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 r53460 = x;
        double r53461 = exp(r53460);
        double r53462 = -r53460;
        double r53463 = exp(r53462);
        double r53464 = r53461 - r53463;
        double r53465 = 2.0;
        double r53466 = r53464 / r53465;
        return r53466;
}
↓
double f(double x) {
        double r53467 = 0.3333333333333333;
        double r53468 = x;
        double r53469 = 3.0;
        double r53470 = pow(r53468, r53469);
        double r53471 = 0.016666666666666666;
        double r53472 = 5.0;
        double r53473 = pow(r53468, r53472);
        double r53474 = 2.0;
        double r53475 = r53474 * r53468;
        double r53476 = fma(r53471, r53473, r53475);
        double r53477 = fma(r53467, r53470, r53476);
        double r53478 = 2.0;
        double r53479 = r53477 / r53478;
        return r53479;
}

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 2019347 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))