Average Error: 29.2 → 0.6
Time: 9.3s
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 r160146 = x;
        double r160147 = exp(r160146);
        double r160148 = 2.0;
        double r160149 = r160147 - r160148;
        double r160150 = -r160146;
        double r160151 = exp(r160150);
        double r160152 = r160149 + r160151;
        return r160152;
}
↓
double f(double x) {
        double r160153 = x;
        double r160154 = 0.002777777777777778;
        double r160155 = 6.0;
        double r160156 = pow(r160153, r160155);
        double r160157 = 0.08333333333333333;
        double r160158 = 4.0;
        double r160159 = pow(r160153, r160158);
        double r160160 = r160157 * r160159;
        double r160161 = fma(r160154, r160156, r160160);
        double r160162 = fma(r160153, r160153, r160161);
        return r160162;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.2

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

  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 2020046 +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))))