Average Error: 58.0 → 0.7
Time: 4.5s
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 r72862 = x;
        double r72863 = exp(r72862);
        double r72864 = -r72862;
        double r72865 = exp(r72864);
        double r72866 = r72863 - r72865;
        double r72867 = 2.0;
        double r72868 = r72866 / r72867;
        return r72868;
}
↓
double f(double x) {
        double r72869 = 0.3333333333333333;
        double r72870 = x;
        double r72871 = 3.0;
        double r72872 = pow(r72870, r72871);
        double r72873 = 0.016666666666666666;
        double r72874 = 5.0;
        double r72875 = pow(r72870, r72874);
        double r72876 = 2.0;
        double r72877 = r72876 * r72870;
        double r72878 = fma(r72873, r72875, r72877);
        double r72879 = fma(r72869, r72872, r72878);
        double r72880 = 2.0;
        double r72881 = r72879 / r72880;
        return r72881;
}

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 2019353 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))