Average Error: 3.0 → 1.2
Time: 2.5s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y} \le 6.3928653876400416 \cdot 10^{200}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y} \le 6.3928653876400416 \cdot 10^{200}:\\
\;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r470367 = x;
        double r470368 = y;
        double r470369 = 1.1283791670955126;
        double r470370 = z;
        double r470371 = exp(r470370);
        double r470372 = r470369 * r470371;
        double r470373 = r470367 * r470368;
        double r470374 = r470372 - r470373;
        double r470375 = r470368 / r470374;
        double r470376 = r470367 + r470375;
        return r470376;
}
↓
double f(double x, double y, double z) {
        double r470377 = x;
        double r470378 = y;
        double r470379 = 1.1283791670955126;
        double r470380 = z;
        double r470381 = exp(r470380);
        double r470382 = r470379 * r470381;
        double r470383 = r470377 * r470378;
        double r470384 = r470382 - r470383;
        double r470385 = r470378 / r470384;
        double r470386 = r470377 + r470385;
        double r470387 = 6.392865387640042e+200;
        bool r470388 = r470386 <= r470387;
        double r470389 = 1.0;
        double r470390 = r470389 / r470377;
        double r470391 = r470377 - r470390;
        double r470392 = r470388 ? r470386 : r470391;
        return r470392;
}

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

Original3.0
Target0.0
Herbie1.2
\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < 6.392865387640042e+200

    1. Initial program 1.1

      \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]

    if 6.392865387640042e+200 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))

    1. Initial program 15.2

      \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]

    2. Taylor expanded around inf 1.9

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y} \le 6.3928653876400416 \cdot 10^{200}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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)))))