Average Error: 30.1 → 0.7
Time: 6.4s
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 r100811 = x;
        double r100812 = exp(r100811);
        double r100813 = 2.0;
        double r100814 = r100812 - r100813;
        double r100815 = -r100811;
        double r100816 = exp(r100815);
        double r100817 = r100814 + r100816;
        return r100817;
}
↓
double f(double x) {
        double r100818 = x;
        double r100819 = 0.002777777777777778;
        double r100820 = 6.0;
        double r100821 = pow(r100818, r100820);
        double r100822 = 0.08333333333333333;
        double r100823 = 4.0;
        double r100824 = pow(r100818, r100823);
        double r100825 = r100822 * r100824;
        double r100826 = fma(r100819, r100821, r100825);
        double r100827 = fma(r100818, r100818, r100826);
        return r100827;
}

Error

Bits error versus x

Target

Original30.1
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  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 2020034 +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))))