Average Error: 3.4 → 1.3
Time: 17.9s
Precision: 64
\[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 2.289166001615931520939634610774789889377 \cdot 10^{-308}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \end{array}\]
x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 2.289166001615931520939634610774789889377 \cdot 10^{-308}:\\
\;\;\;\;x - \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\

\end{array}
double f(double x, double y, double z) {
        double r29520460 = x;
        double r29520461 = y;
        double r29520462 = 1.1283791670955126;
        double r29520463 = z;
        double r29520464 = exp(r29520463);
        double r29520465 = r29520462 * r29520464;
        double r29520466 = r29520460 * r29520461;
        double r29520467 = r29520465 - r29520466;
        double r29520468 = r29520461 / r29520467;
        double r29520469 = r29520460 + r29520468;
        return r29520469;
}
↓
double f(double x, double y, double z) {
        double r29520470 = z;
        double r29520471 = exp(r29520470);
        double r29520472 = 2.2891660016159315e-308;
        bool r29520473 = r29520471 <= r29520472;
        double r29520474 = x;
        double r29520475 = 1.0;
        double r29520476 = r29520475 / r29520474;
        double r29520477 = r29520474 - r29520476;
        double r29520478 = y;
        double r29520479 = 1.1283791670955126;
        double r29520480 = r29520479 * r29520471;
        double r29520481 = r29520474 * r29520478;
        double r29520482 = r29520480 - r29520481;
        double r29520483 = r29520478 / r29520482;
        double r29520484 = r29520474 + r29520483;
        double r29520485 = r29520473 ? r29520477 : r29520484;
        return r29520485;
}

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

Original3.4
Target0.1
Herbie1.3
\[x + \frac{1}{\frac{1.128379167095512558560699289955664426088}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 2.2891660016159315e-308

    1. Initial program 8.7

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

    2. Taylor expanded around inf 0.0

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

    if 2.2891660016159315e-308 < (exp z)

    1. Initial program 1.7

      \[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 2.289166001615931520939634610774789889377 \cdot 10^{-308}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))))