Average Error: 58.1 → 0.6
Time: 15.9s
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 r48407 = x;
        double r48408 = exp(r48407);
        double r48409 = -r48407;
        double r48410 = exp(r48409);
        double r48411 = r48408 - r48410;
        double r48412 = 2.0;
        double r48413 = r48411 / r48412;
        return r48413;
}
↓
double f(double x) {
        double r48414 = 0.3333333333333333;
        double r48415 = x;
        double r48416 = 3.0;
        double r48417 = pow(r48415, r48416);
        double r48418 = 0.016666666666666666;
        double r48419 = 5.0;
        double r48420 = pow(r48415, r48419);
        double r48421 = 2.0;
        double r48422 = r48421 * r48415;
        double r48423 = fma(r48418, r48420, r48422);
        double r48424 = fma(r48414, r48417, r48423);
        double r48425 = 2.0;
        double r48426 = r48424 / r48425;
        return r48426;
}

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 2019326 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))