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

↓

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

double f(double x) {
        double r11473515 = x;
        double r11473516 = exp(r11473515);
        double r11473517 = -r11473515;
        double r11473518 = exp(r11473517);
        double r11473519 = r11473516 - r11473518;
        double r11473520 = 2.0;
        double r11473521 = r11473519 / r11473520;
        return r11473521;
}

↓

double f(double x) {
        double r11473522 = 0.016666666666666666;
        double r11473523 = x;
        double r11473524 = 5.0;
        double r11473525 = pow(r11473523, r11473524);
        double r11473526 = 2.0;
        double r11473527 = r11473526 * r11473523;
        double r11473528 = 0.3333333333333333;
        double r11473529 = r11473523 * r11473523;
        double r11473530 = r11473528 * r11473529;
        double r11473531 = r11473530 * r11473523;
        double r11473532 = r11473527 + r11473531;
        double r11473533 = fma(r11473522, r11473525, r11473532);
        double r11473534 = r11473533 / r11473526;
        return r11473534;
}

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

↓

\frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\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
\[\leadsto \frac{\color{blue}{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(x \cdot (\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right))_*}}{2}\]
Using strategy rm
Applied fma-udef0.7
\[\leadsto \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(x \cdot \color{blue}{\left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right)}\right))_*}{2}\]
Applied distribute-lft-in0.7
\[\leadsto \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \color{blue}{\left(x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) + x \cdot 2\right)})_*}{2}\]
Final simplification0.7
\[\leadsto \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right))_*}{2}\]

Reproduce

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

Error

Derivation

Reproduce