Average Error: 29.5 → 0.5
Time: 18.7s
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 r81661 = x;
        double r81662 = exp(r81661);
        double r81663 = 2.0;
        double r81664 = r81662 - r81663;
        double r81665 = -r81661;
        double r81666 = exp(r81665);
        double r81667 = r81664 + r81666;
        return r81667;
}
↓
double f(double x) {
        double r81668 = x;
        double r81669 = 0.002777777777777778;
        double r81670 = 6.0;
        double r81671 = pow(r81668, r81670);
        double r81672 = 0.08333333333333333;
        double r81673 = 4.0;
        double r81674 = pow(r81668, r81673);
        double r81675 = r81672 * r81674;
        double r81676 = fma(r81669, r81671, r81675);
        double r81677 = fma(r81668, r81668, r81676);
        return r81677;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.5

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

  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 2019235 +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))))