Average Error: 29.9 → 0.3
Time: 18.1s
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 r92074 = x;
        double r92075 = exp(r92074);
        double r92076 = 2.0;
        double r92077 = r92075 - r92076;
        double r92078 = -r92074;
        double r92079 = exp(r92078);
        double r92080 = r92077 + r92079;
        return r92080;
}
↓
double f(double x) {
        double r92081 = x;
        double r92082 = -0.03372358286109268;
        bool r92083 = r92081 <= r92082;
        double r92084 = exp(r92081);
        double r92085 = 2.0;
        double r92086 = r92084 - r92085;
        double r92087 = r92086 * r92086;
        double r92088 = -2.0;
        double r92089 = r92081 * r92088;
        double r92090 = exp(r92089);
        double r92091 = r92087 - r92090;
        double r92092 = -r92081;
        double r92093 = exp(r92092);
        double r92094 = r92086 - r92093;
        double r92095 = r92091 / r92094;
        double r92096 = 0.002777777777777778;
        double r92097 = 6.0;
        double r92098 = pow(r92081, r92097);
        double r92099 = 0.08333333333333333;
        double r92100 = 4.0;
        double r92101 = pow(r92081, r92100);
        double r92102 = r92099 * r92101;
        double r92103 = fma(r92096, r92098, r92102);
        double r92104 = fma(r92081, r92081, r92103);
        double r92105 = r92083 ? r92095 : r92104;
        return r92105;
}

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))))