Average Error: 29.6 → 0.5
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 r122297 = x;
        double r122298 = exp(r122297);
        double r122299 = 2.0;
        double r122300 = r122298 - r122299;
        double r122301 = -r122297;
        double r122302 = exp(r122301);
        double r122303 = r122300 + r122302;
        return r122303;
}
↓
double f(double x) {
        double r122304 = x;
        double r122305 = 0.002777777777777778;
        double r122306 = 6.0;
        double r122307 = pow(r122304, r122306);
        double r122308 = 0.08333333333333333;
        double r122309 = 4.0;
        double r122310 = pow(r122304, r122309);
        double r122311 = r122308 * r122310;
        double r122312 = fma(r122305, r122307, r122311);
        double r122313 = fma(r122304, r122304, r122312);
        return r122313;
}

Error

Bits error versus x

Target

Original29.6
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.6

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

  2. Taylor expanded around 0 0.5

    \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]

  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 2019352 +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))))