Average Error: 58.2 → 0.5
Time: 14.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(x, 2, {x}^{3} \cdot \frac{1}{3}\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(x, 2, {x}^{3} \cdot \frac{1}{3}\right)\right)}{2}
double f(double x) {
        double r50914 = x;
        double r50915 = exp(r50914);
        double r50916 = -r50914;
        double r50917 = exp(r50916);
        double r50918 = r50915 - r50917;
        double r50919 = 2.0;
        double r50920 = r50918 / r50919;
        return r50920;
}
↓
double f(double x) {
        double r50921 = x;
        double r50922 = 5.0;
        double r50923 = pow(r50921, r50922);
        double r50924 = 0.016666666666666666;
        double r50925 = 2.0;
        double r50926 = 3.0;
        double r50927 = pow(r50921, r50926);
        double r50928 = 0.3333333333333333;
        double r50929 = r50927 * r50928;
        double r50930 = fma(r50921, r50925, r50929);
        double r50931 = fma(r50923, r50924, r50930);
        double r50932 = 2.0;
        double r50933 = r50931 / r50932;
        return r50933;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.2

    \[\frac{e^{x} - e^{-x}}{2}\]

  2. Taylor expanded around 0 0.5

    \[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]

  3. Simplified0.5

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(x, 2, \frac{1}{3} \cdot {x}^{3}\right)\right)}}{2}\]

  4. Final simplification0.5

    \[\leadsto \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(x, 2, {x}^{3} \cdot \frac{1}{3}\right)\right)}{2}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))