Average Error: 57.9 → 0.7
Time: 17.0s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right) \cdot x\right)}{2}
double f(double x) {
        double r2215301 = x;
        double r2215302 = exp(r2215301);
        double r2215303 = -r2215301;
        double r2215304 = exp(r2215303);
        double r2215305 = r2215302 - r2215304;
        double r2215306 = 2.0;
        double r2215307 = r2215305 / r2215306;
        return r2215307;
}
↓
double f(double x) {
        double r2215308 = x;
        double r2215309 = 5.0;
        double r2215310 = pow(r2215308, r2215309);
        double r2215311 = 0.016666666666666666;
        double r2215312 = 0.3333333333333333;
        double r2215313 = r2215308 * r2215308;
        double r2215314 = 2.0;
        double r2215315 = fma(r2215312, r2215313, r2215314);
        double r2215316 = r2215315 * r2215308;
        double r2215317 = fma(r2215310, r2215311, r2215316);
        double r2215318 = r2215317 / r2215314;
        return r2215318;
}

Error

Bits error versus x

Derivation

  1. Initial program 57.9

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Taylor expanded around 0 0.7

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

  5. Simplified0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))