Average Error: 29.5 → 0.6
Time: 54.2s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(\frac{1}{12}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\frac{1}{360}, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), x \cdot x\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(\frac{1}{12}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\frac{1}{360}, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), x \cdot x\right)\right)
double f(double x) {
        double r4549260 = x;
        double r4549261 = exp(r4549260);
        double r4549262 = 2.0;
        double r4549263 = r4549261 - r4549262;
        double r4549264 = -r4549260;
        double r4549265 = exp(r4549264);
        double r4549266 = r4549263 + r4549265;
        return r4549266;
}
↓
double f(double x) {
        double r4549267 = 0.08333333333333333;
        double r4549268 = x;
        double r4549269 = r4549268 * r4549268;
        double r4549270 = r4549269 * r4549269;
        double r4549271 = 0.002777777777777778;
        double r4549272 = r4549269 * r4549270;
        double r4549273 = fma(r4549271, r4549272, r4549269);
        double r4549274 = fma(r4549267, r4549270, r4549273);
        return r4549274;
}

Error

Bits error versus x

Target

Original29.5
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.5

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))