Average Error: 58.0 → 0.8
Time: 4.0s
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 r28834 = x;
        double r28835 = exp(r28834);
        double r28836 = -r28834;
        double r28837 = exp(r28836);
        double r28838 = r28835 - r28837;
        double r28839 = 2.0;
        double r28840 = r28838 / r28839;
        return r28840;
}
↓
double f(double x) {
        double r28841 = 0.3333333333333333;
        double r28842 = x;
        double r28843 = 3.0;
        double r28844 = pow(r28842, r28843);
        double r28845 = 0.016666666666666666;
        double r28846 = 5.0;
        double r28847 = pow(r28842, r28846);
        double r28848 = 2.0;
        double r28849 = r28848 * r28842;
        double r28850 = fma(r28845, r28847, r28849);
        double r28851 = fma(r28841, r28844, r28850);
        double r28852 = 2.0;
        double r28853 = r28851 / r28852;
        return r28853;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))