Average Error: 43.5 → 0.9
Time: 39.3s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{2 \cdot x + \left(\frac{1}{60} \cdot {x}^{5} + \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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{2 \cdot x + \left(\frac{1}{60} \cdot {x}^{5} + \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r3834685 = x;
        double r3834686 = exp(r3834685);
        double r3834687 = -r3834685;
        double r3834688 = exp(r3834687);
        double r3834689 = r3834686 + r3834688;
        double r3834690 = 2.0;
        double r3834691 = r3834689 / r3834690;
        double r3834692 = y;
        double r3834693 = cos(r3834692);
        double r3834694 = r3834691 * r3834693;
        double r3834695 = r3834686 - r3834688;
        double r3834696 = r3834695 / r3834690;
        double r3834697 = sin(r3834692);
        double r3834698 = r3834696 * r3834697;
        double r3834699 = /* ERROR: no complex support in C */;
        double r3834700 = /* ERROR: no complex support in C */;
        return r3834700;
}
↓
double f(double x, double y) {
        double r3834701 = x;
        double r3834702 = exp(r3834701);
        double r3834703 = -r3834701;
        double r3834704 = exp(r3834703);
        double r3834705 = r3834702 + r3834704;
        double r3834706 = 2.0;
        double r3834707 = r3834705 / r3834706;
        double r3834708 = y;
        double r3834709 = cos(r3834708);
        double r3834710 = r3834707 * r3834709;
        double r3834711 = 2.0;
        double r3834712 = r3834711 * r3834701;
        double r3834713 = 0.016666666666666666;
        double r3834714 = 5.0;
        double r3834715 = pow(r3834701, r3834714);
        double r3834716 = r3834713 * r3834715;
        double r3834717 = 0.3333333333333333;
        double r3834718 = r3834701 * r3834701;
        double r3834719 = r3834718 * r3834701;
        double r3834720 = r3834717 * r3834719;
        double r3834721 = r3834716 + r3834720;
        double r3834722 = r3834712 + r3834721;
        double r3834723 = r3834722 / r3834706;
        double r3834724 = sin(r3834708);
        double r3834725 = r3834723 * r3834724;
        double r3834726 = /* ERROR: no complex support in C */;
        double r3834727 = /* ERROR: no complex support in C */;
        return r3834727;
}

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 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))