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

↓

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

double f(double x) {
        double r9325872 = x;
        double r9325873 = exp(r9325872);
        double r9325874 = -r9325872;
        double r9325875 = exp(r9325874);
        double r9325876 = r9325873 - r9325875;
        double r9325877 = 2.0;
        double r9325878 = r9325876 / r9325877;
        return r9325878;
}

↓

double f(double x) {
        double r9325879 = x;
        double r9325880 = 5.0;
        double r9325881 = pow(r9325879, r9325880);
        double r9325882 = 0.016666666666666666;
        double r9325883 = r9325881 * r9325882;
        double r9325884 = 2.0;
        double r9325885 = r9325884 * r9325879;
        double r9325886 = 0.3333333333333333;
        double r9325887 = r9325879 * r9325886;
        double r9325888 = r9325879 * r9325887;
        double r9325889 = r9325888 * r9325879;
        double r9325890 = r9325885 + r9325889;
        double r9325891 = r9325883 + r9325890;
        double r9325892 = r9325891 / r9325884;
        return r9325892;
}

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

↓

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

Derivation

Initial program 58.1
\[\frac{e^{x} - e^{-x}}{2}\]
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
\[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{3} \cdot x\right) + 2\right) + {x}^{5} \cdot \frac{1}{60}}}{2}\]
Using strategy rm
Applied distribute-rgt-in0.7
\[\leadsto \frac{\color{blue}{\left(\left(x \cdot \left(\frac{1}{3} \cdot x\right)\right) \cdot x + 2 \cdot x\right)} + {x}^{5} \cdot \frac{1}{60}}{2}\]
Final simplification0.7
\[\leadsto \frac{{x}^{5} \cdot \frac{1}{60} + \left(2 \cdot x + \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot x\right)}{2}\]

Reproduce

herbie shell --seed 2019102 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Error

Derivation

Reproduce