Average Error: 58.1 → 0.6
Time: 3.5s
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 r62250 = x;
        double r62251 = exp(r62250);
        double r62252 = -r62250;
        double r62253 = exp(r62252);
        double r62254 = r62251 - r62253;
        double r62255 = 2.0;
        double r62256 = r62254 / r62255;
        return r62256;
}
↓
double f(double x) {
        double r62257 = 0.3333333333333333;
        double r62258 = x;
        double r62259 = 3.0;
        double r62260 = pow(r62258, r62259);
        double r62261 = 0.016666666666666666;
        double r62262 = 5.0;
        double r62263 = pow(r62258, r62262);
        double r62264 = 2.0;
        double r62265 = r62264 * r62258;
        double r62266 = fma(r62261, r62263, r62265);
        double r62267 = fma(r62257, r62260, r62266);
        double r62268 = 2.0;
        double r62269 = r62267 / r62268;
        return r62269;
}

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 2020065 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))