Average Error: 57.9 → 0.7
Time: 4.2s
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 r69359 = x;
        double r69360 = exp(r69359);
        double r69361 = -r69359;
        double r69362 = exp(r69361);
        double r69363 = r69360 - r69362;
        double r69364 = 2.0;
        double r69365 = r69363 / r69364;
        return r69365;
}
↓
double f(double x) {
        double r69366 = 0.3333333333333333;
        double r69367 = x;
        double r69368 = 3.0;
        double r69369 = pow(r69367, r69368);
        double r69370 = 0.016666666666666666;
        double r69371 = 5.0;
        double r69372 = pow(r69367, r69371);
        double r69373 = 2.0;
        double r69374 = r69373 * r69367;
        double r69375 = fma(r69370, r69372, r69374);
        double r69376 = fma(r69366, r69369, r69375);
        double r69377 = 2.0;
        double r69378 = r69376 / r69377;
        return r69378;
}

Error

Bits error versus x

Derivation

  1. Initial program 57.9

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

  2. Taylor expanded around 0 0.7

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

  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 2020057 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))