Average Error: 58.2 → 0.6
Time: 30.8s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}{2}
double f(double x) {
        double r1121237 = x;
        double r1121238 = exp(r1121237);
        double r1121239 = -r1121237;
        double r1121240 = exp(r1121239);
        double r1121241 = r1121238 - r1121240;
        double r1121242 = 2.0;
        double r1121243 = r1121241 / r1121242;
        return r1121243;
}
↓
double f(double x) {
        double r1121244 = x;
        double r1121245 = 5.0;
        double r1121246 = pow(r1121244, r1121245);
        double r1121247 = 0.016666666666666666;
        double r1121248 = 0.3333333333333333;
        double r1121249 = r1121248 * r1121244;
        double r1121250 = 2.0;
        double r1121251 = fma(r1121249, r1121244, r1121250);
        double r1121252 = r1121251 * r1121244;
        double r1121253 = fma(r1121246, r1121247, r1121252);
        double r1121254 = r1121253 / r1121250;
        return r1121254;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.2

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))