Average Error: 2.8 → 1.0
Time: 2.7s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.922339974919372252:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \end{array}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 0.922339974919372252:\\
\;\;\;\;x - \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\

\end{array}
double f(double x, double y, double z) {
        double r455570 = x;
        double r455571 = y;
        double r455572 = 1.1283791670955126;
        double r455573 = z;
        double r455574 = exp(r455573);
        double r455575 = r455572 * r455574;
        double r455576 = r455570 * r455571;
        double r455577 = r455575 - r455576;
        double r455578 = r455571 / r455577;
        double r455579 = r455570 + r455578;
        return r455579;
}
↓
double f(double x, double y, double z) {
        double r455580 = z;
        double r455581 = exp(r455580);
        double r455582 = 0.9223399749193723;
        bool r455583 = r455581 <= r455582;
        double r455584 = x;
        double r455585 = 1.0;
        double r455586 = r455585 / r455584;
        double r455587 = r455584 - r455586;
        double r455588 = y;
        double r455589 = 1.1283791670955126;
        double r455590 = r455589 * r455581;
        double r455591 = r455584 * r455588;
        double r455592 = r455590 - r455591;
        double r455593 = r455588 / r455592;
        double r455594 = r455584 + r455593;
        double r455595 = r455583 ? r455587 : r455594;
        return r455595;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.9223399749193723

    1. Initial program 7.5

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

    2. Taylor expanded around inf 0.3

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

    if 0.9223399749193723 < (exp z)

    1. Initial program 1.2

      \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.922339974919372252:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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)))))