Average Error: 58.0 → 0.7
Time: 3.9s
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 r68974 = x;
        double r68975 = exp(r68974);
        double r68976 = -r68974;
        double r68977 = exp(r68976);
        double r68978 = r68975 - r68977;
        double r68979 = 2.0;
        double r68980 = r68978 / r68979;
        return r68980;
}
↓
double f(double x) {
        double r68981 = 0.3333333333333333;
        double r68982 = x;
        double r68983 = 3.0;
        double r68984 = pow(r68982, r68983);
        double r68985 = 0.016666666666666666;
        double r68986 = 5.0;
        double r68987 = pow(r68982, r68986);
        double r68988 = 2.0;
        double r68989 = r68988 * r68982;
        double r68990 = fma(r68985, r68987, r68989);
        double r68991 = fma(r68981, r68984, r68990);
        double r68992 = 2.0;
        double r68993 = r68991 / r68992;
        return r68993;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))