Average Error: 58.2 → 0.5
Time: 16.6s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(x, 2, {x}^{3} \cdot \frac{1}{3}\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(x, 2, {x}^{3} \cdot \frac{1}{3}\right)\right)}{2}
double f(double x) {
        double r62512 = x;
        double r62513 = exp(r62512);
        double r62514 = -r62512;
        double r62515 = exp(r62514);
        double r62516 = r62513 - r62515;
        double r62517 = 2.0;
        double r62518 = r62516 / r62517;
        return r62518;
}
↓
double f(double x) {
        double r62519 = x;
        double r62520 = 5.0;
        double r62521 = pow(r62519, r62520);
        double r62522 = 0.016666666666666666;
        double r62523 = 2.0;
        double r62524 = 3.0;
        double r62525 = pow(r62519, r62524);
        double r62526 = 0.3333333333333333;
        double r62527 = r62525 * r62526;
        double r62528 = fma(r62519, r62523, r62527);
        double r62529 = fma(r62521, r62522, r62528);
        double r62530 = 2.0;
        double r62531 = r62529 / r62530;
        return r62531;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.2

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))