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

Average Error: 3.0 → 0.3

Time: 2.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le -2.38038529208034741010033526046896232449 \cdot 10^{232} \lor \neg \left(x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le 6.146831817986485928975089715892198335691 \cdot 10^{254}\right):\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r392263 = x;
        double r392264 = y;
        double r392265 = 1.1283791670955126;
        double r392266 = z;
        double r392267 = exp(r392266);
        double r392268 = r392265 * r392267;
        double r392269 = r392263 * r392264;
        double r392270 = r392268 - r392269;
        double r392271 = r392264 / r392270;
        double r392272 = r392263 + r392271;
        return r392272;
}

↓

double f(double x, double y, double z) {
        double r392273 = x;
        double r392274 = y;
        double r392275 = 1.1283791670955126;
        double r392276 = z;
        double r392277 = exp(r392276);
        double r392278 = r392275 * r392277;
        double r392279 = r392273 * r392274;
        double r392280 = r392278 - r392279;
        double r392281 = r392274 / r392280;
        double r392282 = r392273 + r392281;
        double r392283 = -2.3803852920803474e+232;
        bool r392284 = r392282 <= r392283;
        double r392285 = 6.146831817986486e+254;
        bool r392286 = r392282 <= r392285;
        double r392287 = !r392286;
        bool r392288 = r392284 || r392287;
        double r392289 = 1.0;
        double r392290 = r392289 / r392273;
        double r392291 = r392273 - r392290;
        double r392292 = r392288 ? r392291 : r392282;
        return r392292;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.0
Target	0.0
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < -2.3803852920803474e+232 or 6.146831817986486e+254 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))

   1. Initial program 18.0

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

   2. Taylor expanded around inf 1.4

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

   if -2.3803852920803474e+232 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < 6.146831817986486e+254

   1. Initial program 0.1

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le -2.38038529208034741010033526046896232449 \cdot 10^{232} \lor \neg \left(x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le 6.146831817986485928975089715892198335691 \cdot 10^{254}\right):\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))