This file is indexed.

/usr/share/pyshared/mpmath/functions/expintegrals.py is in python-mpmath 0.18-1.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
from .functions import defun, defun_wrapped

@defun_wrapped
def _erf_complex(ctx, z):
    z2 = ctx.square_exp_arg(z, -1)
    #z2 = -z**2
    v = (2/ctx.sqrt(ctx.pi))*z * ctx.hyp1f1((1,2),(3,2), z2)
    if not ctx._re(z):
        v = ctx._im(v)*ctx.j
    return v

@defun_wrapped
def _erfc_complex(ctx, z):
    if ctx.re(z) > 2:
        z2 = ctx.square_exp_arg(z)
        nz2 = ctx.fneg(z2, exact=True)
        v = ctx.exp(nz2)/ctx.sqrt(ctx.pi) * ctx.hyperu((1,2),(1,2), z2)
    else:
        v = 1 - ctx._erf_complex(z)
    if not ctx._re(z):
        v = 1+ctx._im(v)*ctx.j
    return v

@defun
def erf(ctx, z):
    z = ctx.convert(z)
    if ctx._is_real_type(z):
        try:
            return ctx._erf(z)
        except NotImplementedError:
            pass
    if ctx._is_complex_type(z) and not z.imag:
        try:
            return type(z)(ctx._erf(z.real))
        except NotImplementedError:
            pass
    return ctx._erf_complex(z)

@defun
def erfc(ctx, z):
    z = ctx.convert(z)
    if ctx._is_real_type(z):
        try:
            return ctx._erfc(z)
        except NotImplementedError:
            pass
    if ctx._is_complex_type(z) and not z.imag:
        try:
            return type(z)(ctx._erfc(z.real))
        except NotImplementedError:
            pass
    return ctx._erfc_complex(z)

@defun
def square_exp_arg(ctx, z, mult=1, reciprocal=False):
    prec = ctx.prec*4+20
    if reciprocal:
        z2 = ctx.fmul(z, z, prec=prec)
        z2 = ctx.fdiv(ctx.one, z2, prec=prec)
    else:
        z2 = ctx.fmul(z, z, prec=prec)
    if mult != 1:
        z2 = ctx.fmul(z2, mult, exact=True)
    return z2

@defun_wrapped
def erfi(ctx, z):
    if not z:
        return z
    z2 = ctx.square_exp_arg(z)
    v = (2/ctx.sqrt(ctx.pi)*z) * ctx.hyp1f1((1,2), (3,2), z2)
    if not ctx._re(z):
        v = ctx._im(v)*ctx.j
    return v

@defun_wrapped
def erfinv(ctx, x):
    xre = ctx._re(x)
    if (xre != x) or (xre < -1) or (xre > 1):
        return ctx.bad_domain("erfinv(x) is defined only for -1 <= x <= 1")
    x = xre
    #if ctx.isnan(x): return x
    if not x: return x
    if x == 1: return ctx.inf
    if x == -1: return ctx.ninf
    if abs(x) < 0.9:
        a = 0.53728*x**3 + 0.813198*x
    else:
        # An asymptotic formula
        u = ctx.ln(2/ctx.pi/(abs(x)-1)**2)
        a = ctx.sign(x) * ctx.sqrt(u - ctx.ln(u))/ctx.sqrt(2)
    ctx.prec += 10
    return ctx.findroot(lambda t: ctx.erf(t)-x, a)

@defun_wrapped
def npdf(ctx, x, mu=0, sigma=1):
    sigma = ctx.convert(sigma)
    return ctx.exp(-(x-mu)**2/(2*sigma**2)) / (sigma*ctx.sqrt(2*ctx.pi))

@defun_wrapped
def ncdf(ctx, x, mu=0, sigma=1):
    a = (x-mu)/(sigma*ctx.sqrt(2))
    if a < 0:
        return ctx.erfc(-a)/2
    else:
        return (1+ctx.erf(a))/2

@defun_wrapped
def betainc(ctx, a, b, x1=0, x2=1, regularized=False):
    if x1 == x2:
        v = 0
    elif not x1:
        if x1 == 0 and x2 == 1:
            v = ctx.beta(a, b)
        else:
            v = x2**a * ctx.hyp2f1(a, 1-b, a+1, x2) / a
    else:
        m, d = ctx.nint_distance(a)
        if m <= 0:
            if d < -ctx.prec:
                h = +ctx.eps
                ctx.prec *= 2
                a += h
            elif d < -4:
                ctx.prec -= d
        s1 = x2**a * ctx.hyp2f1(a,1-b,a+1,x2)
        s2 = x1**a * ctx.hyp2f1(a,1-b,a+1,x1)
        v = (s1 - s2) / a
    if regularized:
        v /= ctx.beta(a,b)
    return v

