Average Error: 58.1 → 0.5
Time: 20.5s
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 r1660147 = x;
        double r1660148 = exp(r1660147);
        double r1660149 = -r1660147;
        double r1660150 = exp(r1660149);
        double r1660151 = r1660148 - r1660150;
        double r1660152 = 2.0;
        double r1660153 = r1660151 / r1660152;
        return r1660153;
}
↓
double f(double x) {
        double r1660154 = x;
        double r1660155 = 5.0;
        double r1660156 = pow(r1660154, r1660155);
        double r1660157 = 0.016666666666666666;
        double r1660158 = 0.3333333333333333;
        double r1660159 = r1660158 * r1660154;
        double r1660160 = 2.0;
        double r1660161 = fma(r1660159, r1660154, r1660160);
        double r1660162 = r1660161 * r1660154;
        double r1660163 = fma(r1660156, r1660157, r1660162);
        double r1660164 = r1660163 / r1660160;
        return r1660164;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  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

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))