Average Error: 58.3 → 0.6
Time: 3.6s
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 r67273 = x;
        double r67274 = exp(r67273);
        double r67275 = -r67273;
        double r67276 = exp(r67275);
        double r67277 = r67274 - r67276;
        double r67278 = 2.0;
        double r67279 = r67277 / r67278;
        return r67279;
}
↓
double f(double x) {
        double r67280 = 0.3333333333333333;
        double r67281 = x;
        double r67282 = 3.0;
        double r67283 = pow(r67281, r67282);
        double r67284 = 0.016666666666666666;
        double r67285 = 5.0;
        double r67286 = pow(r67281, r67285);
        double r67287 = 2.0;
        double r67288 = r67287 * r67281;
        double r67289 = fma(r67284, r67286, r67288);
        double r67290 = fma(r67280, r67283, r67289);
        double r67291 = 2.0;
        double r67292 = r67290 / r67291;
        return r67292;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.3

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

  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 2020001 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))