Average Error: 29.7 → 1.1
Time: 29.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 331.7599664969623631805006880313158035278:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]
\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
↓
\begin{array}{l}
\mathbf{if}\;x \le 331.7599664969623631805006880313158035278:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r71474 = 1.0;
        double r71475 = eps;
        double r71476 = r71474 / r71475;
        double r71477 = r71474 + r71476;
        double r71478 = r71474 - r71475;
        double r71479 = x;
        double r71480 = r71478 * r71479;
        double r71481 = -r71480;
        double r71482 = exp(r71481);
        double r71483 = r71477 * r71482;
        double r71484 = r71476 - r71474;
        double r71485 = r71474 + r71475;
        double r71486 = r71485 * r71479;
        double r71487 = -r71486;
        double r71488 = exp(r71487);
        double r71489 = r71484 * r71488;
        double r71490 = r71483 - r71489;
        double r71491 = 2.0;
        double r71492 = r71490 / r71491;
        return r71492;
}
↓
double f(double x, double eps) {
        double r71493 = x;
        double r71494 = 331.75996649696236;
        bool r71495 = r71493 <= r71494;
        double r71496 = 0.6666666666666667;
        double r71497 = 3.0;
        double r71498 = pow(r71493, r71497);
        double r71499 = 2.0;
        double r71500 = fma(r71496, r71498, r71499);
        double r71501 = 1.0;
        double r71502 = 2.0;
        double r71503 = pow(r71493, r71502);
        double r71504 = r71501 * r71503;
        double r71505 = r71500 - r71504;
        double r71506 = r71505 / r71499;
        double r71507 = eps;
        double r71508 = r71501 / r71507;
        double r71509 = r71501 + r71508;
        double r71510 = r71501 - r71507;
        double r71511 = r71510 * r71493;
        double r71512 = -r71511;
        double r71513 = exp(r71512);
        double r71514 = r71508 - r71501;
        double r71515 = r71501 + r71507;
        double r71516 = r71515 * r71493;
        double r71517 = exp(r71516);
        double r71518 = r71514 / r71517;
        double r71519 = -r71518;
        double r71520 = fma(r71509, r71513, r71519);
        double r71521 = r71520 / r71499;
        double r71522 = r71495 ? r71506 : r71521;
        return r71522;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 331.75996649696236

    1. Initial program 39.4

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Taylor expanded around 0 1.4

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.4

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}}{2}\]

    if 331.75996649696236 < x

    1. Initial program 0.2

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
    2. Using strategy rm

    3. Applied fma-neg0.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\]

    4. Simplified0.2

      \[\leadsto \frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, \color{blue}{-\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}\right)}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 331.7599664969623631805006880313158035278:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))