Average Error: 29.7 → 0.6
Time: 5.9s
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 r107557 = x;
        double r107558 = exp(r107557);
        double r107559 = 2.0;
        double r107560 = r107558 - r107559;
        double r107561 = -r107557;
        double r107562 = exp(r107561);
        double r107563 = r107560 + r107562;
        return r107563;
}
↓
double f(double x) {
        double r107564 = x;
        double r107565 = 0.002777777777777778;
        double r107566 = 6.0;
        double r107567 = pow(r107564, r107566);
        double r107568 = 0.08333333333333333;
        double r107569 = 4.0;
        double r107570 = pow(r107564, r107569);
        double r107571 = r107568 * r107570;
        double r107572 = fma(r107565, r107567, r107571);
        double r107573 = fma(r107564, r107564, r107572);
        return r107573;
}

Error

Bits error versus x

Target

Original29.7
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.7

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

  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 2020081 +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))))