Average Error: 58.1 → 0.7
Time: 13.1s
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 r79422 = x;
        double r79423 = exp(r79422);
        double r79424 = -r79422;
        double r79425 = exp(r79424);
        double r79426 = r79423 - r79425;
        double r79427 = 2.0;
        double r79428 = r79426 / r79427;
        return r79428;
}
↓
double f(double x) {
        double r79429 = 0.3333333333333333;
        double r79430 = x;
        double r79431 = 3.0;
        double r79432 = pow(r79430, r79431);
        double r79433 = 0.016666666666666666;
        double r79434 = 5.0;
        double r79435 = pow(r79430, r79434);
        double r79436 = 2.0;
        double r79437 = r79436 * r79430;
        double r79438 = fma(r79433, r79435, r79437);
        double r79439 = fma(r79429, r79432, r79438);
        double r79440 = 2.0;
        double r79441 = r79439 / r79440;
        return r79441;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  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 2020043 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))