Average Error: 58.1 → 0.7
Time: 4.5s
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 r58166 = x;
        double r58167 = exp(r58166);
        double r58168 = -r58166;
        double r58169 = exp(r58168);
        double r58170 = r58167 - r58169;
        double r58171 = 2.0;
        double r58172 = r58170 / r58171;
        return r58172;
}
↓
double f(double x) {
        double r58173 = 0.3333333333333333;
        double r58174 = x;
        double r58175 = 3.0;
        double r58176 = pow(r58174, r58175);
        double r58177 = 0.016666666666666666;
        double r58178 = 5.0;
        double r58179 = pow(r58174, r58178);
        double r58180 = 2.0;
        double r58181 = r58180 * r58174;
        double r58182 = fma(r58177, r58179, r58181);
        double r58183 = fma(r58173, r58176, r58182);
        double r58184 = 2.0;
        double r58185 = r58183 / r58184;
        return r58185;
}

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 2020039 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))