Average Error: 58.0 → 0.6
Time: 14.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \frac{1}{60} \cdot {x}^{5}\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \frac{1}{60} \cdot {x}^{5}\right)\right)}{2}
double f(double x) {
        double r61990 = x;
        double r61991 = exp(r61990);
        double r61992 = -r61990;
        double r61993 = exp(r61992);
        double r61994 = r61991 - r61993;
        double r61995 = 2.0;
        double r61996 = r61994 / r61995;
        return r61996;
}
↓
double f(double x) {
        double r61997 = 2.0;
        double r61998 = x;
        double r61999 = 0.3333333333333333;
        double r62000 = 3.0;
        double r62001 = pow(r61998, r62000);
        double r62002 = 0.016666666666666666;
        double r62003 = 5.0;
        double r62004 = pow(r61998, r62003);
        double r62005 = r62002 * r62004;
        double r62006 = fma(r61999, r62001, r62005);
        double r62007 = fma(r61997, r61998, r62006);
        double r62008 = 2.0;
        double r62009 = r62007 / r62008;
        return r62009;
}

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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))