Average Error: 29.5 → 0.7
Time: 14.4s
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 r109768 = x;
        double r109769 = exp(r109768);
        double r109770 = 2.0;
        double r109771 = r109769 - r109770;
        double r109772 = -r109768;
        double r109773 = exp(r109772);
        double r109774 = r109771 + r109773;
        return r109774;
}
↓
double f(double x) {
        double r109775 = x;
        double r109776 = 0.002777777777777778;
        double r109777 = 6.0;
        double r109778 = pow(r109775, r109777);
        double r109779 = 0.08333333333333333;
        double r109780 = 4.0;
        double r109781 = pow(r109775, r109780);
        double r109782 = r109779 * r109781;
        double r109783 = fma(r109776, r109778, r109782);
        double r109784 = fma(r109775, r109775, r109783);
        return r109784;
}

Error

Bits error versus x

Target

Original29.5
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020047 +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))))