Average Error: 58.0 → 0.7
Time: 10.6s
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 r43787 = x;
        double r43788 = exp(r43787);
        double r43789 = -r43787;
        double r43790 = exp(r43789);
        double r43791 = r43788 - r43790;
        double r43792 = 2.0;
        double r43793 = r43791 / r43792;
        return r43793;
}
↓
double f(double x) {
        double r43794 = 0.3333333333333333;
        double r43795 = x;
        double r43796 = 3.0;
        double r43797 = pow(r43795, r43796);
        double r43798 = 0.016666666666666666;
        double r43799 = 5.0;
        double r43800 = pow(r43795, r43799);
        double r43801 = 2.0;
        double r43802 = r43801 * r43795;
        double r43803 = fma(r43798, r43800, r43802);
        double r43804 = fma(r43794, r43797, r43803);
        double r43805 = 2.0;
        double r43806 = r43804 / r43805;
        return r43806;
}

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 2019303 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))