@defun
def gammainc(ctx, z, a=0, b=None, regularized=False):
    regularized = bool(regularized)
    z = ctx.convert(z)
    if a is None:
        a = ctx.zero
        lower_modified = False
    else:
        a = ctx.convert(a)
        lower_modified = a != ctx.zero
    if b is None:
        b = ctx.inf
        upper_modified = False
    else:
        b = ctx.convert(b)
        upper_modified = b != ctx.inf
    # Complete gamma function
    if not (upper_modified or lower_modified):
        if regularized:
            if ctx.re(z) < 0:
                return ctx.inf
            elif ctx.re(z) > 0:
                return ctx.one
            else:
                return ctx.nan
        return ctx.gamma(z)
    if a == b:
        return ctx.zero
    # Standardize
    if ctx.re(a) > ctx.re(b):
        return -ctx.gammainc(z, b, a, regularized)
    # Generalized gamma
    if upper_modified and lower_modified:
        return +ctx._gamma3(z, a, b, regularized)
    # Upper gamma
    elif lower_modified:
        return ctx._upper_gamma(z, a, regularized)
    # Lower gamma
    elif upper_modified:
        return ctx._lower_gamma(z, b, regularized)

@defun
def _lower_gamma(ctx, z, b, regularized=False):
    # Pole
    if ctx.isnpint(z):
        return type(z)(ctx.inf)
    G = [z] * regularized
    negb = ctx.fneg(b, exact=True)
    def h(z):
        T1 = [ctx.exp(negb), b, z], [1, z, -1], [], G, [1], [1+z], b
        return (T1,)
    return ctx.hypercomb(h, [z])

@defun
def _upper_gamma(ctx, z, a, regularized=False):
    # Fast integer case, when available
    if ctx.isint(z):
        try:
            if regularized:
                # Gamma pole
                if ctx.isnpint(z):
                    return type(z)(ctx.zero)
                orig = ctx.prec
                try:
                    ctx.prec += 10
                    return ctx._gamma_upper_int(z, a) / ctx.gamma(z)
                finally:
                    ctx.prec = orig
            else:
                return ctx._gamma_upper_int(z, a)
        except NotImplementedError:
            pass
    nega = ctx.fneg(a, exact=True)
    G = [z] * regularized
    # Use 2F0 series when possible; fall back to lower gamma representation
    try:
        def h(z):
            r = z-1
            return [([ctx.exp(nega), a], [1, r], [], G, [1, -r], [], 1/nega)]
        return ctx.hypercomb(h, [z], force_series=True)
    except ctx.NoConvergence:
        def h(z):
            T1 = [], [1, z-1], [z], G, [], [], 0
            T2 = [-ctx.exp(nega), a, z], [1, z, -1], [], G, [1], [1+z], a
            return T1, T2
        return ctx.hypercomb(h, [z])

@defun
def _gamma3(ctx, z, a, b, regularized=False):
    pole = ctx.isnpint(z)
    if regularized and pole:
        return ctx.zero
    try:
        ctx.prec += 15
        # We don't know in advance whether it's better to write as a difference
        # of lower or upper gamma functions, so try both
        T1 = ctx.gammainc(z, a, regularized=regularized)
        T2 = ctx.gammainc(z, b, regularized=regularized)
        R = T1 - T2
        if ctx.mag(R) - max(ctx.mag(T1), ctx.mag(T2)) > -10:
            return R
        if not pole:
            T1 = ctx.gammainc(z, 0, b, regularized=regularized)
            T2 = ctx.gammainc(z, 0, a, regularized=regularized)
            R = T1 - T2
            # May be ok, but should probably at least print a warning
            # about possible cancellation
            if 1: #ctx.mag(R) - max(ctx.mag(T1), ctx.mag(T2)) > -10:
                return R
    finally:
        ctx.prec -= 15
    raise NotImplementedError

@defun_wrapped
def expint(ctx, n, z):
    if ctx.isint(n) and ctx._is_real_type(z):
        try:
            return ctx._expint_int(n, z)
        except NotImplementedError:
            pass
    if ctx.isnan(n) or ctx.isnan(z):
        return z*n
    if z == ctx.inf:
        return 1/z
    if z == 0:
        # integral from 1 to infinity of t^n
        if ctx.re(n) <= 1:
            # TODO: reasonable sign of infinity
            return type(z)(ctx.inf)
        else:
            return ctx.one/(n-1)
    if n == 0:
        return ctx.exp(-z)/z
    if n == -1:
        return ctx.exp(-z)*(z+1)/z**2
    return z**(n-1) * ctx.gammainc(1-n, z)

@defun_wrapped
def li(ctx, z, offset=False):
    if offset:
        if z == 2:
            return ctx.zero
        return ctx.ei(ctx.ln(z)) - ctx.ei(ctx.ln2)
    if not z:
        return z
    if z == 1:
        return ctx.ninf
    return ctx.ei(ctx.ln(z))

