Average Error: 2.8 → 1.1
Time: 3.3s
Precision: 64
\[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le 8.58032715339106299894023200120557440118 \cdot 10^{244}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]
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 8.58032715339106299894023200120557440118 \cdot 10^{244}:\\
\;\;\;\;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 r417458 = x;
        double r417459 = y;
        double r417460 = 1.1283791670955126;
        double r417461 = z;
        double r417462 = exp(r417461);
        double r417463 = r417460 * r417462;
        double r417464 = r417458 * r417459;
        double r417465 = r417463 - r417464;
        double r417466 = r417459 / r417465;
        double r417467 = r417458 + r417466;
        return r417467;
}
↓
double f(double x, double y, double z) {
        double r417468 = x;
        double r417469 = y;
        double r417470 = 1.1283791670955126;
        double r417471 = z;
        double r417472 = exp(r417471);
        double r417473 = r417470 * r417472;
        double r417474 = r417468 * r417469;
        double r417475 = r417473 - r417474;
        double r417476 = r417469 / r417475;
        double r417477 = r417468 + r417476;
        double r417478 = 8.580327153391063e+244;
        bool r417479 = r417477 <= r417478;
        double r417480 = 1.0;
        double r417481 = r417480 / r417468;
        double r417482 = r417468 - r417481;
        double r417483 = r417479 ? r417477 : r417482;
        return r417483;
}

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

Original2.8
Target0.0
Herbie1.1
\[x + \frac{1}{\frac{1.128379167095512558560699289955664426088}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < 8.580327153391063e+244

    1. Initial program 1.2

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

    if 8.580327153391063e+244 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))

    1. Initial program 20.3

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

    2. Taylor expanded around inf 1.0

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le 8.58032715339106299894023200120557440118 \cdot 10^{244}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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)))))