1e100000000', 4).ae((1+j)*mpf('1e25000000')/sqrt(2)) | |
| mp.dps = 15 | |
| def test_mpcfun_real_imag(): | |
| mp.dps = 15 | |
| x = mpf(0.3) | |
| y = mpf(0.4) | |
| assert exp(mpc(x,0)) == exp(x) | |
| assert exp(mpc(0,y)) == mpc(cos(y),sin(y)) | |
| assert cos(mpc(x,0)) == cos(x) | |
| assert sin(mpc(x,0)) == sin(x) | |
| assert cos(mpc(0,y)) == cosh(y) | |
| assert sin(mpc(0,y)) == mpc(0,sinh(y)) | |
| assert cospi(mpc(x,0)) == cospi(x) | |
| assert sinpi(mpc(x,0)) == sinpi(x) | |
| assert cospi(mpc(0,y)).ae(cosh(pi*y)) | |
| assert sinpi(mpc(0,y)).ae(mpc(0,sinh(pi*y))) | |
| c, s = cospi_sinpi(mpc(x,0)) | |
| assert c == cospi(x) | |
| assert s == sinpi(x) | |
| c, s = cospi_sinpi(mpc(0,y)) | |
| assert c.ae(cosh(pi*y)) | |
| assert s.ae(mpc(0,sinh(pi*y))) | |
| c, s = cos_sin(mpc(x,0)) | |
| assert c == cos(x) | |
| assert s == sin(x) | |
| c, s = cos_sin(mpc(0,y)) | |
| assert c == cosh(y) | |
| assert s == mpc(0,sinh(y)) | |
| def test_perturbation_rounding(): | |
| mp.dps = 100 | |
| a = pi/10**50 | |
| b = -pi/10**50 | |
| c = 1 + a | |
| d = 1 + b | |
| mp.dps = 15 | |
| assert exp(a) == 1 | |
| assert exp(a, rounding='c') > 1 | |
| assert exp(b, rounding='c') == 1 | |
| assert exp(a, rounding='f') == 1 | |
| assert exp(b, rounding='f') < 1 | |
| assert cos(a) == 1 | |
| assert cos(a, rounding='c') == 1 | |
| assert cos(b, rounding='c') == 1 | |
| assert cos(a, rounding='f') < 1 | |
| assert cos(b, rounding='f') < 1 | |
| for f in [sin, atan, asinh, tanh]: | |
| assert f(a) == +a | |
| assert f(a, rounding='c') > a | |
| assert f(a, rounding='f') < a | |
| assert f(b) == +b | |
| assert f(b, rounding='c') > b | |
| assert f(b, rounding='f') < b | |
| for f in [asin, tan, sinh, atanh]: | |
| assert f(a) == +a | |
| assert f(b) == +b | |
| assert f(a, rounding='c') > a | |
| assert f(b, rounding='c') > b | |
| assert f(a, rounding='f') < a | |
| assert f(b, rounding='f') < b | |
| assert ln(c) == +a | |
| assert ln(d) == +b | |
| assert ln(c, rounding='c') > a | |
| assert ln(c, rounding='f') < a | |
| assert ln(d, rounding='c') > b | |
| assert ln(d, rounding='f') < b | |
| assert cosh(a) == 1 | |
| assert cosh(b) == 1 | |
| assert cosh(a, rounding='c') > 1 | |
| assert cosh(b, rounding='c') > 1 | |
| assert cosh(a, rounding='f') == 1 | |
| assert cosh(b, rounding='f') == 1 | |
| def test_integer_parts(): | |
| assert floor(3.2) == 3 | |
| assert ceil(3.2) == 4 | |
| assert floor(3.2+5j) == 3+5j | |
| assert ceil(3.2+5j) == 4+5j | |
| def test_complex_parts(): | |
| assert fabs('3') == 3 | |
| assert fabs(3+4j) == 5 | |
| assert re(3) == 3 | |
| assert re(1+4j) == 1 | |
| assert im(3) == 0 | |
| assert im(1+4j) == 4 | |
| assert conj(3) == 3 | |
| assert conj(3+4j) == 3-4j | |
| assert mpf(3).conjugate() == 3 | |
| def test_cospi_sinpi(): | |
| assert sinpi(0) == 0 | |
| assert sinpi(0.5) == 1 | |
| assert sinpi(1) == 0 | |
| assert sinpi(1.5) == -1 | |
| assert sinpi(2) == 0 | |
| assert sinpi(2.5) == 1 | |
| assert sinpi(-0.5) == -1 | |
| assert cospi(0) == 1 | |
| assert cospi(0.5) == 0 | |
| assert cospi(1) == -1 | |
| assert cospi(1.5) == 0 | |
| assert cospi(2) == 1 | |
| assert cospi(2.5) == 0 | |
| assert cospi(-0.5) == 0 | |
| assert cospi(100000000000.25).ae(sqrt(2)/2) | |
