Average Error: 29.5 → 0.3
Time: 1.2m
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.026496043719365127:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(e^{x}\right), \left(\mathsf{fma}\left(\left(e^{x}\right), \left(e^{x}\right), -4\right)\right), \left(e^{x} + 2\right)\right)}{e^{x} \cdot \left(e^{x} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\log \left(e^{\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(x \cdot x\right)\right)\right)\right)\\ \end{array}\]
\left(e^{x} - 2\right) + e^{-x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.026496043719365127:\\
\;\;\;\;\frac{\mathsf{fma}\left(\left(e^{x}\right), \left(\mathsf{fma}\left(\left(e^{x}\right), \left(e^{x}\right), -4\right)\right), \left(e^{x} + 2\right)\right)}{e^{x} \cdot \left(e^{x} + 2\right)}\\

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

\end{array}
double f(double x) {
        double r6057028 = x;
        double r6057029 = exp(r6057028);
        double r6057030 = 2.0;
        double r6057031 = r6057029 - r6057030;
        double r6057032 = -r6057028;
        double r6057033 = exp(r6057032);
        double r6057034 = r6057031 + r6057033;
        return r6057034;
}
↓
double f(double x) {
        double r6057035 = x;
        double r6057036 = -0.026496043719365127;
        bool r6057037 = r6057035 <= r6057036;
        double r6057038 = exp(r6057035);
        double r6057039 = -4.0;
        double r6057040 = fma(r6057038, r6057038, r6057039);
        double r6057041 = 2.0;
        double r6057042 = r6057038 + r6057041;
        double r6057043 = fma(r6057038, r6057040, r6057042);
        double r6057044 = r6057038 * r6057042;
        double r6057045 = r6057043 / r6057044;
        double r6057046 = 0.002777777777777778;
        double r6057047 = r6057035 * r6057035;
        double r6057048 = r6057047 * r6057047;
        double r6057049 = r6057046 * r6057048;
        double r6057050 = exp(r6057049);
        double r6057051 = log(r6057050);
        double r6057052 = 0.08333333333333333;
        double r6057053 = fma(r6057048, r6057052, r6057047);
        double r6057054 = fma(r6057051, r6057047, r6057053);
        double r6057055 = r6057037 ? r6057045 : r6057054;
        return r6057055;
}

Error

Bits error versus x

Target

Original29.5
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 < -0.026496043719365127

    1. Initial program 1.5

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

    2. Simplified1.6

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

    4. Applied flip--1.6

      \[\leadsto \color{blue}{\frac{e^{x} \cdot e^{x} - 2 \cdot 2}{e^{x} + 2}} - \frac{-1}{e^{x}}\]

    5. Applied frac-sub1.4

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

    6. Simplified1.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(e^{x}\right), \left(\mathsf{fma}\left(\left(e^{x}\right), \left(e^{x}\right), -4\right)\right), \left(2 + e^{x}\right)\right)}}{\left(e^{x} + 2\right) \cdot e^{x}}\]

    if -0.026496043719365127 < x

    1. Initial program 29.8

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

    2. Simplified29.8

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

    3. Taylor expanded around 0 0.2

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

    4. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(x \cdot x\right)\right)\right)\right)}\]
    5. Using strategy rm

    6. Applied add-log-exp0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.026496043719365127:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(e^{x}\right), \left(\mathsf{fma}\left(\left(e^{x}\right), \left(e^{x}\right), -4\right)\right), \left(e^{x} + 2\right)\right)}{e^{x} \cdot \left(e^{x} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\log \left(e^{\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(x \cdot x\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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