I'm repeatedly issuing the command

P = InverseCDF[NormalDistribution[0, 1], T]

where T is either a 36- or 64-vector of real numbers (points in the unit hypercubes $[0,1]^{36}$ or $[0,1]^{64}$, quasirandomly distributed according to the "golden-ratio" generalization formula of Martin Roberts given in his answer to How can one generate an open-ended sequence of low-discrepancy points in 3D?). (The points--after conversion by the indicated command--are employed for the generation of "two-rebit" and "two-qubit" $4 \times 4$ density matrices, randomly generated with respect to the Bures [minimal monotone] measure, in accordance with formulas (24) and (28), setting $x =\frac{1}{2}$, of "Random Bures mixed states and the distribution of their purity" https://arxiv.org/abs/0909.5094 .)

An analysis of mine shows that this is by far the most time-consuming step in my program.

So, can I use Compile so that the command "knows" that the vector is composed of reals? I presume/hope that, if so, this would lead to a considerable speed-up.

  • 1
    $\begingroup$ Unfortunately, this is not possible, at least not without further effort. The function InverseErfc is not compilable. That means that the compiled function will contain calls to MainEvaluate which will slow down the execution severely. You can speed it up a little by using the Listable-attribute of InverseCDF to process whole matrices at once: T = RandomReal[{0., 1.}, {10000, 64}]; { First@AbsoluteTiming[ P1 = Table[InverseCDF[NormalDistribution[0, 1], t], {t, T}];], First@AbsoluteTiming[P2 = InverseCDF[NormalDistribution[0, 1], T];] }. $\endgroup$ – Henrik Schumacher Sep 2 '18 at 12:48

Out of curiosity, I tried to write my own version of inverse CDF for the normal distribution. I employ a qualitative approximation of the inverse CDF as initial guess and apply Newton iterations with line search until convergence.

This is the code:

f[x_] := CDF[NormalDistribution[0, 1], x];
finv[y_] := InverseCDF[NormalDistribution[0, 1], y];
p = 1/200;
q = 2/5;
g[x_] = N@Piecewise[{
     {finv[$MachineEpsilon], 0 <= x <= $MachineEpsilon},
     {Simplify[Normal@Series[finv[x], {x, 0, 1}], 0 < x < 1], $MachineEpsilon < x < p},
     {Simplify[PadeApproximant[finv[x], {x, 1/2, {7, 8}}]], q < x < 1 - q},
     {Simplify[Normal@Series[finv[x], {x, 1, 1}], 0 < x < 1], 1 - p < x < 1},
     {finv[1 - $MachineEpsilon], 1 - $MachineEpsilon <= x <= 1}
    Simplify[PadeApproximant[finv[x], {x, 1/2, {17, 18}}]]

cfinv = Block[{T, S, Sτ}, With[{
     S0 = N[g[T]],
     ϕ00 = N[(T - f[S])^2],
     direction = N[Simplify[(T - f[S])/f'[S]]],
     residual = N[(T - f[Sτ])^2],
     σ = 0.0001,
     γ = 0.5,
     TOL = 1. 10^-15
    Compile[{{T, _Real}},
     Block[{S, Sτ, ϕ0, τ, u, ϕτ},
      S = S0;
      ϕ0 = ϕ00;
      While[Sqrt[ϕ0] > TOL,
       u = direction;
       τ = 1.;
       Sτ = S + τ u;
       ϕτ = residual;
       While[ϕτ > (1. - σ τ) ϕ0, 
        τ *= γ;
        Sτ = S + τ u;
        ϕτ = residual;
       ϕ0 = ϕτ;
       S = Sτ;
     CompilationTarget -> "C",
     RuntimeAttributes -> {Listable},
     Parallelization -> True,
     RuntimeOptions -> "Speed"

And here an obligatory test for speed and accuracy (tested on a Haswell Quad Core CPU):

T = RandomReal[{0., 1}, {1000000}];
a = finv[T]; // RepeatedTiming // First
b = cfinv[T]; // RepeatedTiming // First
Max[Abs[a - b]]





So this one is almost three eight times faster than the built-in method.

I also expect a speedup should one replace the function g by a better but also reasonably quick approximation for the initial guess. First I tried an InterpolatingFunction for the (suitably) transformed "true" CDF, but that turned out to be way too slow.

Of course Newton's method has its problems on the extreme tails of the distribution (close to 0 and close 1) where the CDF has derivative close to 0. Maybe a secant method would have been more appropriate?


Using expansions of the inverse CDF at $0$, $1/2$ and $1$, I was able to come up with a way better initial guess function g.