| a = cospi(2+3j) | |
| assert a.real.ae(cos((2+3j)*pi).real) | |
| assert a.imag == 0 | |
| b = sinpi(2+3j) | |
| assert b.imag.ae(sin((2+3j)*pi).imag) | |
| assert b.real == 0 | |
| mp.dps = 35 | |
| x1 = mpf(10000) - mpf('1e-15') | |
| x2 = mpf(10000) + mpf('1e-15') | |
| x3 = mpf(10000.5) - mpf('1e-15') | |
| x4 = mpf(10000.5) + mpf('1e-15') | |
| x5 = mpf(10001) - mpf('1e-15') | |
| x6 = mpf(10001) + mpf('1e-15') | |
| x7 = mpf(10001.5) - mpf('1e-15') | |
| x8 = mpf(10001.5) + mpf('1e-15') | |
| mp.dps = 15 | |
| M = 10**15 | |
| assert (sinpi(x1)*M).ae(-pi) | |
| assert (sinpi(x2)*M).ae(pi) | |
| assert (cospi(x3)*M).ae(pi) | |
| assert (cospi(x4)*M).ae(-pi) | |
| assert (sinpi(x5)*M).ae(pi) | |
| assert (sinpi(x6)*M).ae(-pi) | |
| assert (cospi(x7)*M).ae(-pi) | |
| assert (cospi(x8)*M).ae(pi) | |
| assert 0.999 < cospi(x1, rounding='d') < 1 | |
| assert 0.999 < cospi(x2, rounding='d') < 1 | |
| assert 0.999 < sinpi(x3, rounding='d') < 1 | |
| assert 0.999 < sinpi(x4, rounding='d') < 1 | |
| assert -1 < cospi(x5, rounding='d') < -0.999 | |
| assert -1 < cospi(x6, rounding='d') < -0.999 | |
| assert -1 < sinpi(x7, rounding='d') < -0.999 | |
| assert -1 < sinpi(x8, rounding='d') < -0.999 | |
| assert (sinpi(1e-15)*M).ae(pi) | |
| assert (sinpi(-1e-15)*M).ae(-pi) | |
| assert cospi(1e-15) == 1 | |
| assert cospi(1e-15, rounding='d') < 1 | |
| def test_expj(): | |
| assert expj(0) == 1 | |
| assert expj(1).ae(exp(j)) | |
| assert expj(j).ae(exp(-1)) | |
| assert expj(1+j).ae(exp(j*(1+j))) | |
| assert expjpi(0) == 1 | |
| assert expjpi(1).ae(exp(j*pi)) | |
| assert expjpi(j).ae(exp(-pi)) | |
| assert expjpi(1+j).ae(exp(j*pi*(1+j))) | |
| assert expjpi(-10**15 * j).ae('2.22579818340535731e+1364376353841841') | |
| def test_sinc(): | |
| assert sinc(0) == sincpi(0) == 1 | |
| assert sinc(inf) == sincpi(inf) == 0 | |
| assert sinc(-inf) == sincpi(-inf) == 0 | |
| assert sinc(2).ae(0.45464871341284084770) | |
| assert sinc(2+3j).ae(0.4463290318402435457-2.7539470277436474940j) | |
| assert sincpi(2) == 0 | |
| assert sincpi(1.5).ae(-0.212206590789193781) | |
| def test_fibonacci(): | |
| mp.dps = 15 | |
| assert [fibonacci(n) for n in range(-5, 10)] == \ | |
| [5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34] | |
| assert fib(2.5).ae(1.4893065462657091) | |
| assert fib(3+4j).ae(-5248.51130728372 - 14195.962288353j) | |
| assert fib(1000).ae(4.3466557686937455e+208) | |
| assert str(fib(10**100)) == '6.24499112864607e+2089876402499787337692720892375554168224592399182109535392875613974104853496745963277658556235103534' | |
| mp.dps = 2100 | |
| a = fib(10000) | |
| assert a % 10**10 == 9947366875 | |
| mp.dps = 15 | |
| assert fibonacci(inf) == inf | |
| assert fib(3+0j) == 2 | |
| def test_call_with_dps(): | |
| mp.dps = 15 | |
| assert abs(exp(1, dps=30)-e(dps=35)) < 1e-29 | |
| def test_tanh(): | |
| mp.dps = 15 | |
| assert tanh(0) == 0 | |
| assert tanh(inf) == 1 | |
| assert tanh(-inf) == -1 | |
| assert isnan(tanh(nan)) | |
| assert tanh(mpc('inf', '0')) == 1 | |
| def test_atanh(): | |
| mp.dps = 15 | |
| assert atanh(0) == 0 | |
| assert atanh(0.5).ae(0.54930614433405484570) | |
| assert atanh(-0.5).ae(-0.54930614433405484570) | |
| assert atanh(1) == inf | |
| assert atanh(-1) == -inf | |
