Average Error: 58.0 → 0.6
Time: 16.7s
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 r45298 = x;
        double r45299 = exp(r45298);
        double r45300 = -r45298;
        double r45301 = exp(r45300);
        double r45302 = r45299 - r45301;
        double r45303 = 2.0;
        double r45304 = r45302 / r45303;
        return r45304;
}
↓
double f(double x) {
        double r45305 = 0.3333333333333333;
        double r45306 = x;
        double r45307 = 3.0;
        double r45308 = pow(r45306, r45307);
        double r45309 = 0.016666666666666666;
        double r45310 = 5.0;
        double r45311 = pow(r45306, r45310);
        double r45312 = 2.0;
        double r45313 = r45312 * r45306;
        double r45314 = fma(r45309, r45311, r45313);
        double r45315 = fma(r45305, r45308, r45314);
        double r45316 = 2.0;
        double r45317 = r45315 / r45316;
        return r45317;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  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 2019322 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))