Average Error: 58.1 → 0.6
Time: 15.3s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{2}
double f(double x) {
        double r4412668 = x;
        double r4412669 = exp(r4412668);
        double r4412670 = -r4412668;
        double r4412671 = exp(r4412670);
        double r4412672 = r4412669 - r4412671;
        double r4412673 = 2.0;
        double r4412674 = r4412672 / r4412673;
        return r4412674;
}
↓
double f(double x) {
        double r4412675 = 0.016666666666666666;
        double r4412676 = x;
        double r4412677 = 5.0;
        double r4412678 = pow(r4412676, r4412677);
        double r4412679 = 0.3333333333333333;
        double r4412680 = r4412676 * r4412676;
        double r4412681 = 2.0;
        double r4412682 = fma(r4412679, r4412680, r4412681);
        double r4412683 = r4412676 * r4412682;
        double r4412684 = fma(r4412675, r4412678, r4412683);
        double r4412685 = r4412684 / r4412681;
        return r4412685;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))