Average Error: 30.2 → 0.7
Time: 5.1s
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 r115999 = x;
        double r116000 = exp(r115999);
        double r116001 = 2.0;
        double r116002 = r116000 - r116001;
        double r116003 = -r115999;
        double r116004 = exp(r116003);
        double r116005 = r116002 + r116004;
        return r116005;
}
↓
double f(double x) {
        double r116006 = x;
        double r116007 = 0.002777777777777778;
        double r116008 = 6.0;
        double r116009 = pow(r116006, r116008);
        double r116010 = 0.08333333333333333;
        double r116011 = 4.0;
        double r116012 = pow(r116006, r116011);
        double r116013 = r116010 * r116012;
        double r116014 = fma(r116007, r116009, r116013);
        double r116015 = fma(r116006, r116006, r116014);
        return r116015;
}

Error

Bits error versus x

Target

Original30.2
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.2

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019353 +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))))