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

↓

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

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

↓

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

double f(double x) {
        double r2127487 = x;
        double r2127488 = exp(r2127487);
        double r2127489 = -r2127487;
        double r2127490 = exp(r2127489);
        double r2127491 = r2127488 - r2127490;
        double r2127492 = 2.0;
        double r2127493 = r2127491 / r2127492;
        return r2127493;
}

↓

double f(double x) {
        double r2127494 = 0.016666666666666666;
        double r2127495 = x;
        double r2127496 = 5.0;
        double r2127497 = pow(r2127495, r2127496);
        double r2127498 = 0.3333333333333333;
        double r2127499 = r2127495 * r2127495;
        double r2127500 = 2.0;
        double r2127501 = fma(r2127498, r2127499, r2127500);
        double r2127502 = r2127495 * r2127501;
        double r2127503 = fma(r2127494, r2127497, r2127502);
        double r2127504 = r2127503 / r2127500;
        return r2127504;
}

Derivation

Initial program 57.9
\[\frac{e^{x} - e^{-x}}{2}\]
Taylor expanded around 0 0.7
\[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]
Simplified0.7
\[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(\mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right) \cdot x\right)\right)}}{2}\]
Final simplification0.7
\[\leadsto \frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{2}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Error

Derivation

Reproduce