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 r471414 = x;
        double r471415 = y;
        double r471416 = 1.1283791670955126;
        double r471417 = z;
        double r471418 = exp(r471417);
        double r471419 = r471416 * r471418;
        double r471420 = r471414 * r471415;
        double r471421 = r471419 - r471420;
        double r471422 = r471415 / r471421;
        double r471423 = r471414 + r471422;
        return r471423;
}
↓
double f(double x, double y, double z) {
        double r471424 = x;
        double r471425 = y;
        double r471426 = 1.1283791670955126;
        double r471427 = z;
        double r471428 = exp(r471427);
        double r471429 = r471426 * r471428;
        double r471430 = r471424 * r471425;
        double r471431 = r471429 - r471430;
        double r471432 = r471425 / r471431;
        double r471433 = r471424 + r471432;
        double r471434 = 8.580327153391063e+244;
        bool r471435 = r471433 <= r471434;
        double r471436 = 1.0;
        double r471437 = r471436 / r471424;
        double r471438 = r471424 - r471437;
        double r471439 = r471435 ? r471433 : r471438;
        return r471439;
}

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 
(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)))))