Average Error: 29.9 → 0.3
Time: 17.8s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03372358286109267827557189889375877100974:\\ \;\;\;\;\frac{\left(e^{x} - 2\right) \cdot \left(e^{x} - 2\right) - e^{x \cdot -2}}{\left(e^{x} - 2\right) - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\\ \end{array}\]
\left(e^{x} - 2\right) + e^{-x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03372358286109267827557189889375877100974:\\
\;\;\;\;\frac{\left(e^{x} - 2\right) \cdot \left(e^{x} - 2\right) - e^{x \cdot -2}}{\left(e^{x} - 2\right) - e^{-x}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\\

\end{array}
double f(double x) {
        double r81478 = x;
        double r81479 = exp(r81478);
        double r81480 = 2.0;
        double r81481 = r81479 - r81480;
        double r81482 = -r81478;
        double r81483 = exp(r81482);
        double r81484 = r81481 + r81483;
        return r81484;
}
↓
double f(double x) {
        double r81485 = x;
        double r81486 = -0.03372358286109268;
        bool r81487 = r81485 <= r81486;
        double r81488 = exp(r81485);
        double r81489 = 2.0;
        double r81490 = r81488 - r81489;
        double r81491 = r81490 * r81490;
        double r81492 = -2.0;
        double r81493 = r81485 * r81492;
        double r81494 = exp(r81493);
        double r81495 = r81491 - r81494;
        double r81496 = -r81485;
        double r81497 = exp(r81496);
        double r81498 = r81490 - r81497;
        double r81499 = r81495 / r81498;
        double r81500 = 0.002777777777777778;
        double r81501 = 6.0;
        double r81502 = pow(r81485, r81501);
        double r81503 = 0.08333333333333333;
        double r81504 = 4.0;
        double r81505 = pow(r81485, r81504);
        double r81506 = r81503 * r81505;
        double r81507 = fma(r81500, r81502, r81506);
        double r81508 = fma(r81485, r81485, r81507);
        double r81509 = r81487 ? r81499 : r81508;
        return r81509;
}

Error

Bits error versus x

Target

Original29.9
Target0.0
Herbie0.3
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03372358286109268

    1. Initial program 1.3

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

    3. Applied flip-+6.4

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

    4. Simplified6.3

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

    if -0.03372358286109268 < x

    1. Initial program 30.1

      \[\left(e^{x} - 2\right) + e^{-x}\]

    2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]

    3. Simplified0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03372358286109267827557189889375877100974:\\ \;\;\;\;\frac{\left(e^{x} - 2\right) \cdot \left(e^{x} - 2\right) - e^{x \cdot -2}}{\left(e^{x} - 2\right) - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))