Average Error: 29.2 → 0.6
Time: 19.9s
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 r111082 = x;
        double r111083 = exp(r111082);
        double r111084 = 2.0;
        double r111085 = r111083 - r111084;
        double r111086 = -r111082;
        double r111087 = exp(r111086);
        double r111088 = r111085 + r111087;
        return r111088;
}
↓
double f(double x) {
        double r111089 = x;
        double r111090 = 0.002777777777777778;
        double r111091 = 6.0;
        double r111092 = pow(r111089, r111091);
        double r111093 = 0.08333333333333333;
        double r111094 = 4.0;
        double r111095 = pow(r111089, r111094);
        double r111096 = r111093 * r111095;
        double r111097 = fma(r111090, r111092, r111096);
        double r111098 = fma(r111089, r111089, r111097);
        return r111098;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.2

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

  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

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

  4. Final simplification0.6

    \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\]

Reproduce

herbie shell --seed 2019208 +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))))