| assert isnan(atanh(nan)) | |
| assert isinstance(atanh(1), mpf) | |
| assert isinstance(atanh(-1), mpf) | |
| # Limits at infinity | |
| jpi2 = j*pi/2 | |
| assert atanh(inf).ae(-jpi2) | |
| assert atanh(-inf).ae(jpi2) | |
| assert atanh(mpc(inf,-1)).ae(-jpi2) | |
| assert atanh(mpc(inf,0)).ae(-jpi2) | |
| assert atanh(mpc(inf,1)).ae(jpi2) | |
| assert atanh(mpc(1,inf)).ae(jpi2) | |
| assert atanh(mpc(0,inf)).ae(jpi2) | |
| assert atanh(mpc(-1,inf)).ae(jpi2) | |
| assert atanh(mpc(-inf,1)).ae(jpi2) | |
| assert atanh(mpc(-inf,0)).ae(jpi2) | |
| assert atanh(mpc(-inf,-1)).ae(-jpi2) | |
| assert atanh(mpc(-1,-inf)).ae(-jpi2) | |
| assert atanh(mpc(0,-inf)).ae(-jpi2) | |
| assert atanh(mpc(1,-inf)).ae(-jpi2) | |
| def test_expm1(): | |
| mp.dps = 15 | |
| assert expm1(0) == 0 | |
| assert expm1(3).ae(exp(3)-1) | |
| assert expm1(inf) == inf | |
| assert expm1(1e-50).ae(1e-50) | |
| assert (expm1(1e-10)*1e10).ae(1.00000000005) | |
| def test_log1p(): | |
| mp.dps = 15 | |
| assert log1p(0) == 0 | |
| assert log1p(3).ae(log(1+3)) | |
| assert log1p(inf) == inf | |
| assert log1p(1e-50).ae(1e-50) | |
| assert (log1p(1e-10)*1e10).ae(0.99999999995) | |
| def test_powm1(): | |
| mp.dps = 15 | |
| assert powm1(2,3) == 7 | |
| assert powm1(-1,2) == 0 | |
| assert powm1(-1,0) == 0 | |
| assert powm1(-2,0) == 0 | |
| assert powm1(3+4j,0) == 0 | |
| assert powm1(0,1) == -1 | |
| assert powm1(0,0) == 0 | |
| assert powm1(1,0) == 0 | |
| assert powm1(1,2) == 0 | |
| assert powm1(1,3+4j) == 0 | |
| assert powm1(1,5) == 0 | |
| assert powm1(j,4) == 0 | |
| assert powm1(-j,4) == 0 | |
| assert (powm1(2,1e-100)*1e100).ae(ln2) | |
| assert powm1(2,'1e-100000000000') != 0 | |
| assert (powm1(fadd(1,1e-100,exact=True), 5)*1e100).ae(5) | |
| def test_unitroots(): | |
| assert unitroots(1) == [1] | |
| assert unitroots(2) == [1, -1] | |
| a, b, c = unitroots(3) | |
| assert a == 1 | |
| assert b.ae(-0.5 + 0.86602540378443864676j) | |
| assert c.ae(-0.5 - 0.86602540378443864676j) | |
| assert unitroots(1, primitive=True) == [1] | |
| assert unitroots(2, primitive=True) == [-1] | |
| assert unitroots(3, primitive=True) == unitroots(3)[1:] | |
| assert unitroots(4, primitive=True) == [j, -j] | |
| assert len(unitroots(17, primitive=True)) == 16 | |
| assert len(unitroots(16, primitive=True)) == 8 | |
| def test_cyclotomic(): | |
| mp.dps = 15 | |
| assert [cyclotomic(n,1) for n in range(31)] == [1,0,2,3,2,5,1,7,2,3,1,11,1,13,1,1,2,17,1,19,1,1,1,23,1,5,1,3,1,29,1] | |
| assert [cyclotomic(n,-1) for n in range(31)] == [1,-2,0,1,2,1,3,1,2,1,5,1,1,1,7,1,2,1,3,1,1,1,11,1,1,1,13,1,1,1,1] | |
| assert [cyclotomic(n,j) for n in range(21)] == [1,-1+j,1+j,j,0,1,-j,j,2,-j,1,j,3,1,-j,1,2,1,j,j,5] | |
| assert [cyclotomic(n,-j) for n in range(21)] == [1,-1-j,1-j,-j,0,1,j,-j,2,j,1,-j,3,1,j,1,2,1,-j,-j,5] | |
| assert cyclotomic(1624,j) == 1 | |
| assert cyclotomic(33600,j) == 1 | |
| u = sqrt(j, prec=500) | |
| assert cyclotomic(8, u).ae(0) | |
| assert cyclotomic(30, u).ae(5.8284271247461900976) | |
| assert cyclotomic(2040, u).ae(1) | |
| assert cyclotomic(0,2.5) == 1 | |
| assert cyclotomic(1,2.5) == 2.5-1 | |
| assert cyclotomic(2,2.5) == 2.5+1 | |
| assert cyclotomic(3,2.5) == 2.5**2 + 2.5 + 1 | |
| assert cyclotomic(7,2.5) == 406.234375 | |