  • $\begingroup$ A very quick test indicates that my program seems to run at least five times faster on my (very basic) MacBookAir using the new cfinv command than with the (default/built-in) finv[y_] := InverseCDF[NormalDistribution[0, 1], y] command. $\endgroup$ – Paul B. Slater Sep 2 '18 at 17:49
  • $\begingroup$ That great to hear. Interesting that this gets relatively faster on a Duo-Core machine. Maybe InverseCDF is better parallelizable: Newton's methods has the potential to make computations a bit assynchronous, so that the organization overhead rises. That makes it potentially less scalable. $\endgroup$ – Henrik Schumacher Sep 2 '18 at 18:14
  • $\begingroup$ @PaulB.Slater In the meantime, I have been able to come up with a further considerable speedup. Have a try. $\endgroup$ – Henrik Schumacher Sep 2 '18 at 19:08
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    $\begingroup$ Quick test indicates now about twice as quick as the previous improvement--so about ten times faster than the InverseCDF[NormalDistribution[0, 1], T] command itself. (Incidentally, I first was directed to perform a software installation to be able to implement these Schumacher codes.) $\endgroup$ – Paul B. Slater Sep 3 '18 at 5:15
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    $\begingroup$ Oh, interesting (preceding/intervening) comment of Schumacher. So, since Infinity or -Infinity is now never returned, I guess I can discard the Check operation used in the matrix operations step. Maybe this will make the two programs comparable in this regard. $\endgroup$ – Paul B. Slater Sep 3 '18 at 12:49

This is a little long for a comment, beside which it points out an unexpected difference probably resulting from subsystems evolving independently at different times. In addition to Henrik's comment that the code is uncompilable, there are two things to add. The most significant is how the quantity is computed, and the other is the usual difference between packed and unpacked arrays.

If InverseCDF[NormalDistribution[0, 1], t] is evaluated symbolically first, it simplifies to a scaled InverseErfc:

p = InverseCDF[NormalDistribution[0, 1], t]
pv = First[p]
  ConditionalExpression[-Sqrt[2] InverseErfc[2 t], 0 <= t <= 1]
  -Sqrt[2] InverseErfc[2 t]

Surprisingly, the simplified version is over 100 times slower. The ConditionalExpression is not Listable, and one might think that the seemingly "vectorized" pv would be an improvement. However pv is only slightly faster, probably all due to the condition check being removed.

Here are some data for timing tests. The array tt is packed and the array uu is the unpacked version of tt.

SeedRandom[0];    (* so everyone uses the same data *)
tt = RandomReal[1, 100];
uu = Developer`FromPackedArray[tt];

Clearly from the timings below, InverseCDF[NormalDistribution[0, 1], tt] calls some internal vectorized code for the computation, where as both p and pv, which use InverseErfc, are equally slow on packed and unpacked arrays. Only the InverseCDF is significantly faster on packed arrays. It is still 3.5 times faster than InverseErfc on unpacked arrays. The only case in which using the Listable attribute matters is the first one, which uses InverseCDF on packed arrays.

InverseCDF[NormalDistribution[0, 1], tt]; // RepeatedTiming
pv /. t -> tt; // RepeatedTiming       (* InverseErfc *)
Table[p, {t, tt}]; // RepeatedTiming   (* Henrik's 1st example *)
  {0.000054, Null}
  {0.0071, Null}
  {0.0075, Null}

InverseCDF[NormalDistribution[0, 1], uu]; // RepeatedTiming
Table[InverseCDF[NormalDistribution[0, 1], t], {t, uu}]; // RepeatedTiming  
pv /. t -> uu; // RepeatedTiming       (* InverseErfc *)
Table[p, {t, uu}]; // RepeatedTiming   (* Henrik's 1st example *)
  {0.0023, Null}
  {0.0021, Null}
  {0.0074, Null}
  {0.0076, Null}

Thus the fastest method available is InverseCDF[NormalDistribution[0, 1], t] on numeric t; if you have many such computations, packed the input values into an array t. If it is difficult to generate t as a packed array, you can use t = Developer`ToPackedArray[t, Real].


Let me demonstrate two approaches in this answer.

P.J. Acklam (WayBack archive of his page) devised a not-too-long method to approximate the quantile of the Gaussian distribution, with absolute errors on the order of $10^{-9}$. I have tweaked the Mathematica implementation of Acklam's approximation by William Shaw so that one can obtain a compilable function:

