Average Error: 29.6 → 0.7
Time: 12.9s
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 r117483 = x;
        double r117484 = exp(r117483);
        double r117485 = 2.0;
        double r117486 = r117484 - r117485;
        double r117487 = -r117483;
        double r117488 = exp(r117487);
        double r117489 = r117486 + r117488;
        return r117489;
}
↓
double f(double x) {
        double r117490 = x;
        double r117491 = 0.002777777777777778;
        double r117492 = 6.0;
        double r117493 = pow(r117490, r117492);
        double r117494 = 0.08333333333333333;
        double r117495 = 4.0;
        double r117496 = pow(r117490, r117495);
        double r117497 = r117494 * r117496;
        double r117498 = fma(r117491, r117493, r117497);
        double r117499 = fma(r117490, r117490, r117498);
        return r117499;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.6

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020043 +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))))