Average Error: 58.0 → 0.7
Time: 11.5s
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 r30233 = x;
        double r30234 = exp(r30233);
        double r30235 = -r30233;
        double r30236 = exp(r30235);
        double r30237 = r30234 - r30236;
        double r30238 = 2.0;
        double r30239 = r30237 / r30238;
        return r30239;
}
↓
double f(double x) {
        double r30240 = 0.3333333333333333;
        double r30241 = x;
        double r30242 = 3.0;
        double r30243 = pow(r30241, r30242);
        double r30244 = 0.016666666666666666;
        double r30245 = 5.0;
        double r30246 = pow(r30241, r30245);
        double r30247 = 2.0;
        double r30248 = r30247 * r30241;
        double r30249 = fma(r30244, r30246, r30248);
        double r30250 = fma(r30240, r30243, r30249);
        double r30251 = 2.0;
        double r30252 = r30250 / r30251;
        return r30252;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))