Average Error: 29.3 → 1.0
Time: 6.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 154.4482937142435:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left({x}^{3}, 8, 8 - 12 \cdot {x}^{2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - \frac{1}{e^{\mathsf{fma}\left(x, \varepsilon, 1 \cdot x\right)}}\right)}{2}\\ \end{array}\]
\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 154.4482937142435:\\
\;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left({x}^{3}, 8, 8 - 12 \cdot {x}^{2}\right)}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r28400 = 1.0;
        double r28401 = eps;
        double r28402 = r28400 / r28401;
        double r28403 = r28400 + r28402;
        double r28404 = r28400 - r28401;
        double r28405 = x;
        double r28406 = r28404 * r28405;
        double r28407 = -r28406;
        double r28408 = exp(r28407);
        double r28409 = r28403 * r28408;
        double r28410 = r28402 - r28400;
        double r28411 = r28400 + r28401;
        double r28412 = r28411 * r28405;
        double r28413 = -r28412;
        double r28414 = exp(r28413);
        double r28415 = r28410 * r28414;
        double r28416 = r28409 - r28415;
        double r28417 = 2.0;
        double r28418 = r28416 / r28417;
        return r28418;
}
↓
double f(double x, double eps) {
        double r28419 = x;
        double r28420 = 154.4482937142435;
        bool r28421 = r28419 <= r28420;
        double r28422 = 3.0;
        double r28423 = pow(r28419, r28422);
        double r28424 = 8.0;
        double r28425 = 12.0;
        double r28426 = 2.0;
        double r28427 = pow(r28419, r28426);
        double r28428 = r28425 * r28427;
        double r28429 = r28424 - r28428;
        double r28430 = fma(r28423, r28424, r28429);
        double r28431 = cbrt(r28430);
        double r28432 = 2.0;
        double r28433 = r28431 / r28432;
        double r28434 = 1.0;
        double r28435 = eps;
        double r28436 = r28434 / r28435;
        double r28437 = r28434 + r28436;
        double r28438 = r28434 - r28435;
        double r28439 = r28438 * r28419;
        double r28440 = -r28439;
        double r28441 = exp(r28440);
        double r28442 = r28437 * r28441;
        double r28443 = r28419 * r28435;
        double r28444 = r28434 * r28419;
        double r28445 = r28443 + r28444;
        double r28446 = -r28445;
        double r28447 = exp(r28446);
        double r28448 = r28447 / r28435;
        double r28449 = 1.0;
        double r28450 = fma(r28419, r28435, r28444);
        double r28451 = exp(r28450);
        double r28452 = r28449 / r28451;
        double r28453 = r28448 - r28452;
        double r28454 = r28434 * r28453;
        double r28455 = r28442 - r28454;
        double r28456 = r28455 / r28432;
        double r28457 = r28421 ? r28433 : r28456;
        return r28457;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 154.4482937142435

    1. Initial program 38.9

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

    2. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}}{2}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube1.2

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right) \cdot \mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)\right) \cdot \mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}}}{2}\]

    6. Simplified1.2

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)\right)}^{3}}}}{2}\]

    7. Taylor expanded around 0 1.2

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(8 \cdot {x}^{3} + 8\right) - 12 \cdot {x}^{2}}}}{2}\]

    8. Simplified1.2

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\mathsf{fma}\left({x}^{3}, 8, 8 - 12 \cdot {x}^{2}\right)}}}{2}\]

    if 154.4482937142435 < 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. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 154.4482937142435:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left({x}^{3}, 8, 8 - 12 \cdot {x}^{2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - \frac{1}{e^{\mathsf{fma}\left(x, \varepsilon, 1 \cdot x\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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))