Average Error: 43.6 → 0.8
Time: 30.6s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + \left(\left(x + x\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))\]
\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 {x}^{5} + \left(\left(x + x\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r957473 = x;
        double r957474 = exp(r957473);
        double r957475 = -r957473;
        double r957476 = exp(r957475);
        double r957477 = r957474 + r957476;
        double r957478 = 2.0;
        double r957479 = r957477 / r957478;
        double r957480 = y;
        double r957481 = cos(r957480);
        double r957482 = r957479 * r957481;
        double r957483 = r957474 - r957476;
        double r957484 = r957483 / r957478;
        double r957485 = sin(r957480);
        double r957486 = r957484 * r957485;
        double r957487 = /* ERROR: no complex support in C */;
        double r957488 = /* ERROR: no complex support in C */;
        return r957488;
}
↓
double f(double x, double y) {
        double r957489 = x;
        double r957490 = exp(r957489);
        double r957491 = -r957489;
        double r957492 = exp(r957491);
        double r957493 = r957490 + r957492;
        double r957494 = 2.0;
        double r957495 = r957493 / r957494;
        double r957496 = y;
        double r957497 = cos(r957496);
        double r957498 = r957495 * r957497;
        double r957499 = 0.016666666666666666;
        double r957500 = 5.0;
        double r957501 = pow(r957489, r957500);
        double r957502 = r957499 * r957501;
        double r957503 = r957489 + r957489;
        double r957504 = r957489 * r957489;
        double r957505 = 0.3333333333333333;
        double r957506 = r957504 * r957505;
        double r957507 = r957506 * r957489;
        double r957508 = r957503 + r957507;
        double r957509 = r957502 + r957508;
        double r957510 = r957509 / r957494;
        double r957511 = sin(r957496);
        double r957512 = r957510 * r957511;
        double r957513 = /* ERROR: no complex support in C */;
        double r957514 = /* ERROR: no complex support in C */;
        return r957514;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019153 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))