@defun
def ei(ctx, z):
    try:
        return ctx._ei(z)
    except NotImplementedError:
        return ctx._ei_generic(z)

@defun_wrapped
def _ei_generic(ctx, z):
    # Note: the following is currently untested because mp and fp
    # both use special-case ei code
    if z == ctx.inf:
        return z
    if z == ctx.ninf:
        return ctx.zero
    if ctx.mag(z) > 1:
        try:
            r = ctx.one/z
            v = ctx.exp(z)*ctx.hyper([1,1],[],r,
                maxterms=ctx.prec, force_series=True)/z
            im = ctx._im(z)
            if im > 0:
                v += ctx.pi*ctx.j
            if im < 0:
                v -= ctx.pi*ctx.j
            return v
        except ctx.NoConvergence:
            pass
    v = z*ctx.hyp2f2(1,1,2,2,z) + ctx.euler
    if ctx._im(z):
        v += 0.5*(ctx.log(z) - ctx.log(ctx.one/z))
    else:
        v += ctx.log(abs(z))
    return v

@defun
def e1(ctx, z):
    try:
        return ctx._e1(z)
    except NotImplementedError:
        return ctx.expint(1, z)

@defun
def ci(ctx, z):
    try:
        return ctx._ci(z)
    except NotImplementedError:
        return ctx._ci_generic(z)

@defun_wrapped
def _ci_generic(ctx, z):
    if ctx.isinf(z):
        if z == ctx.inf: return ctx.zero
        if z == ctx.ninf: return ctx.pi*1j
    jz = ctx.fmul(ctx.j,z,exact=True)
    njz = ctx.fneg(jz,exact=True)
    v = 0.5*(ctx.ei(jz) + ctx.ei(njz))
    zreal = ctx._re(z)
    zimag = ctx._im(z)
    if zreal == 0:
        if zimag > 0: v += ctx.pi*0.5j
        if zimag < 0: v -= ctx.pi*0.5j
    if zreal < 0:
        if zimag >= 0: v += ctx.pi*1j
        if zimag <  0: v -= ctx.pi*1j
    if ctx._is_real_type(z) and zreal > 0:
        v = ctx._re(v)
    return v

@defun
def si(ctx, z):
    try:
        return ctx._si(z)
    except NotImplementedError:
        return ctx._si_generic(z)

@defun_wrapped
def _si_generic(ctx, z):
    if ctx.isinf(z):
        if z == ctx.inf: return 0.5*ctx.pi
        if z == ctx.ninf: return -0.5*ctx.pi
    # Suffers from cancellation near 0
    if ctx.mag(z) >= -1:
        jz = ctx.fmul(ctx.j,z,exact=True)
        njz = ctx.fneg(jz,exact=True)
        v = (-0.5j)*(ctx.ei(jz) - ctx.ei(njz))
        zreal = ctx._re(z)
        if zreal > 0:
            v -= 0.5*ctx.pi
        if zreal < 0:
            v += 0.5*ctx.pi
        if ctx._is_real_type(z):
            v = ctx._re(v)
        return v
    else:
        return z*ctx.hyp1f2((1,2),(3,2),(3,2),-0.25*z*z)

@defun_wrapped
def chi(ctx, z):
    nz = ctx.fneg(z, exact=True)
    v = 0.5*(ctx.ei(z) + ctx.ei(nz))
    zreal = ctx._re(z)
    zimag = ctx._im(z)
    if zimag > 0:
        v += ctx.pi*0.5j
    elif zimag < 0:
        v -= ctx.pi*0.5j
    elif zreal < 0:
        v += ctx.pi*1j
    return v

@defun_wrapped
def shi(ctx, z):
    # Suffers from cancellation near 0
    if ctx.mag(z) >= -1:
        nz = ctx.fneg(z, exact=True)
        v = 0.5*(ctx.ei(z) - ctx.ei(nz))
        zimag = ctx._im(z)
        if zimag > 0: v -= 0.5j*ctx.pi
        if zimag < 0: v += 0.5j*ctx.pi
        return v
    else:
        return z * ctx.hyp1f2((1,2),(3,2),(3,2),0.25*z*z)

@defun_wrapped
def fresnels(ctx, z):
    if z == ctx.inf:
        return ctx.mpf(0.5)
    if z == ctx.ninf:
        return ctx.mpf(-0.5)
    return ctx.pi*z**3/6*ctx.hyp1f2((3,4),(3,2),(7,4),-ctx.pi**2*z**4/16)

@defun_wrapped
def fresnelc(ctx, z):
    if z == ctx.inf:
        return ctx.mpf(0.5)
    if z == ctx.ninf:
        return ctx.mpf(-0.5)
    return z*ctx.hyp1f2((1,4),(1,2),(5,4),-ctx.pi**2*z**4/16)