Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A

Average Error: 2.9 → 0.4

Time: 10.4s

Precision: 64

Math

\[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} = -\infty \lor \neg \left(x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le 3.585600393933963942295649381360364716234 \cdot 10^{207}\right):\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r247395 = x;
        double r247396 = y;
        double r247397 = 1.1283791670955126;
        double r247398 = z;
        double r247399 = exp(r247398);
        double r247400 = r247397 * r247399;
        double r247401 = r247395 * r247396;
        double r247402 = r247400 - r247401;
        double r247403 = r247396 / r247402;
        double r247404 = r247395 + r247403;
        return r247404;
}

↓

double f(double x, double y, double z) {
        double r247405 = x;
        double r247406 = y;
        double r247407 = 1.1283791670955126;
        double r247408 = z;
        double r247409 = exp(r247408);
        double r247410 = r247407 * r247409;
        double r247411 = r247405 * r247406;
        double r247412 = r247410 - r247411;
        double r247413 = r247406 / r247412;
        double r247414 = r247405 + r247413;
        double r247415 = -inf.0;
        bool r247416 = r247414 <= r247415;
        double r247417 = 3.585600393933964e+207;
        bool r247418 = r247414 <= r247417;
        double r247419 = !r247418;
        bool r247420 = r247416 || r247419;
        double r247421 = 1.0;
        double r247422 = r247421 / r247405;
        double r247423 = r247405 - r247422;
        double r247424 = r247420 ? r247423 : r247414;
        return r247424;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.9
Target	0.0
Herbie	0.4

\[x + \frac{1}{\frac{1.128379167095512558560699289955664426088}{y} \cdot e^{z} - x}\]

Derivation

1. Split input into 2 regimes

2. if (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < -inf.0 or 3.585600393933964e+207 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))

   1. Initial program 19.8

      \[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]

   2. Taylor expanded around inf 1.6

      \[\leadsto \color{blue}{x - \frac{1}{x}}\]

   if -inf.0 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < 3.585600393933964e+207

   1. Initial program 0.2

      \[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} = -\infty \lor \neg \left(x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le 3.585600393933963942295649381360364716234 \cdot 10^{207}\right):\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1 (- (* (/ 1.12837916709551256 y) (exp z)) x)))

  (+ x (/ y (- (* 1.12837916709551256 (exp z)) (* x y)))))