Average Error: 57.9 → 0.6
Time: 3.3s
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 r58969 = x;
        double r58970 = exp(r58969);
        double r58971 = -r58969;
        double r58972 = exp(r58971);
        double r58973 = r58970 - r58972;
        double r58974 = 2.0;
        double r58975 = r58973 / r58974;
        return r58975;
}
↓
double f(double x) {
        double r58976 = 0.3333333333333333;
        double r58977 = x;
        double r58978 = 3.0;
        double r58979 = pow(r58977, r58978);
        double r58980 = 0.016666666666666666;
        double r58981 = 5.0;
        double r58982 = pow(r58977, r58981);
        double r58983 = 2.0;
        double r58984 = r58983 * r58977;
        double r58985 = fma(r58980, r58982, r58984);
        double r58986 = fma(r58976, r58979, r58985);
        double r58987 = 2.0;
        double r58988 = r58986 / r58987;
        return r58988;
}

Error

Bits error versus x

Derivation

  1. Initial program 57.9

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))