Average Error: 58.2 → 0.6
Time: 13.5s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}
double f(double x) {
        double r46922 = x;
        double r46923 = exp(r46922);
        double r46924 = -r46922;
        double r46925 = exp(r46924);
        double r46926 = r46923 - r46925;
        double r46927 = 2.0;
        double r46928 = r46926 / r46927;
        return r46928;
}
↓
double f(double x) {
        double r46929 = 0.3333333333333333;
        double r46930 = x;
        double r46931 = 3.0;
        double r46932 = pow(r46930, r46931);
        double r46933 = 0.016666666666666666;
        double r46934 = 5.0;
        double r46935 = pow(r46930, r46934);
        double r46936 = 2.0;
        double r46937 = r46936 * r46930;
        double r46938 = fma(r46933, r46935, r46937);
        double r46939 = fma(r46929, r46932, r46938);
        double r46940 = 2.0;
        double r46941 = r46939 / r46940;
        return r46941;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.2

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))