Average Error: 29.8 → 0.1
Time: 5.6s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \le 1.2761800916 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array}\]
\left(e^{x} - 2\right) + e^{-x}
↓
\begin{array}{l}
\mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \le 1.2761800916 \cdot 10^{-7}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\

\end{array}
double f(double x) {
        double r78449 = x;
        double r78450 = exp(r78449);
        double r78451 = 2.0;
        double r78452 = r78450 - r78451;
        double r78453 = -r78449;
        double r78454 = exp(r78453);
        double r78455 = r78452 + r78454;
        return r78455;
}
↓
double f(double x) {
        double r78456 = x;
        double r78457 = exp(r78456);
        double r78458 = 2.0;
        double r78459 = r78457 - r78458;
        double r78460 = -r78456;
        double r78461 = exp(r78460);
        double r78462 = r78459 + r78461;
        double r78463 = 1.2761800916027966e-07;
        bool r78464 = r78462 <= r78463;
        double r78465 = 0.002777777777777778;
        double r78466 = 6.0;
        double r78467 = pow(r78456, r78466);
        double r78468 = 0.08333333333333333;
        double r78469 = 4.0;
        double r78470 = pow(r78456, r78469);
        double r78471 = r78468 * r78470;
        double r78472 = fma(r78465, r78467, r78471);
        double r78473 = fma(r78456, r78456, r78472);
        double r78474 = r78464 ? r78473 : r78462;
        return r78474;
}

Error

Bits error versus x

Target

Original29.8
Target0.0
Herbie0.1
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (- (exp x) 2.0) (exp (- x))) < 1.2761800916027966e-07

    1. Initial program 30.3

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0

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

    if 1.2761800916027966e-07 < (+ (- (exp x) 2.0) (exp (- x)))

    1. Initial program 5.1

      \[\left(e^{x} - 2\right) + e^{-x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \le 1.2761800916 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))))