Average Error: 43.1 → 0.8
Time: 1.0m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)\right)}{2} \cdot \sin y i\right))\]
\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{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1571673 = x;
        double r1571674 = exp(r1571673);
        double r1571675 = -r1571673;
        double r1571676 = exp(r1571675);
        double r1571677 = r1571674 + r1571676;
        double r1571678 = 2.0;
        double r1571679 = r1571677 / r1571678;
        double r1571680 = y;
        double r1571681 = cos(r1571680);
        double r1571682 = r1571679 * r1571681;
        double r1571683 = r1571674 - r1571676;
        double r1571684 = r1571683 / r1571678;
        double r1571685 = sin(r1571680);
        double r1571686 = r1571684 * r1571685;
        double r1571687 = /* ERROR: no complex support in C */;
        double r1571688 = /* ERROR: no complex support in C */;
        return r1571688;
}
↓
double f(double x, double y) {
        double r1571689 = x;
        double r1571690 = exp(r1571689);
        double r1571691 = -r1571689;
        double r1571692 = exp(r1571691);
        double r1571693 = r1571690 + r1571692;
        double r1571694 = 2.0;
        double r1571695 = r1571693 / r1571694;
        double r1571696 = y;
        double r1571697 = cos(r1571696);
        double r1571698 = r1571695 * r1571697;
        double r1571699 = 0.016666666666666666;
        double r1571700 = 5.0;
        double r1571701 = pow(r1571689, r1571700);
        double r1571702 = r1571689 * r1571689;
        double r1571703 = 0.3333333333333333;
        double r1571704 = r1571702 * r1571703;
        double r1571705 = r1571704 + r1571694;
        double r1571706 = r1571689 * r1571705;
        double r1571707 = fma(r1571699, r1571701, r1571706);
        double r1571708 = r1571707 / r1571694;
        double r1571709 = sin(r1571696);
        double r1571710 = r1571708 * r1571709;
        double r1571711 = /* ERROR: no complex support in C */;
        double r1571712 = /* ERROR: no complex support in C */;
        return r1571712;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.1

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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)))))