Average Error: 43.7 → 0.8
Time: 12.4s
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 r43585 = x;
        double r43586 = exp(r43585);
        double r43587 = -r43585;
        double r43588 = exp(r43587);
        double r43589 = r43586 + r43588;
        double r43590 = 2.0;
        double r43591 = r43589 / r43590;
        double r43592 = y;
        double r43593 = cos(r43592);
        double r43594 = r43591 * r43593;
        double r43595 = r43586 - r43588;
        double r43596 = r43595 / r43590;
        double r43597 = sin(r43592);
        double r43598 = r43596 * r43597;
        double r43599 = /* ERROR: no complex support in C */;
        double r43600 = /* ERROR: no complex support in C */;
        return r43600;
}
↓
double f(double x, double y) {
        double r43601 = 0.3333333333333333;
        double r43602 = x;
        double r43603 = 3.0;
        double r43604 = pow(r43602, r43603);
        double r43605 = r43601 * r43604;
        double r43606 = 0.016666666666666666;
        double r43607 = 5.0;
        double r43608 = pow(r43602, r43607);
        double r43609 = r43606 * r43608;
        double r43610 = 2.0;
        double r43611 = r43610 * r43602;
        double r43612 = r43609 + r43611;
        double r43613 = r43605 + r43612;
        double r43614 = 2.0;
        double r43615 = r43613 / r43614;
        double r43616 = y;
        double r43617 = sin(r43616);
        double r43618 = r43615 * r43617;
        return r43618;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

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

  2. Simplified43.7

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

  3. Taylor expanded around 0 0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020021 
(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)))))