Average Error: 58.0 → 0.6
Time: 12.4s
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 r70429 = x;
        double r70430 = exp(r70429);
        double r70431 = -r70429;
        double r70432 = exp(r70431);
        double r70433 = r70430 - r70432;
        double r70434 = 2.0;
        double r70435 = r70433 / r70434;
        return r70435;
}
↓
double f(double x) {
        double r70436 = x;
        double r70437 = 5.0;
        double r70438 = pow(r70436, r70437);
        double r70439 = 0.016666666666666666;
        double r70440 = 2.0;
        double r70441 = 3.0;
        double r70442 = pow(r70436, r70441);
        double r70443 = 0.3333333333333333;
        double r70444 = r70442 * r70443;
        double r70445 = fma(r70436, r70440, r70444);
        double r70446 = fma(r70438, r70439, r70445);
        double r70447 = 2.0;
        double r70448 = r70446 / r70447;
        return r70448;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))