Average Error: 29.6 → 0.6
Time: 16.7s
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 r77206 = x;
        double r77207 = exp(r77206);
        double r77208 = 2.0;
        double r77209 = r77207 - r77208;
        double r77210 = -r77206;
        double r77211 = exp(r77210);
        double r77212 = r77209 + r77211;
        return r77212;
}
↓
double f(double x) {
        double r77213 = x;
        double r77214 = 0.002777777777777778;
        double r77215 = 6.0;
        double r77216 = pow(r77213, r77215);
        double r77217 = 0.08333333333333333;
        double r77218 = 4.0;
        double r77219 = pow(r77213, r77218);
        double r77220 = r77217 * r77219;
        double r77221 = fma(r77214, r77216, r77220);
        double r77222 = fma(r77213, r77213, r77221);
        return r77222;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.6

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

  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 2019325 +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))))