Average Error: 43.2 → 0.9
Time: 20.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r40505 = x;
        double r40506 = exp(r40505);
        double r40507 = -r40505;
        double r40508 = exp(r40507);
        double r40509 = r40506 + r40508;
        double r40510 = 2.0;
        double r40511 = r40509 / r40510;
        double r40512 = y;
        double r40513 = cos(r40512);
        double r40514 = r40511 * r40513;
        double r40515 = r40506 - r40508;
        double r40516 = r40515 / r40510;
        double r40517 = sin(r40512);
        double r40518 = r40516 * r40517;
        double r40519 = /* ERROR: no complex support in C */;
        double r40520 = /* ERROR: no complex support in C */;
        return r40520;
}
↓
double f(double x, double y) {
        double r40521 = 0.3333333333333333;
        double r40522 = x;
        double r40523 = 3.0;
        double r40524 = pow(r40522, r40523);
        double r40525 = r40521 * r40524;
        double r40526 = 0.016666666666666666;
        double r40527 = 5.0;
        double r40528 = pow(r40522, r40527);
        double r40529 = r40526 * r40528;
        double r40530 = 2.0;
        double r40531 = r40530 * r40522;
        double r40532 = r40529 + r40531;
        double r40533 = r40525 + r40532;
        double r40534 = 2.0;
        double r40535 = r40533 / r40534;
        double r40536 = y;
        double r40537 = sin(r40536);
        double r40538 = r40535 * r40537;
        return r40538;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  2. Simplified43.2

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019350 
(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)))))