Average Error: 30.2 → 0.6
Time: 4.6s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)
double f(double x) {
        double r82702 = x;
        double r82703 = exp(r82702);
        double r82704 = 2.0;
        double r82705 = r82703 - r82704;
        double r82706 = -r82702;
        double r82707 = exp(r82706);
        double r82708 = r82705 + r82707;
        return r82708;
}
↓
double f(double x) {
        double r82709 = x;
        double r82710 = 0.002777777777777778;
        double r82711 = 6.0;
        double r82712 = pow(r82709, r82711);
        double r82713 = 0.08333333333333333;
        double r82714 = 4.0;
        double r82715 = pow(r82709, r82714);
        double r82716 = r82713 * r82715;
        double r82717 = fma(r82710, r82712, r82716);
        double r82718 = fma(r82709, r82709, r82717);
        return r82718;
}

Error

Bits error versus x

Target

Original30.2
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.2

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020001 +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))))