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

Average Error: 3.0 → 0.5

Time: 11.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y} \le -1.760267047790503409639426851947972538871 \cdot 10^{224}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y} \le 3.139871723518476025304133244597234031702 \cdot 10^{196}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r20215367 = x;
        double r20215368 = y;
        double r20215369 = 1.1283791670955126;
        double r20215370 = z;
        double r20215371 = exp(r20215370);
        double r20215372 = r20215369 * r20215371;
        double r20215373 = r20215367 * r20215368;
        double r20215374 = r20215372 - r20215373;
        double r20215375 = r20215368 / r20215374;
        double r20215376 = r20215367 + r20215375;
        return r20215376;
}

↓

double f(double x, double y, double z) {
        double r20215377 = x;
        double r20215378 = y;
        double r20215379 = z;
        double r20215380 = exp(r20215379);
        double r20215381 = 1.1283791670955126;
        double r20215382 = r20215380 * r20215381;
        double r20215383 = r20215377 * r20215378;
        double r20215384 = r20215382 - r20215383;
        double r20215385 = r20215378 / r20215384;
        double r20215386 = r20215377 + r20215385;
        double r20215387 = -1.7602670477905034e+224;
        bool r20215388 = r20215386 <= r20215387;
        double r20215389 = 1.0;
        double r20215390 = r20215389 / r20215377;
        double r20215391 = r20215377 - r20215390;
        double r20215392 = 3.139871723518476e+196;
        bool r20215393 = r20215386 <= r20215392;
        double r20215394 = r20215393 ? r20215386 : r20215391;
        double r20215395 = r20215388 ? r20215391 : r20215394;
        return r20215395;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.0
Target	0.1
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < -1.7602670477905034e+224 or 3.139871723518476e+196 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))

   1. Initial program 12.2

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

   2. Taylor expanded around inf 1.8

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

   if -1.7602670477905034e+224 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < 3.139871723518476e+196

   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 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y} \le -1.760267047790503409639426851947972538871 \cdot 10^{224}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y} \le 3.139871723518476025304133244597234031702 \cdot 10^{196}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"

  :herbie-target
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

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