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

↓

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

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

↓

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

double f(double x) {
        double r4106558 = x;
        double r4106559 = exp(r4106558);
        double r4106560 = -r4106558;
        double r4106561 = exp(r4106560);
        double r4106562 = r4106559 - r4106561;
        double r4106563 = 2.0;
        double r4106564 = r4106562 / r4106563;
        return r4106564;
}

↓

double f(double x) {
        double r4106565 = 2.0;
        double r4106566 = x;
        double r4106567 = 0.3333333333333333;
        double r4106568 = r4106566 * r4106566;
        double r4106569 = r4106568 * r4106566;
        double r4106570 = 0.016666666666666666;
        double r4106571 = 5.0;
        double r4106572 = pow(r4106566, r4106571);
        double r4106573 = r4106570 * r4106572;
        double r4106574 = fma(r4106567, r4106569, r4106573);
        double r4106575 = fma(r4106565, r4106566, r4106574);
        double r4106576 = 2.0;
        double r4106577 = r4106575 / r4106576;
        return r4106577;
}

Derivation

Initial program 57.9
\[\frac{e^{x} - e^{-x}}{2}\]
Taylor expanded around 0 0.6
\[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]
Simplified0.6
\[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right)\right)}}{2}\]
Taylor expanded around 0 0.6
\[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]
Simplified0.6
\[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right) \cdot x, \frac{1}{60} \cdot {x}^{5}\right)\right)}}{2}\]
Final simplification0.6
\[\leadsto \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right) \cdot x, \frac{1}{60} \cdot {x}^{5}\right)\right)}{2}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))

Error

Derivation

Reproduce