Average Error: 58.0 → 0.6
Time: 17.5s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right) \cdot x + {x}^{5} \cdot \frac{1}{60}}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right) \cdot x + {x}^{5} \cdot \frac{1}{60}}{2}
double f(double x) {
        double r2089854 = x;
        double r2089855 = exp(r2089854);
        double r2089856 = -r2089854;
        double r2089857 = exp(r2089856);
        double r2089858 = r2089855 - r2089857;
        double r2089859 = 2.0;
        double r2089860 = r2089858 / r2089859;
        return r2089860;
}
↓
double f(double x) {
        double r2089861 = 0.3333333333333333;
        double r2089862 = x;
        double r2089863 = r2089862 * r2089862;
        double r2089864 = 2.0;
        double r2089865 = fma(r2089861, r2089863, r2089864);
        double r2089866 = r2089865 * r2089862;
        double r2089867 = 5.0;
        double r2089868 = pow(r2089862, r2089867);
        double r2089869 = 0.016666666666666666;
        double r2089870 = r2089868 * r2089869;
        double r2089871 = r2089866 + r2089870;
        double r2089872 = r2089871 / r2089864;
        return r2089872;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  5. Applied fma-udef0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))