Average Error: 58.1 → 0.6
Time: 11.1s
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 r54970 = x;
        double r54971 = exp(r54970);
        double r54972 = -r54970;
        double r54973 = exp(r54972);
        double r54974 = r54971 - r54973;
        double r54975 = 2.0;
        double r54976 = r54974 / r54975;
        return r54976;
}
↓
double f(double x) {
        double r54977 = 0.3333333333333333;
        double r54978 = x;
        double r54979 = 3.0;
        double r54980 = pow(r54978, r54979);
        double r54981 = 0.016666666666666666;
        double r54982 = 5.0;
        double r54983 = pow(r54978, r54982);
        double r54984 = 2.0;
        double r54985 = r54984 * r54978;
        double r54986 = fma(r54981, r54983, r54985);
        double r54987 = fma(r54977, r54980, r54986);
        double r54988 = 2.0;
        double r54989 = r54987 / r54988;
        return r54989;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  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 2019350 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))