Average Error: 58.0 → 0.5
Time: 4.6s
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 r69508 = x;
        double r69509 = exp(r69508);
        double r69510 = -r69508;
        double r69511 = exp(r69510);
        double r69512 = r69509 - r69511;
        double r69513 = 2.0;
        double r69514 = r69512 / r69513;
        return r69514;
}
↓
double f(double x) {
        double r69515 = 0.3333333333333333;
        double r69516 = x;
        double r69517 = 3.0;
        double r69518 = pow(r69516, r69517);
        double r69519 = 0.016666666666666666;
        double r69520 = 5.0;
        double r69521 = pow(r69516, r69520);
        double r69522 = 2.0;
        double r69523 = r69522 * r69516;
        double r69524 = fma(r69519, r69521, r69523);
        double r69525 = fma(r69515, r69518, r69524);
        double r69526 = 2.0;
        double r69527 = r69525 / r69526;
        return r69527;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))