\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

↓

\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_* \cdot x\right))_*}{2} \cdot \sin y i\right))\]

double f(double x, double y) {
        double r8337464 = x;
        double r8337465 = exp(r8337464);
        double r8337466 = -r8337464;
        double r8337467 = exp(r8337466);
        double r8337468 = r8337465 + r8337467;
        double r8337469 = 2.0;
        double r8337470 = r8337468 / r8337469;
        double r8337471 = y;
        double r8337472 = cos(r8337471);
        double r8337473 = r8337470 * r8337472;
        double r8337474 = r8337465 - r8337467;
        double r8337475 = r8337474 / r8337469;
        double r8337476 = sin(r8337471);
        double r8337477 = r8337475 * r8337476;
        double r8337478 = /* ERROR: no complex support in C */;
        double r8337479 = /* ERROR: no complex support in C */;
        return r8337479;
}

↓

double f(double x, double y) {
        double r8337480 = x;
        double r8337481 = exp(r8337480);
        double r8337482 = -r8337480;
        double r8337483 = exp(r8337482);
        double r8337484 = r8337481 + r8337483;
        double r8337485 = 2.0;
        double r8337486 = r8337484 / r8337485;
        double r8337487 = y;
        double r8337488 = cos(r8337487);
        double r8337489 = r8337486 * r8337488;
        double r8337490 = 0.016666666666666666;
        double r8337491 = 5.0;
        double r8337492 = pow(r8337480, r8337491);
        double r8337493 = 0.3333333333333333;
        double r8337494 = r8337480 * r8337480;
        double r8337495 = fma(r8337493, r8337494, r8337485);
        double r8337496 = r8337495 * r8337480;
        double r8337497 = fma(r8337490, r8337492, r8337496);
        double r8337498 = r8337497 / r8337485;
        double r8337499 = sin(r8337487);
        double r8337500 = r8337498 * r8337499;
        double r8337501 = /* ERROR: no complex support in C */;
        double r8337502 = /* ERROR: no complex support in C */;
        return r8337502;
}

\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))

↓

\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_* \cdot x\right))_*}{2} \cdot \sin y i\right))

Derivation

Initial program 43.6
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
Taylor expanded around 0 0.8
\[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2} \cdot \sin y i\right))\]
Simplified0.8
\[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(x \cdot (\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right))_*}}{2} \cdot \sin y i\right))\]
Final simplification0.8
\[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_* \cdot x\right))_*}{2} \cdot \sin y i\right))\]

Error

Derivation

Reproduce