Average Error: 29.0 → 0.6
Time: 28.7s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(x \cdot x\right)\right)\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(x \cdot x\right)\right)\right)\right)
double f(double x) {
        double r10634837 = x;
        double r10634838 = exp(r10634837);
        double r10634839 = 2.0;
        double r10634840 = r10634838 - r10634839;
        double r10634841 = -r10634837;
        double r10634842 = exp(r10634841);
        double r10634843 = r10634840 + r10634842;
        return r10634843;
}
↓
double f(double x) {
        double r10634844 = x;
        double r10634845 = r10634844 * r10634844;
        double r10634846 = r10634845 * r10634845;
        double r10634847 = 0.002777777777777778;
        double r10634848 = r10634846 * r10634847;
        double r10634849 = 0.08333333333333333;
        double r10634850 = fma(r10634846, r10634849, r10634845);
        double r10634851 = fma(r10634848, r10634845, r10634850);
        return r10634851;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.0

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))