Average Error: 29.4 → 0.8
Time: 41.5s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*\]
\left(e^{x} - 2\right) + e^{-x}
↓
(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*
double f(double x) {
        double r9529902 = x;
        double r9529903 = exp(r9529902);
        double r9529904 = 2.0;
        double r9529905 = r9529903 - r9529904;
        double r9529906 = -r9529902;
        double r9529907 = exp(r9529906);
        double r9529908 = r9529905 + r9529907;
        return r9529908;
}
↓
double f(double x) {
        double r9529909 = x;
        double r9529910 = r9529909 * r9529909;
        double r9529911 = r9529910 * r9529910;
        double r9529912 = 0.002777777777777778;
        double r9529913 = r9529911 * r9529912;
        double r9529914 = 0.08333333333333333;
        double r9529915 = fma(r9529911, r9529914, r9529910);
        double r9529916 = fma(r9529913, r9529910, r9529915);
        return r9529916;
}

Error

Bits error versus x

Target

Original29.4
Target0.0
Herbie0.8
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.4

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

    \[\leadsto \color{blue}{(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*}\]

  4. Final simplification0.8

    \[\leadsto (\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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