Average Error: 29.7 → 0.6
Time: 5.6s
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 r104416 = x;
        double r104417 = exp(r104416);
        double r104418 = 2.0;
        double r104419 = r104417 - r104418;
        double r104420 = -r104416;
        double r104421 = exp(r104420);
        double r104422 = r104419 + r104421;
        return r104422;
}
↓
double f(double x) {
        double r104423 = x;
        double r104424 = 0.002777777777777778;
        double r104425 = 6.0;
        double r104426 = pow(r104423, r104425);
        double r104427 = 0.08333333333333333;
        double r104428 = 4.0;
        double r104429 = pow(r104423, r104428);
        double r104430 = r104427 * r104429;
        double r104431 = fma(r104424, r104426, r104430);
        double r104432 = fma(r104423, r104423, r104431);
        return r104432;
}

Error

Bits error versus x

Target

Original29.7
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

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. Simplified0.6

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

  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 2019347 +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))))