Average Error: 30.3 → 0.5
Time: 17.5s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left({x}^{6}, \frac{1}{360}, \mathsf{fma}\left({x}^{4}, \frac{1}{12}, x \cdot x\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left({x}^{6}, \frac{1}{360}, \mathsf{fma}\left({x}^{4}, \frac{1}{12}, x \cdot x\right)\right)
double f(double x) {
        double r113455 = x;
        double r113456 = exp(r113455);
        double r113457 = 2.0;
        double r113458 = r113456 - r113457;
        double r113459 = -r113455;
        double r113460 = exp(r113459);
        double r113461 = r113458 + r113460;
        return r113461;
}
↓
double f(double x) {
        double r113462 = x;
        double r113463 = 6.0;
        double r113464 = pow(r113462, r113463);
        double r113465 = 0.002777777777777778;
        double r113466 = 4.0;
        double r113467 = pow(r113462, r113466);
        double r113468 = 0.08333333333333333;
        double r113469 = r113462 * r113462;
        double r113470 = fma(r113467, r113468, r113469);
        double r113471 = fma(r113464, r113465, r113470);
        return r113471;
}

Error

Bits error versus x

Target

Original30.3
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.3

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

  2. Simplified30.3

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

  3. Taylor expanded around 0 0.5

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

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))