Average Error: 58.1 → 0.6
Time: 15.4s
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 r36032 = x;
        double r36033 = exp(r36032);
        double r36034 = -r36032;
        double r36035 = exp(r36034);
        double r36036 = r36033 - r36035;
        double r36037 = 2.0;
        double r36038 = r36036 / r36037;
        return r36038;
}
↓
double f(double x) {
        double r36039 = 0.3333333333333333;
        double r36040 = x;
        double r36041 = 3.0;
        double r36042 = pow(r36040, r36041);
        double r36043 = 0.016666666666666666;
        double r36044 = 5.0;
        double r36045 = pow(r36040, r36044);
        double r36046 = 2.0;
        double r36047 = r36046 * r36040;
        double r36048 = fma(r36043, r36045, r36047);
        double r36049 = fma(r36039, r36042, r36048);
        double r36050 = 2.0;
        double r36051 = r36049 / r36050;
        return r36051;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  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 2019326 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))