Average Error: 2.7 → 1.2
Time: 3.0s
Precision: 64
\[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le 7.375980088346894603541968916273812509677 \cdot 10^{198}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]
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 7.375980088346894603541968916273812509677 \cdot 10^{198}:\\
\;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{1}{x}\\

\end{array}
double f(double x, double y, double z) {
        double r662451 = x;
        double r662452 = y;
        double r662453 = 1.1283791670955126;
        double r662454 = z;
        double r662455 = exp(r662454);
        double r662456 = r662453 * r662455;
        double r662457 = r662451 * r662452;
        double r662458 = r662456 - r662457;
        double r662459 = r662452 / r662458;
        double r662460 = r662451 + r662459;
        return r662460;
}
↓
double f(double x, double y, double z) {
        double r662461 = x;
        double r662462 = y;
        double r662463 = 1.1283791670955126;
        double r662464 = z;
        double r662465 = exp(r662464);
        double r662466 = r662463 * r662465;
        double r662467 = r662461 * r662462;
        double r662468 = r662466 - r662467;
        double r662469 = r662462 / r662468;
        double r662470 = r662461 + r662469;
        double r662471 = 7.375980088346895e+198;
        bool r662472 = r662470 <= r662471;
        double r662473 = 1.0;
        double r662474 = r662473 / r662461;
        double r662475 = r662461 - r662474;
        double r662476 = r662472 ? r662470 : r662475;
        return r662476;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.7
Target0.0
Herbie1.2
\[x + \frac{1}{\frac{1.128379167095512558560699289955664426088}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < 7.375980088346895e+198

    1. Initial program 1.0

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

    if 7.375980088346895e+198 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))

    1. Initial program 13.5

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

    2. Taylor expanded around inf 2.4

      \[\leadsto \color{blue}{x - \frac{1}{x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le 7.375980088346894603541968916273812509677 \cdot 10^{198}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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)))))