Average Error: 29.9 → 0.5
Time: 5.5s
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 r93092 = x;
        double r93093 = exp(r93092);
        double r93094 = 2.0;
        double r93095 = r93093 - r93094;
        double r93096 = -r93092;
        double r93097 = exp(r93096);
        double r93098 = r93095 + r93097;
        return r93098;
}
↓
double f(double x) {
        double r93099 = x;
        double r93100 = 0.002777777777777778;
        double r93101 = 6.0;
        double r93102 = pow(r93099, r93101);
        double r93103 = 0.08333333333333333;
        double r93104 = 4.0;
        double r93105 = pow(r93099, r93104);
        double r93106 = r93103 * r93105;
        double r93107 = fma(r93100, r93102, r93106);
        double r93108 = fma(r93099, r93099, r93107);
        return r93108;
}

Error

Bits error versus x

Target

Original29.9
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  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 2019346 +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))))