Average Error: 29.4 → 0.6
Time: 22.4s
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 r67394 = x;
        double r67395 = exp(r67394);
        double r67396 = 2.0;
        double r67397 = r67395 - r67396;
        double r67398 = -r67394;
        double r67399 = exp(r67398);
        double r67400 = r67397 + r67399;
        return r67400;
}
↓
double f(double x) {
        double r67401 = x;
        double r67402 = 0.002777777777777778;
        double r67403 = 6.0;
        double r67404 = pow(r67401, r67403);
        double r67405 = 0.08333333333333333;
        double r67406 = 4.0;
        double r67407 = pow(r67401, r67406);
        double r67408 = r67405 * r67407;
        double r67409 = fma(r67402, r67404, r67408);
        double r67410 = fma(r67401, r67401, r67409);
        return r67410;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.4

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

  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 2019198 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))