Average Error: 43.4 → 0.8
Time: 40.3s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r55508 = x;
        double r55509 = exp(r55508);
        double r55510 = -r55508;
        double r55511 = exp(r55510);
        double r55512 = r55509 + r55511;
        double r55513 = 2.0;
        double r55514 = r55512 / r55513;
        double r55515 = y;
        double r55516 = cos(r55515);
        double r55517 = r55514 * r55516;
        double r55518 = r55509 - r55511;
        double r55519 = r55518 / r55513;
        double r55520 = sin(r55515);
        double r55521 = r55519 * r55520;
        double r55522 = /* ERROR: no complex support in C */;
        double r55523 = /* ERROR: no complex support in C */;
        return r55523;
}
↓
double f(double x, double y) {
        double r55524 = x;
        double r55525 = exp(r55524);
        double r55526 = -r55524;
        double r55527 = exp(r55526);
        double r55528 = r55525 + r55527;
        double r55529 = 2.0;
        double r55530 = r55528 / r55529;
        double r55531 = y;
        double r55532 = cos(r55531);
        double r55533 = r55530 * r55532;
        double r55534 = 0.3333333333333333;
        double r55535 = 3.0;
        double r55536 = pow(r55524, r55535);
        double r55537 = r55534 * r55536;
        double r55538 = 0.016666666666666666;
        double r55539 = 5.0;
        double r55540 = pow(r55524, r55539);
        double r55541 = r55538 * r55540;
        double r55542 = 2.0;
        double r55543 = r55542 * r55524;
        double r55544 = r55541 + r55543;
        double r55545 = r55537 + r55544;
        double r55546 = r55545 / r55529;
        double r55547 = sin(r55531);
        double r55548 = r55546 * r55547;
        double r55549 = /* ERROR: no complex support in C */;
        double r55550 = /* ERROR: no complex support in C */;
        return r55550;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2019306 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))