Average Error: 58.0 → 0.7
Time: 9.6s
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 r72601 = x;
        double r72602 = exp(r72601);
        double r72603 = -r72601;
        double r72604 = exp(r72603);
        double r72605 = r72602 - r72604;
        double r72606 = 2.0;
        double r72607 = r72605 / r72606;
        return r72607;
}
↓
double f(double x) {
        double r72608 = 0.3333333333333333;
        double r72609 = x;
        double r72610 = 3.0;
        double r72611 = pow(r72609, r72610);
        double r72612 = 0.016666666666666666;
        double r72613 = 5.0;
        double r72614 = pow(r72609, r72613);
        double r72615 = 2.0;
        double r72616 = r72615 * r72609;
        double r72617 = fma(r72612, r72614, r72616);
        double r72618 = fma(r72608, r72611, r72617);
        double r72619 = 2.0;
        double r72620 = r72618 / r72619;
        return r72620;
}

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 2020047 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))