Average Error: 58.1 → 0.5
Time: 3.7s
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 r54203 = x;
        double r54204 = exp(r54203);
        double r54205 = -r54203;
        double r54206 = exp(r54205);
        double r54207 = r54204 - r54206;
        double r54208 = 2.0;
        double r54209 = r54207 / r54208;
        return r54209;
}
↓
double f(double x) {
        double r54210 = 0.3333333333333333;
        double r54211 = x;
        double r54212 = 3.0;
        double r54213 = pow(r54211, r54212);
        double r54214 = 0.016666666666666666;
        double r54215 = 5.0;
        double r54216 = pow(r54211, r54215);
        double r54217 = 2.0;
        double r54218 = r54217 * r54211;
        double r54219 = fma(r54214, r54216, r54218);
        double r54220 = fma(r54210, r54213, r54219);
        double r54221 = 2.0;
        double r54222 = r54220 / r54221;
        return r54222;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  2. Taylor expanded around 0 0.5

    \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2}\]

  3. Simplified0.5

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}}{2}\]

  4. Final simplification0.5

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))