Average Error: 58.0 → 0.7
Time: 3.9s
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 r58413 = x;
        double r58414 = exp(r58413);
        double r58415 = -r58413;
        double r58416 = exp(r58415);
        double r58417 = r58414 - r58416;
        double r58418 = 2.0;
        double r58419 = r58417 / r58418;
        return r58419;
}
↓
double f(double x) {
        double r58420 = 0.3333333333333333;
        double r58421 = x;
        double r58422 = 3.0;
        double r58423 = pow(r58421, r58422);
        double r58424 = 0.016666666666666666;
        double r58425 = 5.0;
        double r58426 = pow(r58421, r58425);
        double r58427 = 2.0;
        double r58428 = r58427 * r58421;
        double r58429 = fma(r58424, r58426, r58428);
        double r58430 = fma(r58420, r58423, r58429);
        double r58431 = 2.0;
        double r58432 = r58430 / r58431;
        return r58432;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  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 2020060 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))