Average Error: 29.6 → 0.9
Time: 1.2m
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 90.24341034286276:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right), \frac{2}{3}, \left(2 - x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{e^{\left(-1 - \varepsilon\right) \cdot x}}\right), \left(\sqrt{e^{\left(-1 - \varepsilon\right) \cdot x}}\right), \left(e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \frac{-1}{\varepsilon}\right)\right) + \left(\left(e^{\varepsilon \cdot x - x} + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right) + \mathsf{fma}\left(\left(\frac{-1}{\varepsilon}\right), \left(e^{\left(-1 - \varepsilon\right) \cdot x}\right), \left(e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \frac{1}{\varepsilon}\right)\right)\right)}{2}\\ \end{array}\]
\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 90.24341034286276:\\
\;\;\;\;\frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right), \frac{2}{3}, \left(2 - x \cdot x\right)\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1544473 = 1.0;
        double r1544474 = eps;
        double r1544475 = r1544473 / r1544474;
        double r1544476 = r1544473 + r1544475;
        double r1544477 = r1544473 - r1544474;
        double r1544478 = x;
        double r1544479 = r1544477 * r1544478;
        double r1544480 = -r1544479;
        double r1544481 = exp(r1544480);
        double r1544482 = r1544476 * r1544481;
        double r1544483 = r1544475 - r1544473;
        double r1544484 = r1544473 + r1544474;
        double r1544485 = r1544484 * r1544478;
        double r1544486 = -r1544485;
        double r1544487 = exp(r1544486);
        double r1544488 = r1544483 * r1544487;
        double r1544489 = r1544482 - r1544488;
        double r1544490 = 2.0;
        double r1544491 = r1544489 / r1544490;
        return r1544491;
}
↓
double f(double x, double eps) {
        double r1544492 = x;
        double r1544493 = 90.24341034286276;
        bool r1544494 = r1544492 <= r1544493;
        double r1544495 = r1544492 * r1544492;
        double r1544496 = r1544495 * r1544492;
        double r1544497 = 0.6666666666666666;
        double r1544498 = 2.0;
        double r1544499 = r1544498 - r1544495;
        double r1544500 = fma(r1544496, r1544497, r1544499);
        double r1544501 = r1544500 / r1544498;
        double r1544502 = -1.0;
        double r1544503 = eps;
        double r1544504 = r1544502 - r1544503;
        double r1544505 = r1544504 * r1544492;
        double r1544506 = exp(r1544505);
        double r1544507 = sqrt(r1544506);
        double r1544508 = r1544502 / r1544503;
        double r1544509 = r1544506 * r1544508;
        double r1544510 = fma(r1544507, r1544507, r1544509);
        double r1544511 = r1544503 * r1544492;
        double r1544512 = r1544511 - r1544492;
        double r1544513 = exp(r1544512);
        double r1544514 = r1544513 / r1544503;
        double r1544515 = r1544513 + r1544514;
        double r1544516 = 1.0;
        double r1544517 = r1544516 / r1544503;
        double r1544518 = r1544506 * r1544517;
        double r1544519 = fma(r1544508, r1544506, r1544518);
        double r1544520 = r1544515 + r1544519;
        double r1544521 = r1544510 + r1544520;
        double r1544522 = r1544521 / r1544498;
        double r1544523 = r1544494 ? r1544501 : r1544522;
        return r1544523;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 90.24341034286276

    1. Initial program 39.5

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

    2. Simplified39.5

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

    3. Taylor expanded around 0 1.1

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

    4. Simplified1.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot \left(x \cdot x\right)\right), 2\right) - x \cdot x}}{2}\]
    5. Using strategy rm

    6. Applied fma-udef1.1

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

    7. Applied associate--l+1.1

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

    8. Taylor expanded around 0 1.1

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

    9. Simplified1.1

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

    if 90.24341034286276 < 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. Simplified0.2

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

    4. Applied div-inv0.2

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

    5. Applied add-sqr-sqrt0.2

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

    6. Applied prod-diff0.2

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

    7. Applied associate-+l+0.2

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 90.24341034286276:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right), \frac{2}{3}, \left(2 - x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{e^{\left(-1 - \varepsilon\right) \cdot x}}\right), \left(\sqrt{e^{\left(-1 - \varepsilon\right) \cdot x}}\right), \left(e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \frac{-1}{\varepsilon}\right)\right) + \left(\left(e^{\varepsilon \cdot x - x} + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right) + \mathsf{fma}\left(\left(\frac{-1}{\varepsilon}\right), \left(e^{\left(-1 - \varepsilon\right) \cdot x}\right), \left(e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \frac{1}{\varepsilon}\right)\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))