Average Error: 29.7 → 0.5
Time: 5.2s
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 r167153 = x;
        double r167154 = exp(r167153);
        double r167155 = 2.0;
        double r167156 = r167154 - r167155;
        double r167157 = -r167153;
        double r167158 = exp(r167157);
        double r167159 = r167156 + r167158;
        return r167159;
}
↓
double f(double x) {
        double r167160 = x;
        double r167161 = 0.002777777777777778;
        double r167162 = 6.0;
        double r167163 = pow(r167160, r167162);
        double r167164 = 0.08333333333333333;
        double r167165 = 4.0;
        double r167166 = pow(r167160, r167165);
        double r167167 = r167164 * r167166;
        double r167168 = fma(r167161, r167163, r167167);
        double r167169 = fma(r167160, r167160, r167168);
        return r167169;
}

Error

Bits error versus x

Target

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

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. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020056 +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))))