Average Error: 29.8 → 0.8
Time: 5.5s
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 r110622 = x;
        double r110623 = exp(r110622);
        double r110624 = 2.0;
        double r110625 = r110623 - r110624;
        double r110626 = -r110622;
        double r110627 = exp(r110626);
        double r110628 = r110625 + r110627;
        return r110628;
}
↓
double f(double x) {
        double r110629 = x;
        double r110630 = 0.002777777777777778;
        double r110631 = 6.0;
        double r110632 = pow(r110629, r110631);
        double r110633 = 0.08333333333333333;
        double r110634 = 4.0;
        double r110635 = pow(r110629, r110634);
        double r110636 = r110633 * r110635;
        double r110637 = fma(r110630, r110632, r110636);
        double r110638 = fma(r110629, r110629, r110637);
        return r110638;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.8

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020049 +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))))