Average Error: 58.1 → 0.6
Time: 2.8s
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 r69150 = x;
        double r69151 = exp(r69150);
        double r69152 = -r69150;
        double r69153 = exp(r69152);
        double r69154 = r69151 - r69153;
        double r69155 = 2.0;
        double r69156 = r69154 / r69155;
        return r69156;
}
↓
double f(double x) {
        double r69157 = 0.3333333333333333;
        double r69158 = x;
        double r69159 = 3.0;
        double r69160 = pow(r69158, r69159);
        double r69161 = 0.016666666666666666;
        double r69162 = 5.0;
        double r69163 = pow(r69158, r69162);
        double r69164 = 2.0;
        double r69165 = r69164 * r69158;
        double r69166 = fma(r69161, r69163, r69165);
        double r69167 = fma(r69157, r69160, r69166);
        double r69168 = 2.0;
        double r69169 = r69167 / r69168;
        return r69169;
}

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 2020020 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))