Average Error: 58.1 → 0.6
Time: 10.7s
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 r45342 = x;
        double r45343 = exp(r45342);
        double r45344 = -r45342;
        double r45345 = exp(r45344);
        double r45346 = r45343 - r45345;
        double r45347 = 2.0;
        double r45348 = r45346 / r45347;
        return r45348;
}
↓
double f(double x) {
        double r45349 = 0.3333333333333333;
        double r45350 = x;
        double r45351 = 3.0;
        double r45352 = pow(r45350, r45351);
        double r45353 = 0.016666666666666666;
        double r45354 = 5.0;
        double r45355 = pow(r45350, r45354);
        double r45356 = 2.0;
        double r45357 = r45356 * r45350;
        double r45358 = fma(r45353, r45355, r45357);
        double r45359 = fma(r45349, r45352, r45358);
        double r45360 = 2.0;
        double r45361 = r45359 / r45360;
        return r45361;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))