Average Error: 29.1 → 0.7
Time: 4.8s
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 r119672 = x;
        double r119673 = exp(r119672);
        double r119674 = 2.0;
        double r119675 = r119673 - r119674;
        double r119676 = -r119672;
        double r119677 = exp(r119676);
        double r119678 = r119675 + r119677;
        return r119678;
}
↓
double f(double x) {
        double r119679 = x;
        double r119680 = 0.002777777777777778;
        double r119681 = 6.0;
        double r119682 = pow(r119679, r119681);
        double r119683 = 0.08333333333333333;
        double r119684 = 4.0;
        double r119685 = pow(r119679, r119684);
        double r119686 = r119683 * r119685;
        double r119687 = fma(r119680, r119682, r119686);
        double r119688 = fma(r119679, r119679, r119687);
        return r119688;
}

Error

Bits error versus x

Target

Original29.1
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.1

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

  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 2019354 +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))))