Average Error: 43.5 → 0.8
Time: 13.3s
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 r48612 = x;
        double r48613 = exp(r48612);
        double r48614 = -r48612;
        double r48615 = exp(r48614);
        double r48616 = r48613 + r48615;
        double r48617 = 2.0;
        double r48618 = r48616 / r48617;
        double r48619 = y;
        double r48620 = cos(r48619);
        double r48621 = r48618 * r48620;
        double r48622 = r48613 - r48615;
        double r48623 = r48622 / r48617;
        double r48624 = sin(r48619);
        double r48625 = r48623 * r48624;
        double r48626 = /* ERROR: no complex support in C */;
        double r48627 = /* ERROR: no complex support in C */;
        return r48627;
}
↓
double f(double x, double y) {
        double r48628 = x;
        double r48629 = exp(r48628);
        double r48630 = -r48628;
        double r48631 = exp(r48630);
        double r48632 = r48629 + r48631;
        double r48633 = 2.0;
        double r48634 = r48632 / r48633;
        double r48635 = y;
        double r48636 = cos(r48635);
        double r48637 = r48634 * r48636;
        double r48638 = 0.3333333333333333;
        double r48639 = 3.0;
        double r48640 = pow(r48628, r48639);
        double r48641 = r48638 * r48640;
        double r48642 = 0.016666666666666666;
        double r48643 = 5.0;
        double r48644 = pow(r48628, r48643);
        double r48645 = r48642 * r48644;
        double r48646 = 2.0;
        double r48647 = r48646 * r48628;
        double r48648 = r48645 + r48647;
        double r48649 = r48641 + r48648;
        double r48650 = r48649 / r48633;
        double r48651 = sin(r48635);
        double r48652 = r48650 * r48651;
        double r48653 = /* ERROR: no complex support in C */;
        double r48654 = /* ERROR: no complex support in C */;
        return r48654;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.5

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

  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 2020049 
(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)))))