AcklamQuantile = Block[{u}, 
      Compile[{{u, _Real}}, #, RuntimeAttributes -> {Listable}] & @@ 
      Hold[With[{a = {-39.69683028665376, 220.9460984245205, -275.9285104469687,
                      138.3577518672690, -30.66479806614716, 2.506628277459239}, 
                 b = {-54.47609879822406, 161.5858368580409, -155.6989798598866,
                      66.80131188771972, -13.28068155288572, 1}, 
                 c = {-0.007784894002430293, -0.3223964580411365, -2.400758277161838,
                      -2.549732539343734, 4.374664141464968, 2.938163982698783}, 
                 d = {0.007784695709041462, 0.3224671290700398, 2.445134137142996,
                      3.754408661907416, 1.}},
                Which[0.02435 <= u <= 0.97575, 
                      With[{v = u - 1/2}, 
                           v Fold[(#1 v^2 + #2) &, 0, a]/
                           Fold[(#1 v^2 + #2) &, 0, b]] // Evaluate,
                      u > 0.97575, 
                      With[{q = Sqrt[-2 Log[1 - u]]},
                           -Fold[(#1 q + #2) &, 0, c]/
                           Fold[(#1 q + #2) &, 0, d]] // Evaluate,
                      With[{q = Sqrt[-2 Log[u]]}, 
                           Fold[(#1 q + #2) &, 0, c]/
                           Fold[(#1 q + #2) &, 0, d]] // Evaluate]]]];

You can use CompiledFunctionTools`CompilePrint[] to check that the function was compiled properly.

Here is a plot of the absolute error over $(0,1)$:

nq[p_] = InverseCDF[NormalDistribution[], p];
Plot[AcklamQuantile[p] - nq[p], {p, 0, 1}, PlotRange -> All]

absolute error for Acklam's approximation

Of course, one can polish this further with a few steps of Newton-Raphson or Halley, if seen fit.

As it turns out, however, some spelunking shows that Mathematica does provide a compilable implementation of the normal distribution quantile. The undocumented function SpecialFunctions`Probit[] is the routine of interest, but there is a minor botch in its implementation that I will show how to fix.

Spelunking the code of SpecialFunctions`Probit[] shows that the function relies on a compiled function, System`StatisticalFunctionsDump`CompiledProbit[] for numerical implementation. Using CompilePrint[] to inspect its code, however, one sees a number of unsightly MainEvaluate calls in the code. Thus, I offer a fix to this routine that yields a fully compiled function:

SpecialFunctions`Probit; (* force autoload *)
probit = With[{d = N[17/40],
               cf1 = System`StatisticalFunctionsDump`CompiledProbitCentralMinimax,
               cf2 = System`StatisticalFunctionsDump`CompiledProbitAsymptotic}, 
              Compile[{{u, _Real}}, If[Abs[u - 0.5] <= d, cf1[u], cf2[u]], 
                      CompilationOptions -> {"InlineCompiledFunctions" -> True,
                                             "InlineExternalDefinitions" -> True}, 
                      RuntimeAttributes -> {Listable}]];

where the two compiled sub-functions System`StatisticalFunctionsDump`CompiledProbitCentralMinimax and System`StatisticalFunctionsDump`CompiledProbitAsymptotic implement different approximations; for hopefully obvious reasons, I will not be reproducing their code here.

(Thanks to Henrik Schumacher for suggesting a better reformulation.)

CompilePrint[probit] will show that this version has compiled properly. Here is a plot of the absolute error:

Plot[probit[p] - nq[p], {p, 0, 1}, PlotRange -> All]

absolute error for modified built-in approximation

  • 2
    $\begingroup$ A Necromancer badge on your first day back seem appropriate. $\endgroup$ – Michael E2 Sep 24 '18 at 18:29
  • $\begingroup$ In terms of my ongoing “production” runs, should I just continue with my use of the Henrik Schumacher code or could either AcklamQuantile[p] or probit[p] accelerate computations considerably faster than the Schumacher code (which I found to be roughly ten times faster than the uncompiled InverseCDF[NormalDistribution[0, 1], T] command)? For probit[p] (which seems to be the more precise of the two), I take it that I would require the code for the two compiled sub-functions. I’m also applying the command in question to vectors T of either length 36 or 64. $\endgroup$ – Paul B. Slater Sep 25 '18 at 2:29
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    $\begingroup$ "could either AcklamQuantile[p] or probit[p] accelerate computations considerably faster than the Schumacher code" - I don't have your code, so I wholly encourage you to do your own experiments. I gave the Acklam routine because some applications don't need a quantile accurate to machine precision, and that it can still be used to start up a Newton-Raphson iteration. In the case of probit, the two sub-functions are already built-in but hidden, so all you need to access them is to autoload the built-in probit function by evaluating SpecialFunctions`Probit;. $\endgroup$ – J. M.'s ennui Sep 25 '18 at 2:48
  • $\begingroup$ OK, I will certainly try comparing the performances of the three codes (AcklamQuantile[p], profit[p] and the Schumacher one given above), when I (shortly) have the opportunity. But since the Schumacher code is given above, can not any immediate assertions be advanced in terms of their relative performances? (I now see that the Schumacher code also uses the Compile command.) $\endgroup$ – Paul B. Slater Sep 25 '18 at 3:57
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    $\begingroup$ @J.M. I ran both your probit and my cfinv implementations on my machine (see here). Your code turned out to be much faster on my machine, but Paul did not observe any major speed-up. $\endgroup$ – Henrik Schumacher Sep 26 '18 at 18:53

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