Average Error: 29.3 → 0.6
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 r67660 = x;
        double r67661 = exp(r67660);
        double r67662 = 2.0;
        double r67663 = r67661 - r67662;
        double r67664 = -r67660;
        double r67665 = exp(r67664);
        double r67666 = r67663 + r67665;
        return r67666;
}
↓
double f(double x) {
        double r67667 = x;
        double r67668 = 6.0;
        double r67669 = pow(r67667, r67668);
        double r67670 = 0.002777777777777778;
        double r67671 = 4.0;
        double r67672 = pow(r67667, r67671);
        double r67673 = 0.08333333333333333;
        double r67674 = r67667 * r67667;
        double r67675 = fma(r67672, r67673, r67674);
        double r67676 = fma(r67669, r67670, r67675);
        return r67676;
}

Error

Bits error versus x

Target

Original29.3
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  2. Simplified29.3

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

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019195 +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))))